On Polyanalytic Functions in Several Complex Variables

Abstract

We give a characterisation of the polyanalytic type subspaces of the Hilbert spaces $\mathcal {H}$, being the weighted $L_2$ function spaces on a connected simply connected domains $D \subset \mathbb {C}^n$. The typical examples considered in the paper are the unit ball $\mathbb {B}^n$ and the whole space $\mathbb {C}^n$. Our approach is based on the use of the two tuples of operators

$$\begin{aligned} \mathbf {\mathfrak {a}}= (\mathfrak {a}_1, \ \mathfrak {a}_2, \ \ldots , \ \mathfrak {a}_n) \quad \textrm{and} \quad \mathbf {\mathfrak {b}}= (\mathfrak {b}_1, \ \mathfrak {b}_2, \ \ldots , \ \mathfrak {b}_n), \end{aligned}$$

which act invariantly in some linear space and satisfy therein the commutation relations

$$\begin{aligned} \left[ \mathfrak {a}_j, \mathfrak {b}_{\ell }\right] = \delta _{j,\ell }I, \quad \left[ \mathfrak {a}_j, \mathfrak {a}_{\ell }\right] = 0, \quad \left[ \mathfrak {b}_j, \mathfrak {b}_{\ell }\right] = 0, \quad j,\ell = 1,2,...,n. \end{aligned}$$

We assume further that a common invariant domain of the above operators is a dense linear subspace in a Hilbert space $\mathcal {H}$, and impose several additional conditions. The exact formulation is given in the Extended Fock space construction. Further we specify the obtained description to different concrete realizations of the above operators and Hilbert spaces $\mathcal {H}$, illustrating a variety of possibilities that may occur in the characterisation of the polyanalytic type spaces in several complex variables.

1 Introduction

The paper is devoted to the characterisation of the polyanalytic type subspaces of the Hilbert spaces $\mathcal {H}$, being the weighted $L_2$ function spaces on a connected simply connected domains $D \subset \mathbb {C}^n$. The typical examples considered in the paper are the unit ball $\mathbb {B}^n$ and the whole space $\mathbb {C}^n$.

Recall that, given a tuple $\textbf{k}= (k_1,k_2,...,k_n) \in \mathbb {N}^n$, a smooth function $\varphi$ is called $\textbf{k}$-polyanalytic if it satisfies the equations

$$\begin{aligned} \frac{\partial ^{k_1}\varphi }{\partial \overline{z}_1^{k_1}}= 0, \quad \frac{\partial ^{k_2}\varphi }{\partial \overline{z}_2^{k_2}} = 0, \quad \ldots , \quad \ \frac{\partial ^{k_n}\varphi }{\partial \overline{z}_n^{k_n}} = 0. \end{aligned}$$

For a purpose of Introduction, we denote by $\mathcal {A}_{\textbf{k}}$ the closed subspace of $\mathcal {H}$ which consists of all $\textbf{k}$-polyanalytic functions.

Note that there is a big quantitative difference between multidimensional case $n > 1$, studied in the paper and already well-understood case of $n=1$. Indeed, in one-dimensional case the k-polyanalytic spaces are linearly ordered by inclusion, $\mathcal {A}_q \subsetneq \mathcal {A}_k$ if and only if $q < k$. This suggests us to consider consecutive orthogonal differences, the so-called true-k-polyanalytic spaces

$$\begin{aligned} \mathcal {A}_{(k)} = \mathcal {A}_k \ominus \mathcal {A}_{k-1} = \mathcal {A}_k \cap \mathcal {A}_{k-1}^{\perp }. \end{aligned}$$

All of them are orthogonal and isomorphic to each other and

$$\begin{aligned} \mathcal {H} = \bigoplus _{k \in \mathbb {N}} \mathcal {A}_{(k)}. \end{aligned}$$

In particular, the orthonormal basis of $\mathcal {H}$ can be formed by the union of the orthonormal bases of all true-polyanalytic spaces.

If $n >1$, the $\textbf{k}$-polyanalytic subspaces $\mathcal {A}_{\textbf{k}}$ are only partially ordered by inclusion, $\mathcal {A}_{\textbf{q}} \subsetneq \mathcal {A}_{\textbf{k}}$ if and only if $\textbf{q}\prec \textbf{k}$, where the order $\textbf{q}\prec \textbf{k}$ means that $q_j \le k_j$, for all $j = 1,2...,n$, and at least for one $j = 1,2...,n$, $q_j < k_j$.

The true-$\textbf{k}$-polyanalytic spaces are defined then as

$$\begin{aligned} \mathcal {A}_{(\textbf{k})} = \mathcal {A}_{\textbf{k}} \cap \left( \dot{\sum }_{j=1}^n \mathcal {A}_{\textbf{k}-\textbf{e}_j}\right) ^{\perp }, \end{aligned}$$

where

$$\begin{aligned} \textbf{e}_j = (0,\ldots , 0, \underset{j^{th}- place}{1}, 0,\ldots , 0 ). \end{aligned}$$

Surprisingly for $n > 1$ it turns out that not all true-polyanalytic spaces are orthogonal to each other. To be more precise, in case the tuples $\textbf{q}$ and $\textbf{k}$ are comparable (i.e., either $\textbf{q}\prec \textbf{k}$ or $\textbf{k}\prec \textbf{q}$), the true-polyanalytic spaces $\mathcal {A}_{(\textbf{q})}$ and $\mathcal {A}_{(\textbf{k})}$ are orthogonal, while in case the tuples $\textbf{q}$ and $\textbf{k}$ are incomparable, the true-polyanalytic spaces $\mathcal {A}_{(\textbf{q})}$ and $\mathcal {A}_{(\textbf{k})}$ are not necessarily orthogonal.

Such a situation is described in Lemma 5.4, where even the minimal angle between two true-poly-Bergman spaces on the unit ball $\mathbb {B}^n$ is zero, and in Lemma 6.3, where two true-poly-Fock spaces in $\mathbb {C}^2$ (with non-standard generalized Gaussian measure) are not orthogonal with positive minimal angle between them. In particular the above implies that an orthonormal basis in the corresponding Hilbert space cannot be formed by the union of the orthonormal bases all true-polyanalytic spaces.

At the same time, Sects. 4 and 6.1 provide us with the examples when all true-polyanalytic spaces are orthogonal to each other, the corresponding Hilbert space is recovered as the orthogonal sum of all its true-polyanalytic subspaces, and the union of the orthonormal bases all true-polyanalytic spaces forms an orthonormal basis in the Hilbert space in question.

Let us mention that under study the polyanalytic functions (and the corresponding spaces) two tuples of operators are naturally appear: multiplication by $\overline{z}_j$ and $\frac{\partial }{\partial \overline{z}_{j}}$, with $j=1,2,...,n$, which act invariantly on the linear space of all smooth function in D, and satisfy therein the commutation relations

$$\begin{aligned} \left[ \frac{\partial }{\partial \overline{z}_j}, \overline{z}_{\ell }\right] = \delta _{j,\ell }I, \quad \left[ \frac{\partial }{\partial \overline{z}_j}, \frac{\partial }{\partial \overline{z}_{\ell }}\right] =0, \quad \left[ \overline{z}_j, \overline{z}_{\ell }\right] = 0, \quad j,\ell = 1,2,...,n. \end{aligned}$$

These relations, in fact, essentially determine the majority of the properties of the spaces under study.

To make then our study more case independent and applicable to a variety of different situations, we consider the following abstract generalization of the above operators. Let two tuples of operators

$$\begin{aligned} \mathbf {\mathfrak {a}}= (\mathfrak {a}_1, \ \mathfrak {a}_2, \ \ldots , \ \mathfrak {a}_n) \quad \textrm{and} \quad \mathbf {\mathfrak {b}}= (\mathfrak {b}_1, \ \mathfrak {b}_2, \ \ldots , \ \mathfrak {b}_n), \end{aligned}$$

act invariantly in some linear space and satisfy therein the commutation relations

$$\begin{aligned} \left[ \mathfrak {a}_j, \mathfrak {b}_{\ell }\right] = \delta _{j,\ell }I, \quad \left[ \mathfrak {a}_j, \mathfrak {a}_{\ell }\right] = 0, \quad \left[ \mathfrak {b}_j, \mathfrak {b}_{\ell }\right] = 0, \quad j,\ell = 1,2,...,n. \end{aligned}$$

Several examples that fit to the above setting are given in Example 2.2. We assume further that a common invariant domain of the above operators is a dense linear subspace in a Hilbert space $\mathcal {H}$, and impose several additional conditions. The exact formulation is given in the Extended Fock space construction, see Definition 3.1. Our approach extends the classical Fock space formalism (see e.g. [3, Chapter 5, Section 5.2]) in two directions: first, the space $\bigcap _{j=1}^n \ker \mathfrak {a}_j$ is not anymore necessarily one-dimensional and, second, the operators $\mathfrak {b}_j$ are not necessarily formally adjoint to $\mathfrak {a}_j$. Although we pay also special attention to the case when $\mathfrak {b}_j$ are formally adjoint to $\mathfrak {a}_j$.

After such a general approach has been developed (Sects. 2 and 3), in the remaining part of the paper (Sects. 4–6) we consider different particular realizations of the above tuples of operators $\mathbf {\mathfrak {a}}$ and $\mathbf {\mathfrak {b}}$, and the corresponding Hilbert spaces $\mathcal {H}$, which shows a variety of possibilities that may occur in the characterisation of the polyanalytic type spaces in several complex variables.

2 Polyanalytic Linear Spaces

In what follows we will consider connected simply connected domains $D \subset \mathbb {C}^n$ (the typical examples are the unit ball $\mathbb {B}^n$ and the whole space $\mathbb {C}^n$), and use the standard notations: $\varvec{z}=(z_1,z_2,...,z_n) \in \mathbb {C}^n$, and for a tuple $\textbf{k}= (k_1,k_2,...,k_n) \in \mathbb {N}^n$,

$$\begin{aligned}{} & {} \textbf{k}! = k_1!k_2!\cdot \ldots \cdot k_n!, \qquad \overline{\varvec{z}}^{\textbf{k}} = \overline{z}_1^{k_1}\overline{z}_2^{k_2}\cdot \ldots \cdot \overline{z}_n^{k_n}, \\{} & {} \frac{\partial ^{\textbf{k}}}{\partial \overline{\varvec{z}}^{\textbf{k}}} =\left( \frac{\partial ^{k_1}}{\partial \overline{z}_1^{k_1}}, \ \frac{\partial ^{k_2}}{\partial \overline{z}_2^{k_2}}, \ \ldots , \ \frac{\partial ^{k_n}}{\partial \overline{z}_n^{k_n}} \right) . \end{aligned}$$

Then, given a tuple of operators $\textbf{A}= (A_1,\ldots , A_n)$ acting invariantly in some linear space, if it does not state otherwise, we set $\textbf{A}^{\textbf{k}}= A_1^{k_1}\cdot \ldots \cdot A_n^{k_n}$.

Further, given two tuples $\textbf{q}= (q_1,q_2,...,q_n)$ and $\textbf{k}= (k_1, k_2,...,k_n)$ from $\mathbb {Z}_+^n$, we set $\textbf{q}\preceq \textbf{k}$ if and only if $q_j \le k_j$ for all $j = 1,2...,n$, and $\textbf{q}\prec \textbf{k}$ if and only if $\textbf{q}\preceq \textbf{k}$ and at least for one $j = 1,2...,n$, $q_j < k_j$.

Recall that, given a tuple $\textbf{k}= (k_1,k_2,...,k_n) \in \mathbb {N}^n$, a smooth function $\varphi$ is called $\textbf{k}$-polyanalytic if it satisfies in D the equation

$$\begin{aligned} \frac{\partial ^{\textbf{k}} \varphi }{\partial \overline{\varvec{z}}^{\textbf{k}}} = \textbf{0}. \end{aligned}$$

As stated in [2, Section 6.4], the above condition is equivalent to the following its representations

$$\begin{aligned} \varphi = \sum _{\textbf{q}=\textbf{0}}^{\textbf{k}-\textbf{1}} \overline{\varvec{z}}^{\textbf{q}} f_{\textbf{q}},\end{aligned}$$

(2.1)

where the summation extends to all tuples $\textbf{q}\preceq \textbf{k}-\textbf{1}$, with $\textbf{1}=(1,1,\ldots ,1)$, and where all $f_{\textbf{q}}$ are analytic in D functions.

We denote by $\mathcal {O}_{\textbf{k}} = \mathcal {O}_{\textbf{k}}(D)$ the linear space of all $\textbf{k}$-polyanalytic functions in D.

Observe that the two tuples of operators are involved: multiplication by $\overline{z}_j$ and $\frac{\partial }{\partial \overline{z}_{j}}$, with $j=1,2,...,n$, which act invariantly on the linear space of all smooth function in D, and satisfy therein the relations

$$\begin{aligned} \left[ \frac{\partial }{\partial \overline{z}_{j}}, \overline{z}_{\ell }\right] = \frac{\partial }{\partial \overline{z}_j} \overline{z}_{\ell } - \overline{z}_{\ell } \frac{\partial }{\partial \overline{z}_j} = \delta _{j,\ell }I, \quad \left[ \frac{\partial }{\partial \overline{z}_j}, \frac{\partial }{\partial \overline{z}_{\ell }}\right] =0, \quad \left[ \overline{z}_j, \overline{z}_{\ell }\right] = 0, \end{aligned}$$

where $j,\ell = 1,2,...,n$.

We put now the above into an abstract setting. Let $\mathfrak {L}$ be a linear space where two tuples of operators

$$\begin{aligned} \mathbf {\mathfrak {a}}= (\mathfrak {a}_1, \ \mathfrak {a}_2, \ \ldots , \ \mathfrak {a}_n) \quad \textrm{and} \quad \mathbf {\mathfrak {b}}= (\mathfrak {b}_1, \ \mathfrak {b}_2, \ \ldots , \ \mathfrak {b}_n), \end{aligned}$$

act invariantly and satisfy therein the commutation relations

$$\begin{aligned} \left[ \mathfrak {a}_j, \mathfrak {b}_{\ell }\right] = \delta _{j,\ell }I, \quad \left[ \mathfrak {a}_j, \mathfrak {a}_{\ell }\right] = 0, \quad \left[ \mathfrak {b}_j, \mathfrak {b}_{\ell }\right] = 0, \quad j,\ell = 1,2,...,n. \end{aligned}$$

(2.2)

Similarly to the above, given $\textbf{k}= (k_1,k_2,...,k_n) \in \mathbb {N}^n$, we denote by $\textsf {L}_{\textbf{k}}$ the linear subspace of $\mathfrak {L}$ which consists of all elements h satisfying the equation $\mathfrak {a}^{\textbf{k}} h = \left( \mathfrak {a}_1^{k_1}h, \ldots , \mathfrak {a}_n^{k_n}h \right) = \textbf{0}$, and call them $\textbf{k}$-poly-elements. In particular,

$$\begin{aligned} \textsf {L}_{\textbf{1}} = \bigcap _{j=1}^n\ker \mathfrak {a}_j. \end{aligned}$$

The relation $\textsf {L}_{\textbf{k}'} \subsetneq \textsf {L}_{\textbf{k}''}$ defines a partial order on the set $\mathcal {L} = \left\{ \textsf {L}_{\textbf{k}}: \textbf{k}\in \mathbb {N}^n\right\}$ of all subspaces $\textsf {L}_{\textbf{k}}$. Observe that $\textsf {L}_{\textbf{k}'} \subsetneq \textsf {L}_{\textbf{k}''}$ if and only if $\textbf{k}' \prec \textbf{k}''$. Note that in such an ordering $\mathcal {L}$ is a well-founded ($\mathcal {L}$ has no infinite strictly decreasing sequences) lattice (any two elements $\textsf {L}_{\textbf{k}'}$ and $\textsf {L}_{\textbf{k}''}$ of $\mathcal {L}$ have a least upper bound $\textsf {L}_{\textbf{k}}$, where $k_j= \max \{k_j',k_j''\}$, and a greatest lower bound $\textsf {L}_{\textbf{k}}$, where $k_j= \min \{k_j',k_j''\}$, $j=1,2,\ldots ,n$).

We mention that (2.2) easily implies $[\mathfrak {a}_j,\mathfrak {b}_{\ell }^p] = \delta _{j,\ell }\, p\mathfrak {b}_{\ell }^{p-1}$, for all $j,\ell = 1,\ldots , n$ and $p \in \mathbb {N}$, which, in turn, yields that for each $h \in \textsf {L}_{\textbf{1}}$ and all $p \in \mathbb {N}$,

$$\begin{aligned} \mathfrak {a}_j \mathfrak {b}_{\ell }^{p} h = \delta _{j,\ell }\, p\mathfrak {b}_{\ell }^{p-1}h. \end{aligned}$$

(2.3)

By induction, for all $h \in \textsf {L}_{\textbf{1}}$, we have then

$$\begin{aligned} \mathfrak {a}_{\ell }^s \mathfrak {b}_{\ell }^p h = \delta _{j,\ell } {\left\{ \begin{array}{ll} p(p-1)\cdots (p-s+1) \mathfrak {b}_{\ell }^{p-s} h, &{} \textrm{if} \quad s < p \\ p! h, &{} \textrm{if} \quad s = p \\ 0, &{} \textrm{if} \quad s > p \end{array}\right. }. \end{aligned}$$

(2.4)

Lemma 2.1

An element $h \in \textsf {L}_{\textbf{k}}$, i.e. is a $\textbf{k}$-poly-element, if and only if it admits the representation

$$\begin{aligned} h = \sum _{\textbf{q}=\textbf{0}}^{\textbf{k}-\textbf{1}} \mathbf {\mathfrak {b}}^{\textbf{q}} h_{\textbf{q}}, \qquad \textrm{with} \quad h_{\textbf{q}} \in \textsf {L}_{\textbf{1}}\end{aligned}$$

(2.5)

Proof

($\Rightarrow$) Follows from (2.4).

($\Leftarrow$) We prove it using Noetherian (well-founded) induction, see e.g. [7, Chapter 14, Section 1]. Observe that well-founded partially ordered set $\mathcal {L}$ is single-rooted, i.e., for each $\textsf {L}_{\textbf{k}}$ the set of its descending elements has the same minimal element $\textsf {L}_{\textbf{1}}$, for which the lemma statement is obvious. To prove then (2.5) for any $\textsf {L}_{\textbf{k}}$, we assume that such a representation is valid for each $\textsf {L}_{\textbf{q}}$ with $\textsf {L}_{\textbf{q}}\subsetneq \textsf {L}_{\textbf{k}}$.

We start with h satisfying the relation $\mathbf {\mathfrak {a}}^{\textbf{k}}h = \left( \mathfrak {a}_1^{k_1}h, \ldots , \mathfrak {a}_n^{k_n}h \right) = \textbf{0}$.

To simplify the notation and without loss of generality, we may assume that $k_1= \max \{k_j: j=1,2,\ldots ,n\}$. We have then

$$\begin{aligned} \mathbf {\mathfrak {a}}^{\textbf{k}}h = \mathbf {\mathfrak {a}}^{\textbf{k}-\textbf{e}_1}(\mathfrak {a}_1h) = \textbf{0}, \end{aligned}$$

where $\textbf{e}_1 = (1,0,\ldots , 0)$, implying that $\mathfrak {a}_1h \in \textsf {L}_{\textbf{k}- \textbf{e}_1}$. Thus $\mathfrak {a}_1h$ admits the representation

$$\begin{aligned} \mathfrak {a}_1h = \sum _{\textbf{q}=\textbf{0}}^{\textbf{k}- \textbf{e}_1-\textbf{1}} \mathbf {\mathfrak {b}}^{\textbf{q}} h'_{\textbf{q}}, \qquad \textrm{with} \quad h'_{\textbf{q}} \in \textsf {L}_{\textbf{1}}. \end{aligned}$$

which implies

$$\begin{aligned} \mathfrak {a}_1 \left( h - \sum _{\textbf{q}=\textbf{0}}^{\textbf{k}- \textbf{e}_1-\textbf{1}} \frac{1}{q_1+1}\mathfrak {b}_1\mathbf {\mathfrak {b}}^{\textbf{q}} h'_{\textbf{q}} \right) = \mathfrak {a}_1 \left( h - \sum _{\textbf{q}=\textbf{0}+\textbf{e}_1}^{\textbf{k}-\textbf{1}} \frac{1}{q_1+1}\mathbf {\mathfrak {b}}^{\textbf{q}} h'_{\textbf{q}} \right) = 0. \end{aligned}$$

Thus

$$\begin{aligned} h = \sum _{\textbf{q}=\textbf{0}+\textbf{e}_1}^{\textbf{k}-\textbf{1}} \frac{1}{q_1+1}\mathbf {\mathfrak {b}}^{\textbf{q}} h'_{\textbf{q}} + h_0, \quad \textrm{where} \quad h_0 \in \ker \mathfrak {a}_1. \end{aligned}$$

(2.6)

Then, for each $j=2,3,\ldots ,n$,

$$\begin{aligned} 0 = \mathfrak {b}^{k_j}h = \mathfrak {b}^{k_j}h_0, \end{aligned}$$

which implies that $h_0 \in \textsf {L}_{(1,k_2,\ldots , k_n)}$. Thus $h_0$ admits the representation

$$\begin{aligned} h_0 = \sum _{\textbf{q}=\textbf{0}}^{\textbf{k}-(k_1, 1,\ldots ,1)} \mathbf {\mathfrak {b}}^{\textbf{q}} h''_{\textbf{q}}, \qquad \textrm{with} \quad h''_{\textbf{q}} \in \textsf {L}_{\textbf{1}}, \end{aligned}$$

(2.7)

and the result follows from (2.6) and (2.7). $\square$

Before we move forward, let us give several examples that fit to the above setting.

Example 2.2

The following two tuples of operators $\mathfrak {a}_j$ and $\mathfrak {b}_j$, with $j=1,2,\ldots ,n$, satisfy the conditions (2.2) on a linear set of smooth functions in $D \subset \mathbb {C}^n$:

• (1) $\mathfrak {a}= \frac{\partial }{\partial \overline{z}_j}$ and $\mathfrak {b}_j = \overline{z}_j$;

• (2) $\mathfrak {a}= \frac{\partial }{\partial \overline{z}_j}$ and $\mathfrak {b}_j = - \frac{\partial }{\partial z_j} + \overline{z}_j$;

• (3) $\mathfrak {a}_j = \frac{\partial }{\partial \overline{z}_j} - \omega _j z_j$ and $\mathfrak {b}_j = \overline{z}_j$, with $\omega _j \in \mathbb {R}$ and $D = \mathbb {C}^n$;

• (4) $\mathfrak {a}_j = \frac{\partial }{\partial \overline{z}_j} + \mu \frac{z_j}{1-|z|^2}$ and $\mathfrak {b}_j = \overline{z}_j$, with $\mu \in \mathbb {R}_+$ and $D = \mathbb {B}^n$;

• (5) all above items with $z_j$ and $\overline{z}_j$ interchanged.

For each $\textbf{k}= (k_1,k_2,\ldots , k_n)$, we introduce the following subspaces of $\mathfrak {L}$:

$$\begin{aligned} \textsf {L}_{[\textbf{1}]} = \textsf {L}_{\textbf{1}} =\bigcap _{j=1}^n\ker \mathfrak {a}_j \qquad \textrm{and} \qquad \textsf {L}_{[\textbf{k}]} = \mathbf {\mathfrak {b}}^{\textbf{k}- \textbf{1}}\textsf {L}_{[\textbf{1}]}, \end{aligned}$$

and consider some their properties.

Lemma 2.3

All subspaces $\textsf {L}_{[\textbf{k}]}$ are isomorphic among each other.

Proof

It is sufficient to prove that each $\textsf {L}_{[\textbf{k}]}$ is isomorphic to $\textsf {L}_{[\textbf{1}]}$. Indeed, the mapping

$$\begin{aligned} \mathbf {\mathfrak {b}}^{\textbf{k}- \textbf{1}}: h \in \textsf {L}_{[\textbf{1}]} \ \longmapsto \ \mathbf {\mathfrak {b}}^{\textbf{k}- \textbf{1}}h \in \textsf {L}_{[\textbf{k}]} \end{aligned}$$

is an isomorphism, whose inverse is given by

$$\begin{aligned} {\textstyle \frac{1}{(\textbf{k}- \textbf{1})!}}\mathbf {\mathfrak {a}}^{\textbf{k}- \textbf{1}}: \ \mathbf {\mathfrak {b}}^{\textbf{k}- \textbf{1}}h \in \textsf {L}_{[\textbf{k}]} \ \longmapsto \ h \in \textsf {L}_{[\textbf{1}]}. \end{aligned}$$

$\square$

Lemma 2.4

Each finite number of subspaces $\textsf {L}_{[\textbf{k}]}$ are linearly independent.

Proof

Assume contrary that for some $h_{\textbf{k}^{(p)}} = \mathbf {\mathfrak {b}}^{\textbf{k}^{(p)} - \varvec{1}}h^{(p)} \in \textsf {L}_{[\textbf{k}^{(p)}]}$, $p=1,2,\ldots ,m$, we have

$$\begin{aligned} g=\sum _{p=1}^m \mathbf {\mathfrak {b}}^{\textbf{k}^{(p)} - \textbf{1}}h^{(p)} =0. \end{aligned}$$

(2.8)

We prove then recursively that all $h^{(p)} = 0$. We start with

$$\begin{aligned} k'_{j_1}= & {} \max \{k^{(p)}_j: \ p=1,\ldots ,m, \ j=1,\ldots ,n\}, \\ k'_{j_2}= & {} \max \{k^{(p)}_j: \ k^{(p)}_{j_1} = k'_{j_1}, \ j \ne j_1\}, \\ k'_{j_3}= & {} \max \{k^{(p)}_j: \ k^{(p)}_{j_1} = k'_{j_1}, \ k^{(p)}_{j_2} = k'_{j_2}, \ j \ne j_1, \ j \ne j_2 \},\\ {}&\vdots&\end{aligned}$$

As all $\textbf{k}^{(p)}$ are different, this process ends at no more then $n-1$ steps, resulting the unique tuple $\textbf{k}^{(p')}$ such that $k^{(p')}_{j_1}=k'_{j_1}$, $k^{(p')}_{j_2}=k'_{j_2}$, $\ldots$ Then

$$\begin{aligned} 0 = \mathbf {\mathfrak {a}}{\textbf{k}^{(p') - \textbf{1}}}\, g = (\textbf{k}^{(p')} - \textbf{1})!\, h^{(p')}, \end{aligned}$$

which implies that $h^{(p')} = 0$ and expression (2.8) has one summand less.

Repeating this procedure m times, we conclude that all $h^{(p)} = 0$. $\square$

Corollary 2.5

We have that

$$\begin{aligned} \textsf {L}_{\textbf{k}} = \sum _{\textbf{q}=\textbf{1}}^{\textbf{k}} \textsf {L}_{[\textbf{q}]}. \end{aligned}$$

Proof

By Lemma 2.1 the space $\textsf {L}_{\textbf{k}}$ consists of all elements h of the form

$$\begin{aligned} h = \sum _{\textbf{q}=\textbf{0}}^{\textbf{k}-\textbf{1}} \mathbf {\mathfrak {b}}^{\textbf{q}} h'_{\textbf{q}} = \sum _{\textbf{q}=\textbf{1}}^{\textbf{k}} \mathbf {\mathfrak {b}}^{\textbf{q}- \textbf{1}} h_{\textbf{q}}, \qquad \textrm{with} \quad h_{\textbf{q}}=h'_{\textbf{q}- \textbf{1}} \in \textsf {L}_{\textbf{1}}.\end{aligned}$$

Then each element of $\textsf {L}_{[\textbf{q}]}$ is of the form $\mathbf {\mathfrak {b}}^{\textbf{q}- \textbf{1}} h_{\textbf{q}}$, with $h_{\textbf{q}} \in \textsf {L}_{\textbf{1}}$, and, by Lemma 2.4, the subspaces $\textsf {L}_{[\textbf{k}]}$ are linearly independent. This finishes the proof. $\square$

3 Extended Fock Space Construction

We add here a Hilbert space structure to a pure linear algebra considerations of the previous section. We extend the classical multi-operator Fork space formalism (see e.g. [3, Chapter 5, Section 5.2]) so that it will cover many important cases, including e.g. the case of polyanalytic functions.

3.1 Generic Case

Definition 3.1

(Extended Fock Space Construction) There are a separable Hilbert space $\mathcal {H}$ and two tuples of operators

$$\begin{aligned} \mathbf {\mathfrak {a}}= (\mathfrak {a}_1, \ \mathfrak {a}_2, \ \ldots , \ \mathfrak {a}_n) \quad \textrm{and} \quad \mathbf {\mathfrak {b}}= (\mathfrak {b}_1, \ \mathfrak {b}_2, \ \ldots , \ \mathfrak {b}_n), \end{aligned}$$

defined on their natural (maximal) domains dense in $\mathcal {H}$ and such that

• 1. there exists a linear subspace $\mathcal {D}$ dense in $\mathcal {H}$, where the operators $\mathfrak {a}_j$ and $\mathfrak {b}_j$, $j = 1,2,...,n$, act invariantly, and on which they satisfy the commutation relations

  $$\begin{aligned} \left[ \mathfrak {a}_j, \mathfrak {b}_{\ell }\right] = \delta _{j,\ell }I, \quad \left[ \mathfrak {a}_j, \mathfrak {a}_{\ell }\right] = 0, \quad \left[ \mathfrak {b}_j, \mathfrak {b}_{\ell }\right] = 0, \quad j,\ell = 1,2,...,n. \end{aligned}$$

  (3.1)

• 2. the linear subspace

  $$\begin{aligned} \textsf {L}_{[\textbf{1}]} = \bigcap _{j=1}^n\ker \mathfrak {a}_j = \left\{ h \in \mathcal {D}: \, \mathfrak {a}h = \left( \mathfrak {a}_1 h, \ldots , \mathfrak {a}_n h \right) = \textbf{0} \right\} \end{aligned}$$

  is non-trivial with $\dim \textsf {L}_{[1]} \ge 1$;

• 3. the set $\mathcal {D}_0$, formed by finite linear combinations of elements from the linear subspaces $\textsf {L}_{[\textbf{k}]}:= \mathbf {\mathfrak {b}}^{\textbf{k}-\textbf{1}} \textsf {L}_{[\textbf{1}]}$, $\textbf{k}\in \mathbb {N}^n$, is dense in $\mathcal {H}$.

Note that the classical Fock formalism of [3, Chapter 5, Section 5.2] corresponds to the case when $\textsf {L}_{[\textbf{1}]}$ is generated by a single element $\Phi _0 \in \bigcap _{j=1}^n\ker \mathfrak {a}_j$, called vacuum vector, and all $\mathfrak {b}_j =\mathfrak {a}_J^{\dag }$ are formally adjoint to $\mathfrak {a}_j$. That is, we extend the classical Fock formalism in two directions: first, the space $\textsf {L}_{[\textbf{1}]}$ is not anymore necessarily one-dimensional and, second, the operators $\mathfrak {b}_j$ are not necessarily formally adjoint to $\mathfrak {a}_j$.

We mention also that, as examples show, the subspaces $\textsf {L}_{[\textbf{k}]}$ may or may not be closed, as well as, they may or may not be mutually orthogonal to each other.

Let us give now just two examples to the above construction related to the first item of Example 2.2, the other ones will be considered later on.

Example 3.2

We consider two classical cases of the Hilbert space $\mathcal {H}$:

• (i) $\mathcal {H} = L_2(\mathbb {\mathbb {B}}^n, d\nu _{\lambda })$, where the weighted measure $v_{\lambda }$, $\lambda > -1$, is absolutely continuous with respect to the Lebesgue volume form $dv(\varvec{z})$ on $\mathbb {B}^n$ and given by:

  $$\begin{aligned} dv_{\lambda }(\varvec{z})=c_{\lambda }^{(n)} (1-|\varvec{z}|^2)^{\lambda } dv(\varvec{z}) \hspace{3ex} \text { with} \hspace{3ex} c_{\lambda }^{(n)}:= \frac{\Gamma (n+\lambda +1)}{\pi ^n \Gamma (\lambda +1)}; \end{aligned}$$

• (ii) $\mathcal {H} = L_2(\mathbb {\mathbb {C}}^n, d\mu _{\alpha })$, where the Gaussian measure $d\mu _{\alpha }$ is given by

  $$\begin{aligned} d\mu _{\alpha }(\varvec{z}) = \frac{\alpha ^n}{\pi ^n} e^{-\alpha |\varvec{z}|^2}dv(\varvec{z}), \qquad \textrm{with} \quad \alpha >0; \end{aligned}$$

  (3.2)

In both cases, $\mathfrak {a}_j = \frac{\partial }{\partial \overline{z_j}}$ and $\mathfrak {b}_j = \overline{z}_j$ are the multiplication by $\overline{z}_j$ operators, $j=1,\ldots ,n$; $\mathcal {D} = \mathcal {D}_0$ consists of finite linear combinations of the monomials $m_{p,q}= \overline{\varvec{z}}^{\textbf{q}}\varvec{z}^{\textbf{p}}$, $\textbf{p},\,\textbf{q}\in \mathbb {Z}_+^n$, and $\textsf {L}_{[\textbf{1}]}= \bigcap _{j=1}^n\ker \mathfrak {a}_j$ coincides with the set of all analytic polynomials (polynomials on $\varvec{z}^{\textbf{p}}$).

Note that the linear set $\mathcal {D} = \mathcal {D}_0$ is nothing but the set of all polyanalytic polynomials, i.e., the set of all functions (2.1),

$$\begin{aligned} \varphi = \sum _{\textbf{q}} \overline{\varvec{z}}^{\textbf{q}} f_{\textbf{q}}, \end{aligned}$$

where all $f_{\textbf{q}}$ are analytic polynomials.

Having at hand the Hilbert space structure, we extend now the properties of the spaces $\textsf {L}_{[\textbf{k}]}$ of the previous section, setting $\mathfrak {L} = \mathcal {D}_0$.

Lemma 3.3

For each $\textbf{k}\in \mathbb {N}^n$, the operators $\mathfrak {a}_j$ and $\mathfrak {b}_j$, $j = 1,\ldots ,n$, restricted correspondingly on $\textsf {L}_{[\textbf{k}+\textbf{e}_j]}$ and $\textsf {L}_{[\textbf{k}]}$, act as isomorphisms between the following spaces

$$\begin{aligned} \mathfrak {a}_j|_{\textsf {L}_{[\textbf{k}+\textbf{e}_j]}}: \ \textsf {L}_{[\textbf{k}+\textbf{e}_j]} \ \longrightarrow \ \textsf {L}_{[\textbf{k}]} \qquad \textrm{and} \qquad \mathfrak {b}_j|_{\textsf {L}_{[\textbf{k}]}}: \ \textsf {L}_{[\textbf{k}]} \ \longrightarrow \ \textsf {L}_{[\textbf{k}+\textbf{e}_j]}. \end{aligned}$$

Moreover, the operators $\mathfrak {a}_j\mathfrak {b}_j$ and $\mathfrak {b}_j\mathfrak {a}_j$, being restricted on $\textsf {L}_{[\textbf{k}]}$ and $L_{[\textbf{k}+\textbf{e}_j]}$, respectively, act as the scalar operators,

$$\begin{aligned} \mathfrak {a}_j\mathfrak {b}_j|_{\textsf {L}_{[\textbf{k}]}}{} & {} = k_jI: \ \textsf {L}_{[\textbf{k}]} \ \rightarrow \ \textsf {L}_{[\textbf{k}]} \quad \textrm{and} \quad \mathfrak {b}_j\mathfrak {a}_j|_{\textsf {L}_{[\textbf{k}+\textbf{e}_j]}}\nonumber \\{} & {} = k_jI: \ \textsf {L}_{[\textbf{k}+\textbf{e}_j]} \ \rightarrow \ \textsf {L}_{[\textbf{k}+\textbf{e}_j]}. \end{aligned}$$

(3.3)

Proof

Recall that $\textsf {L}_{[\textbf{k}]} = \{\mathbf {\mathfrak {b}}^{\textbf{k}-\textsf {1}}h: \ \mathrm {for \ all} \ \ h \in \textsf {L}_{[\textsf {1}]}\}$. The lemma assertions follow then from

$$\begin{aligned} \mathfrak {a}_j|_{\textsf {L}_{[\textbf{k}+\textbf{e}_j]]}}&:&\ \mathbf {\mathfrak {b}}^{\textbf{k}-\textsf {1}+\textsf {e}_j}h \ \longmapsto \ \mathfrak {a}_j\mathbf {\mathfrak {b}}^{\textbf{k}-\textsf {1}+\textsf {e}_j}h = k_j \mathbf {\mathfrak {b}}^{\textbf{k}-\textsf {1}}h = \mathbf {\mathfrak {b}}^{\textbf{k}-\textsf {1}}(k_jh), \\ \mathfrak {b}_j|_{\textsf {L}_{[\textbf{k}]}}&:&\ \mathbf {\mathfrak {b}}^{\textbf{k}-\textsf {1}}h \ \longmapsto \ \mathbf {\mathfrak {b}}^{\textbf{k}-\textsf {1}+\textsf {e}_j}h. \qquad \qquad \qquad \qquad \qquad \qquad \qquad \end{aligned}$$

$\square$

By (3.3), for all $\textbf{k}\in \mathbb {N}^n$, each operator $\mathfrak {a}_j\mathfrak {b}_j$, $j=1,\ldots ,n$, can be extended by continuity from $\textsf {L}_{[\textbf{k}]}$ to its closure $\overline{\textsf {L}_{[\textbf{k}]}} = \textrm{clos}\,\textsf {L}_{[\textbf{k}]}$, with $\mathfrak {a}_j\mathfrak {b}_j|_{\overline{\textsf {L}_{[\textbf{k}]}}} = k_jI$. That is, the operators $\mathfrak {a}_j\mathfrak {b}_j$, initially defined on $\mathcal {D}_0$, can be extended to a wider common domain which consists of all finite linear combinations of elements from different $\overline{\textsf {L}_{[\textbf{k}]}}$.

They can be even extended to a more wide common domain $\mathcal {D}^{\#}$ which consists of all elements $f = {\sum _{\textbf{k}\in \mathbb {N}^n}} f_{\textbf{k}} \in \mathcal {H}$, with $f_{\textbf{k}} \in \overline{\textsf {L}_{[\textbf{k}]}}$, such that ${\sum _{\textbf{k}\in \mathbb {N}^n}} {\max _j}\{k_j\}f_{\textbf{k}} \in \mathcal {H}$.

Corollary 3.4

Each space $\overline{\textsf {L}_{[\textbf{k}]}}$ is a common invariant subspace of all operators $\mathfrak {a}_j\mathfrak {b}_j$, $j=1,\ldots ,n$, defined on $\mathcal {D}^{\#}$. All of them are eigenspaces of the operator $\sum _{j=1}^n\mathfrak {a}_j\mathfrak {b}_j$, whose corresponding eigenvalues are $\sum _{j=1}^nk_j$.

Corollary 3.5

For all $\textbf{k}' \ne \textbf{k}''$, the intersection of the closed subspaces $\overline{\textsf {L}_{[\textbf{k}']}}$ and $\overline{\textsf {L}_{[\textbf{k}'']}}$ is trivial, $\overline{\textsf {L}_{[\textbf{k}']}} \cap \overline{\textsf {L}_{[\textbf{k}'']}} = \{0\}$.

Proof

The condition $\textbf{k}' \ne \textbf{k}''$ implies that there is j with $k_j' \ne k_l''$. Assuming the existence of $f \in \overline{\textsf {L}_{[\textbf{k}']}} \cap \overline{\textsf {L}_{[\textbf{k}'']}} \ne \{0\}$, we have $k_j'f = \mathfrak {a}_j\mathfrak {b}_j|_{\overline{\textsf {L}_{[\textbf{k}']}}} f = \mathfrak {a}_j\mathfrak {b}_j|_{\overline{\textsf {L}_{[\textbf{k}'']}}} f = k_j''f$, implying $f=0$. $\square$

Corollary 3.6

Any finite number of spaces $\overline{\textsf {L}_{[\textbf{k}^{(1)}]}}$, $\overline{\textsf {L}_{[\textbf{k}^{(2)}]}}$,..., $\overline{\textsf {L}_{[\textbf{k}^{(p)}]}}$ are linearly independent.

Proof

Let us assume that: there exist elements $h^{[\textbf{k}^{(\ell )}]} \in \overline{\textsf {L}_{[\textbf{k}^{{(\ell )}}]}}$, $\ell = 1,2,\ldots , p$, not all of them equal to 0, such that

$$\begin{aligned} h^{[\textbf{k}^{(1)}]} + \cdots + h^{[\textbf{k}^{(p)}]} = 0. \end{aligned}$$

(3.4)

By a kind of recursion we prove that all $h^{[\textbf{k}^{(\ell )}]} = 0$. We start with an analysis of the first components of the multi-indices $\textbf{k}^{(\ell )}$, $\ell = 1,\ldots ,p$. They take at most p different values, say, $k_1^{(1)},\ldots ,k_1^{(s_1)}$, where $1 \lneq s_1 \lneq p$. Divide then elements in (3.4) in $s_1$ groups:

$$\begin{aligned} h_{k_1^{(m)}} = \sum _{k_1^{[\textbf{k}^{(\ell )}]} = k_1^{(m)}} h^{[\textbf{k}^{(\ell )}]}, \qquad m = 1,\ldots , s_1, \end{aligned}$$

obtaining thus

$$\begin{aligned} h_{k_1^{(1)}} + \dots h_{k_1^{(s_1)}} = 0. \end{aligned}$$

Applying consequently the powers $(\mathfrak {a}_1\mathfrak {b}_1)^j$, $j = 1,2,\ldots , s_1-1$, we obtain other $s_1-1$ relations

$$\begin{aligned} (k_1^{(1)})^jh_{k_1^{(1)}} + \cdots + (k_1^{(s_1)})^jh_{k_1^{(s_1)}} = 0. \end{aligned}$$

We get thus the system of linear equations on $h_{k_1^{(m)}}$, $m = 1,2,\ldots , s_1$, whose determinant is Vandermonde,

$$\begin{aligned} \begin{vmatrix} 1&1&\cdots&1 \\ k_1^{(1)}&k_1^{(2)}&\cdots&k_1^{(s_1)} \\ \vdots&\vdots&\ddots&\vdots \\ (k_1^{(1)})^{s_1-1}&(k_1^{(2)})^{s_1-1}&\cdots&(k_1^{(s_1)})^{s_1-1} \end{vmatrix} = \prod _{1\le i < j \le s_1} (k_1^{(j)} - k_1^{(i)}) \ne 0, \end{aligned}$$

implying that all $h_{k_1^{(m)}}$ are 0.

We repeat then the same procedure to each one of $h_{k_1^{(m)}} = \sum _{k_1^{[\textbf{k}^{(\ell )}]} = k_1^{(m)}} h^{[\textbf{k}^{(\ell )}]}$ analysing the second component of the multi-indices $\textbf{k}^{(\ell )}$, $\ell = 1,\ldots ,p$. Then, as all multi indices $\textbf{k}^{(\ell )}$, $\ell = 1,\ldots ,p$ are different, at no more that $n-1$ steps, we get that all $h^{[\textbf{k}^{(\ell )}]} = 0$. $\square$

We will use the following standard notation. The linear span of finite number $H_1$, $H_2$,...,$H_n$ of linearly independent subspaces of $\mathcal {H}$ is called the direct sum and is denoted by

$$\begin{aligned} H_1 \dotplus H_2 \dotplus \cdots \dotplus H_n, \end{aligned}$$

in case when they are additionally pairwise orthogonal, we write

$$\begin{aligned} H_1 \oplus H_2 \oplus \cdots \oplus H_n, \end{aligned}$$

and call it the orthogonal sum.

We note as well that, given even just two closed subspaces $H_1$ and $H_2$ with trivial intersection, $H_1\cap H_2 = \{0\}$, (equivalently being linearly independent), their direct sum $H_1 \dotplus H_2$ may not be closed. It depends on the so-called minimal angle between them.

Recall (see e.g. [5]) that the minimal angle $\varphi ^{(m)}(H_1,H_2)$ between two closed subspaces $H_1$ and $H_2$ of a Hilbert space H is defined as

$$\begin{aligned} \cos \varphi ^{(m)}(H_1,H_2) = \sup \left\{ |\langle x, y \rangle |: \ x \in H_1, \ y \in H_2 \ \ \textrm{and} \ \ \Vert x\Vert =\Vert y\Vert = 1 \right\} .\nonumber \\ \end{aligned}$$

(3.5)

Recall in this connection the criterion of when the direct sum of two closed spaces is closed.

Lemma 3.7

[5, Lemma 1] The direct sum of two closed subspaces, that intersect only by zero, is closed if and only if the minimal angle between them is greater than zero.

By Lemma 2.1, for each $\textbf{k}\in \mathbb {N}^n$, the subspace $\displaystyle {\textsf {L}_{\textbf{k}} = \dot{\sum }_{\textbf{q}= \varvec{1}}^{\textbf{k}} \textsf {L}_{[\textbf{q}]}}$ coincides with the linear set of elements h satisfying the conditions

$$\begin{aligned} \mathfrak {a}_1^{k_1} h = 0, \quad \ldots , \quad \mathfrak {a}_n^{k_n} h = 0, \end{aligned}$$

(3.6)

or ${\textsf {L}_{\textbf{k}} = \bigcap _{j=1}^k \ker \mathfrak {a}_j^{k_j}|_{\mathfrak {D}_0}}$. By continuity, conditions (3.6) extends to the closure $\overline{\textsf {L}_{\textbf{k}}}$ of $\textsf {L}_{\textbf{k}}$.

Note that, depending on the concrete Hilbert space $\mathcal {H}$ and operators $\mathfrak {a}_j$, $\mathfrak {b}_j$ involved, it can happen that, in spite of $\textsf {L}_{\textbf{k}} = \dot{\sum }_{\textbf{q}= \varvec{1}}^{\textbf{k}} \textsf {L}_{[\textbf{q}]}$,

$$\begin{aligned} \mathop {\dot{\sum }}\limits _{\textbf{q}= \varvec{1}}^{\textbf{k}} \overline{\textsf {L}_{[\textbf{q}]}} \ \subsetneq \ \overline{\textsf {L}_{\textbf{k}}} = \textrm{clos}\,\left( \mathop {\dot{\sum }}\limits _{\textbf{q}= \varvec{1}}^{\textbf{k}} \textsf {L}_{[\textbf{q}]}\right) \end{aligned}$$

Such a situation will be discussed in detail in Subsection 5.1.

We denote then $\overline{\textsf {L}_{\textbf{k}}}$ by $\mathcal {H}_{\textbf{k}}$ and call it $\textbf{k}$-poly-$\mathbf {\mathfrak {a}}$ subspace of $\mathcal {H}$; introduce also

$$\begin{aligned} \mathcal {H}_{(\textbf{k})} = \mathcal {H}_{\textbf{k}} \cap \left( \mathop {\dot{\sum }}\limits _{j=1}^n \mathcal {H}_{\textbf{k}-\textbf{e}_j}\right) ^{\perp }, \end{aligned}$$

and call it true-$\textbf{k}$-poly-$\mathbf {\mathfrak {a}}$ subspace of $\mathcal {H}$.

Note that each $\mathcal {H}_{(\textbf{k})}$ is orthogonal to all $\mathcal {H}_{(\textbf{q})}$ with $\textbf{q}\prec \textbf{k}$, at the same time the subspaces $\mathcal {H}_{(\textbf{k})}$ and $\mathcal {H}_{(\textbf{q})}$ with incomparable $\textbf{k}$ and $\textbf{q}$ are not necessarily orthogonal (see examples in Lemmas 5.4 and 6.3).

The density of $\mathcal {D}_0$ in $\mathcal {H}$ implies then two alternative representation of the Hilbert space $\mathcal {H}$,

$$\begin{aligned} \mathcal {H} = \textrm{clos}\,\left( \mathop {\dot{\sum }}\limits _{\textbf{k}\in \mathbb {N}^n} \mathcal {H}_{(\textbf{k})}\right) = \textrm{clos}\,\left( \mathop {\dot{\sum }}\limits _{\textbf{k}\in \mathbb {N}^n} \textsf {L}_{[\textbf{k}]}\right) . \end{aligned}$$

3.2 Special Case of $\mathfrak {b}= \mathfrak {a}^\dag$

Let now $\mathfrak {b}_j = \mathfrak {a}_j^\dag$ be formally adjoint to $\mathfrak {a}_j$ in $\mathcal {D}_0$ for all $j=1,\ldots ,n$. We discuss here some specific features of this case.

Proposition 3.8

All closed subspaces $\overline{\textsf {L}_{[\textbf{k}]}}$, $\textbf{k}\in \mathbb {N}^n$, are pairwise orthogonal.

Proof

It is sufficient to check the statement for the (non closed) subspaces $\textsf {L}_{[\textbf{k}]}$. Take now any two subspaces $\textsf {L}_{[\textbf{k}']}$ and $\textsf {L}_{[\textbf{k}'']}$. As they are different, there is at least one $j_0$ such that $k'_{j_0} \ne k''_{j_0}$. Let $k'_{j_0} > k''_{j_0}$, considering the scalar product of $(\mathbf {\mathfrak {a}}^{\dag })^{\textbf{k}'-\varvec{1}}h' \in \textsf {L}_{[\textbf{k}']}$ and $(\mathbf {\mathfrak {a}}^{\dag })^{\textbf{k}''-\varvec{1}}h'' \in \textsf {L}_{[\textbf{k}'']}$, and using the last line of (2.4) we have

$$\begin{aligned}{} & {} \langle (\mathbf {\mathfrak {a}}^{\dag })^{\textbf{k}'-\varvec{1}}h', (\mathbf {\mathfrak {a}}^{\dag })^{\textbf{k}''-\varvec{1}}h'' \rangle = \langle (\mathbf {\mathfrak {a}}^{\dag })^{\textbf{k}'-\varvec{1}-k''_{j_0}\textbf{e}_{j_0}}h', \mathfrak {a}_{j_0}^{k''_{j_0}}(\mathbf {\mathfrak {a}}^{\dag })^{\textbf{k}''-\varvec{1}}h'' \rangle \\{} & {} \quad = \ \langle (\mathbf {\mathfrak {a}}^{\dag })^{\textbf{k}'-\varvec{1}-k''_{j_0}\textbf{e}_{j_0}}h', (\mathbf {\mathfrak {a}}^{\dag })^{\textbf{k}''-\varvec{1}-(k''_{j_0}-1)\textbf{e}_{j_0}}\mathfrak {a}_{j_0}^{k''_{j_0}}(\mathfrak {a}_{j_0}^{\dag })^{k''_{j_0}-1}h'' \rangle = 0. \qquad \end{aligned}$$

$\square$

Corollary 3.9

We have

$$\begin{aligned} \mathcal {H}_{\textbf{k}}= & {} \overline{\textsf {L}_{\textbf{k}}} = \bigoplus _{\textbf{q}=\varvec{1}}^{\textbf{k}} \overline{\textsf {L}_{[\textbf{q}]}}, \\ \mathcal {H}_{(\textbf{k})}= & {} \overline{\textsf {L}_{[\textbf{k}]}}, \\ \mathcal {H}= & {} \bigoplus _{\textbf{k}\in \mathbb {N}^n}\mathcal {H}_{(\textbf{k})}. \end{aligned}$$

We introduce further the following subspaces of $\mathcal {H}$. For each $j=1,\ldots ,n$ and $\ell \in \mathbb {N}$, we set

$$\begin{aligned} \mathcal {H}_{\ell }^{(j)} = \bigoplus _{\textbf{q}\in \mathbb {N}^n \ \textrm{with} \ q_j=\ell }\mathcal {H}_{(\textbf{q})}. \end{aligned}$$

Note that

$$\begin{aligned} \mathcal {H} = \bigoplus _{\ell \in \mathbb {N}}\mathcal {H}_{\ell }^{(j)} \qquad \textrm{and} \qquad \mathcal {H}_{(\textbf{k})} = \bigcap _{j=1}^n \mathcal {H}_{k_j}^{(j)}. \end{aligned}$$

Proposition 3.10

For each $j=1,\ldots ,n$ and $\ell \in \mathbb {N}$, the operator

$$\begin{aligned} \textstyle {\frac{1}{\sqrt{\ell }}}\,\mathfrak {a}_j^\dag |_{\mathcal {H}_{\ell }^{(j)}}: \mathcal {H}_{\ell }^{(j)} \ \longrightarrow \ \mathcal {H}_{\ell +1}^{(j)} \end{aligned}$$

(3.7)

is an isometric isomorphism, and the operator

$$\begin{aligned} \textstyle {\frac{1}{\sqrt{\ell }}}\,\mathfrak {a}_j|_{\mathcal {H}_{\ell +1}^{(j)}}: \mathcal {H}_{\ell +1}^{(j)} \ \longrightarrow \ \mathcal {H}_{\ell }^{(j)} \end{aligned}$$

(3.8)

is its inverse. Furthermore, $\ker \mathfrak {a}_j = \mathcal {H}_{1}^{(j)}$.

Proof

It is sufficient to check the proposition statements on dense subspaces of the spaces involved, for example, on the set of finite linear combinations of elements of $\textsf {L}_{[\textbf{q}]}$. Take an element

$$\begin{aligned} h = \sum _{m=1}^p (\mathbf {\mathfrak {a}}^{\dag })^{\textbf{q}_m}h_m \in \mathcal {H}_{\ell }^{(j)}, \quad \mathrm {that \ is} \quad (\textbf{q}_m)_j = \ell \quad \mathrm {for \ all} \quad m=1,\ldots , p. \end{aligned}$$

Then, making use of (2.4) and (3.1), we have

$$\begin{aligned} \left\| \textstyle {\frac{1}{\sqrt{\ell }}}\mathfrak {a}_j^{\dag }h\right\| ^2 = \left\langle \textstyle {\frac{1}{\sqrt{\ell }}}\mathfrak {a}_j^{\dag }h, \textstyle {\frac{1}{\sqrt{\ell }}}\mathfrak {a}_j^{\dag }h \right\rangle = \frac{1}{\ell }\left\langle h, \mathfrak {a}_j\mathfrak {a}_j^{\dag }h \right\rangle = \frac{1}{\ell }\left\langle h, \ell h\right\rangle =\left\langle h,h \right\rangle = \Vert h\Vert ^2. \end{aligned}$$

Analogously, taking

$$\begin{aligned} h = \sum _{m=1}^p (\mathbf {\mathfrak {a}}^{\dag })^{\textbf{q}_m}h_m \in \mathcal {H}_{\ell +1}^{(j)}, \quad \mathrm {that \ is} \ \ (\textbf{q}_m)_j = \ell +1 \ \ \mathrm {for \ all} \ \ m=1,\ldots , p. \end{aligned}$$

we have

$$\begin{aligned} \left\| \textstyle {\frac{1}{\sqrt{\ell }}}\mathfrak {a}_jh\right\| ^2 = \left\langle \textstyle {\frac{1}{\sqrt{\ell }}}\mathfrak {a}_jh, \textstyle {\frac{1}{\sqrt{\ell }}}\mathfrak {a}_jh \right\rangle = \frac{1}{\ell }\left\langle h, \mathfrak {a}_j^{\dag }\mathfrak {a}_jh \right\rangle = \frac{1}{\ell }\left\langle h, \ell h\right\rangle =\left\langle h,h \right\rangle = \Vert h\Vert ^2. \end{aligned}$$

The last statement of the proposition is obvious. $\square$

Corollary 3.11

The operator $V_j$ defined on $\mathcal {H} = \bigoplus _{\ell \in \mathbb {N}}\mathcal {H}_{\ell }^{(j)}$ by

$$\begin{aligned} V_j|_{\mathcal {H}_{\ell }^{(j)}} = \textstyle {\frac{1}{\sqrt{\ell }}}\,\mathfrak {a}_j^\dag |_{\mathcal {H}_{\ell }^{(j)}}: \mathcal {H}_{\ell }^{(j)} \ \longrightarrow \ \mathcal {H}_{\ell +1}^{(j)} \end{aligned}$$

on each summand $\mathcal {H}_{\ell }^{(j)}$ is a pure isometry, and his adjoint $V_j^*$ is defined by

$$\begin{aligned} V_j^*|_{\mathcal {H}_{\ell +1}^{(j)}} = \textstyle {\frac{1}{\sqrt{\ell }}}\,\mathfrak {a}_j|_{\mathcal {H}_{\ell +1}^{(j)}}: \mathcal {H}_{\ell +1}^{(j)} \ \longrightarrow \ \mathcal {H}_{\ell }^{(j)} \end{aligned}$$

for $\ell \ge 1$ and $V_j^*|_{\mathcal {H}_{1}^{(j)}} = 0$.

Corollary 3.12

In the case of $\mathfrak {b}_j = \mathfrak {a}_j^\dag$,

$$\begin{aligned} \ker \mathfrak {a}_j^{m} = \left\{ h \in \mathcal {H}: \mathfrak {a}_j^{m} h = 0\right\} = \ker (V_j^*)^m = \bigoplus _{\ell =1}^{m} \mathcal {H}_{\ell }^{(j)}. \end{aligned}$$

Note that, by (3.1) with $\mathfrak {b}_j = \mathfrak {a}_j^{\dag }$, the pure isometries $V_1$,..., $V_n$ are doubly commuting, that is (see e.g., [10]) $V_{j'} V_{j''} = V_{j''} V_{j'}$ and $V_{j'} V_{j''}^* = V_{j''}^* V_{j'}$ for all $j' \ne j''$.

Recall in his connection the following extension of the classical Wold decomposition.

Proposition 3.13

[10, Corollary 3.2] Let $\varvec{V}=(V_1,\ldots ,V_n)$ be a tuple of doubly commuting pure isometries on $\mathcal {H}$. Then

$$\begin{aligned} \mathcal {W} = \bigcap _{j=1}^n \textrm{ran}\,(I-V_jV_j^*) \end{aligned}$$

is a wondering subspace for $\varvec{V}$ and

$$\begin{aligned} \mathcal {H} = \bigoplus _{\textbf{k}\in \mathbb {N}^n}\varvec{V}^{\textbf{k}- \varvec{1}}\mathcal {W}. \end{aligned}$$

The next result states that the three objects of the theorem to follow are tightly connected to each other, any one of them defines another two. As a consequence, each one of them can be used (and will be used) in the characterization of the polyanalytic type spaces.

Theorem 3.14

Let $\mathcal {H}$ be a separable infinite dimensional Hilbert space. Then the following statements are equivalent:

• 1. there is a tuple of doubly commuting pure isometries $\varvec{V}=(V_1,\ldots ,V_n)$ on $\mathcal {H}$;

• 2. the Hilbert space $\mathcal {H}$ admits the orthogonal sum decomposition

  $$\begin{aligned} \mathcal {H} = \bigoplus _{\textbf{k}\in \mathbb {N}^n} \mathcal {H}_{(\textbf{k})}, \end{aligned}$$

  (3.9)

  where all $\mathcal {H}_{(\textbf{k})}$ have the same (finite or infinite) dimension;

• 3. there are two tuples of operators

  $$\begin{aligned} \mathbf {\mathfrak {a}}= (\mathfrak {a}_1, \ \mathfrak {a}_2, \ \ldots , \ \mathfrak {a}_n) \quad \textrm{and} \quad \mathbf {\mathfrak {a}}^{\dag } = (\mathfrak {a}_1^{\dag }, \ \mathfrak {a}_2^{\dag }, \ \ldots , \ \mathfrak {a}_n^{\dag }), \end{aligned}$$

  (3.10)

  that are formally adjoint and act invariantly on a domain $\mathcal {D}$ dense in $\mathcal {H}$, satisfying therein the commutation relations

  $$\begin{aligned} \left[ \mathfrak {a}_j, \mathfrak {a}^{\dag }_{\ell }\right] = \delta _{j,\ell }I, \quad \left[ \mathfrak {a}_j, \mathfrak {a}_{\ell }\right] = 0, \quad \left[ \mathfrak {a}^{\dag }_j, \mathfrak {a}^{\dag }_{\ell }\right] = 0, \quad j,\ell = 1,2,...,n; \nonumber \\ \end{aligned}$$

  (3.11)

  and the set of finite linear combinations of elements from all spaces $\textsf {L}_{[\textbf{k}]}:= (\mathfrak {a}^\dag )^{n-1} \textsf {L}_{[\textbf{1}]}$, with $\textbf{k}\in \mathbb {N}^n$ and where

  $$\begin{aligned} \textsf {L}_{[\textbf{1}]} = \bigcap _{j=1}^n\ker \mathfrak {a}_j = \left\{ h \in \mathcal {D}: \, \mathbf {\mathfrak {a}}h = \left( \mathfrak {a}_1 h, \ldots , \mathfrak {a}_n h \right) = \textbf{0} \right\} , \end{aligned}$$

  is dense in $\mathcal {H}$.

Moreover, the subspaces $\mathcal {H}_{(k)}$ in (3.9) are related to the operators in $\varvec{V}$, $\mathbf {\mathfrak {a}}$ and $\mathbf {\mathfrak {a}}^\dag$ as follows

$$\begin{aligned} \mathcal {H}_{(\varvec{1})} = \mathcal {W} = \overline{\textsf {L}_{[\textbf{1}]}} \quad \textrm{and}\quad \mathcal {H}_{(\textbf{k})} = \varvec{V}^{\textbf{k}-\varvec{1}}(\overline{\textsf {L}_{[\textbf{1}]}}) = (\mathbf {\mathfrak {a}}^\dag )^{\textbf{k}-\varvec{1}}(\overline{\textsf {L}_{[\textbf{1}]}}), \ \ \textrm{for} \ \ \textbf{k}\in \mathbb {N}^n. \nonumber \\ \end{aligned}$$

(3.12)

Proof

$(1) \Rightarrow (2)$ Follows from Proposition 3.13 with $\mathcal {H}_{(\textbf{k})} = \varvec{V}^{\textbf{k}-\varvec{1}}\mathcal {W}$.

$(2) \Rightarrow (3)$ We denote by $\{e_p^{(\textbf{k})}\}_{p \in N}$, with N being the (finite or infinite) index set, the orthonormal basis of the subspace $\mathcal {H}_{(k)}$ of decomposition (3.9). As all spaces $\mathcal {H}_{(\textbf{k})}$ have the same dimension, the index set N does not depend on $\textbf{k}$. Then, on the dense subset $\mathcal {D}$ formed by finite linear combinations of all basis elements $e_p^{(\textbf{k})}$, $p \in N$, $\textbf{k}\in \mathbb {N}^n$, we introduce the operators (3.10) as follows

$$\begin{aligned} a_j:\, e_p^{(\textbf{k})} \longmapsto {\left\{ \begin{array}{ll} \sqrt{k_j-1}e_p^{(\textbf{k}- \varvec{e}_j)}, &{} k_j > 1 \\ 0, &{} k_j =1 \end{array}\right. } \quad \textrm{and} \quad a_j^{\dag }:\, e_p^{(\textbf{k})} \longmapsto \sqrt{k_j}e_p^{(\textbf{k}+ \varvec{e}_j)}. \end{aligned}$$

It is easily seen that so defined operators $\mathfrak {a}_j$ and $a_j^{\dag }$, $j=1,\ldots ,n$, are formally adjoint and satisfy the commutation relations (3.11). Furthermore, $\mathcal {H}_{(\varvec{1})} =\textrm{clos}\left( \displaystyle {\bigcap _{j=1}^n \ker \mathfrak {a}_j}\right)$.

$(3) \Rightarrow (1)$ Follows from Corollary 3.11 and relations (3.11). $\square$

Then, by [10, Theorem 3.3, item (v)], a pair of tuples $\varvec{V'}=(V'_1,\ldots ,V'_n)$ on $\mathcal {H'}$ and $\varvec{V''}=(V''_1,\ldots ,V''_n)$ on $\mathcal {H''}$ of doubly commuting pure isometries are unitarily equivalent, that is there exists a unitary operator $U: \mathcal {H'} \rightarrow \mathcal {H''}$ such that $UV'_j =V''_jU$ for all $j=1,\ldots ,n$, if and only if their wondering spaces have the same dimension, $\dim W' = \dim W''=: \varvec{d}$. As a canonical representative of doubly commuting pure isometries $\varvec{V}=(V_1,\ldots ,V_n)$ with $\dim W=: \varvec{d}$, being finite or infinite, [10, Theorem 3.3, item (v)] suggests the tuple of multiplication operators $\varvec{M}_{z} = (M_{z_1}, \ldots , M_{z_n})$ acting on $H^2(\mathbb {D}^n) \otimes \mathcal {E}$, where $H^2(\mathbb {D}^n)$ is the Hardy space on the poly-disk $\mathbb {D}^n$, and $\mathcal {E}$ is a $\varvec{d}$-dimensional Hilbert space.

Theorem 3.14 implies then that the only initary equivalent of tuples of formally adjoint operators

$$\begin{aligned} \mathbf {\mathfrak {a}}= (\mathfrak {a}_1, \ \mathfrak {a}_2, \ \ldots , \ \mathfrak {a}_n) \quad \textrm{and} \quad \mathbf {\mathfrak {a}}^{\dag } = (\mathfrak {a}_1^{\dag }, \ \mathfrak {a}_2^{\dag }, \ \ldots , \ \mathfrak {a}_n^{\dag }), \end{aligned}$$

satisfying Definition 3.1 of the extended Fock space construction, is

$$\begin{aligned} \varvec{d}:= \dim \textsf {L}_{[\textbf{1}]} =\dim \bigcap _{j=1}^n\ker \mathfrak {a}_j. \end{aligned}$$

In Sect. 4.2, for any given $\varvec{d}$, we propose a canonical representation of the tuples $\mathbf {\mathfrak {a}}$ and $\mathbf {\mathfrak {a}}^{\dag }$, given in terms of the first order differential operators.

3.3 Tensor Product Case

We consider here a special case of the situation considered in the previous subsection, which is important for the forthcoming study of the polyanalytic function spaces.

For each $j=1,\ldots ,n$, we consider a separable Hilbert space $\textsf {H}_j$ and a pure isometry $\textsf {V}_j$, acting on it. We denote by $\textsf {W}_j = \ker \textsf {V}_j^* = \textrm{ran}\,(\textsf {I}-\textsf {V}_j\textsf {V}_j^*)$ its wondering subspace. Then

$$\begin{aligned} \textsf {H}_j = \bigoplus _{\ell = 0}^{\infty }\textsf {V}_j^{\ell }\textsf {W}_j. \end{aligned}$$

(3.13)

This is equivalent (see e.g. [14, Theorem 5.1]) to the existence of two formally adjoint operators $\textsf {a}_j$ and $\textsf {a}_j^{\dag }$, densely defined on the set of finite linear combinations of elements from all $\textsf {V}_j^{\ell }\textsf {W}_j$, and given on the summands of (3.13) by

$$\begin{aligned} \textsf {a}_j|_{\textsf {V}_j^{\ell }\textsf {W}_j}= & {} {\left\{ \begin{array}{ll} \sqrt{\ell -1}\textsf {V}_j^*: \textsf {V}_j^{\ell }\textsf {W}_j \rightarrow \textsf {V}_j^{\ell -1}\textsf {W}_j, &{} \ell > 0 \\ 0, &{} \ell =0 \end{array}\right. } \nonumber \\&\textrm{and}&\nonumber \\ \textsf {a}_j^{\dag }|_{\textsf {V}_j^{\ell }\textsf {W}_j}= & {} \sqrt{\ell }\textsf {V}_j: \textsf {V}_j^{\ell }\textsf {W}_j \rightarrow \textsf {V}_j^{\ell +1}\textsf {W}_j. \end{aligned}$$

(3.14)

It is just a matter of a simple calculation to show that so defined operators $\textsf {a}_j$ and $\textsf {a}_j^{\dag }$ satisfy the relation $[\textsf {a}_j,\textsf {a}_j^{\dag }] = \textsf {I}$.

We introduce now the Hilbert space $\mathcal {H}$ as the (Hilbert space) tensor product of $\textsf {H}_j$, $j =1,\ldots ,n$,

$$\begin{aligned} \mathcal {H} = \bigotimes _{j=1}^n \textsf {H}_j, \end{aligned}$$

and the tuple of operators $\varvec{V}=(V_1,\ldots ,V_n)$, acting therein, where

$$\begin{aligned} V_j = \textsf {I}\otimes \ldots \otimes \underset{j^{th}- place}{\textsf {V}_j}\otimes \ldots \otimes \textsf {I}. \end{aligned}$$

It is easy to figure out that $V_j$, $j =1,\ldots ,n$, are doubly commuting pure isometries, and that $W = \displaystyle {\bigotimes _{j=1}^n\textsf {W}_j}$ is the wondering subspace for $\varvec{V}$. Then

$$\begin{aligned} \mathcal {H} = \bigoplus _{\textbf{k}\in \mathbb {N}^n} \mathcal {H}_{(\textbf{k})}, \end{aligned}$$

with

$$\begin{aligned} \mathcal {H}_{(\textbf{k})} = \varvec{V}^{\textbf{k}- \varvec{1}} W \ = \ \bigotimes _{j=1}^n \textsf {V}_j^{k_j-1}\textsf {W}_j. \end{aligned}$$

The pure isometries $\varvec{V}=(V_1,\ldots ,V_n)$ define in turn two tuples of formally adjoint operators

$$\begin{aligned} \mathbf {\mathfrak {a}}= (\mathfrak {a}_1, \ \mathfrak {a}_2, \ \ldots , \ \mathfrak {a}_n) \qquad \textrm{and} \qquad \mathbf {\mathfrak {a}}^{\dag } = (\mathfrak {a}_1^{\dag }, \ \mathfrak {a}_2^{\dag }, \ \ldots , \ \mathfrak {a}_n^{\dag }), \end{aligned}$$

densely defined on the finite linear combinations of $\mathcal {H}_{(\textbf{k})}$, $\textbf{k}\in \mathbb {N}^n$, by

$$\begin{aligned} \mathfrak {a}_j = \textsf {I}\otimes \ldots \otimes \underset{j^{th}- place}{\textsf {a}_j}\otimes \ldots \otimes \textsf {I} \qquad \textrm{and} \qquad \mathfrak {a}_j^{\dag } = \textsf {I}\otimes \ldots \otimes \underset{j^{th}- place}{\textsf {a}_j^{\dag }}\otimes \ldots \otimes \textsf {I}, \end{aligned}$$

with $\textsf {a}_j$ and $\textsf {a}_j^{\dag }$ defined via $\textsf {V}_j^*$ and $\textsf {V}_j$ by (3.14).

Corollary 3.9, relations (3.12), together with the above considerations lead then to the following corollary, important for further characterisation of polyanalytic function spaces.

Corollary 3.15

Each true-$\textbf{k}$-poly-$\mathbf {\mathfrak {a}}$ subspace of $\mathcal {H} = \displaystyle {\bigotimes _{j=1}^n \textsf {H}_j}$ has the form

$$\begin{aligned} \mathcal {H}_{(\textbf{k})} = \ \bigotimes _{j=1}^n (\textsf {a}_j^{\dag })^{k_j-1}\textsf {W}_j, \end{aligned}$$

being, in fact, the tensor product of the true-$k_j$-poly-$\textsf {a}_j$ subspaces $\textsf {H}_{j,(k_j)}$ of $\textsf {H}_j$.

All true-$\textbf{k}$-poly-$\mathbf {\mathfrak {a}}$ subspaces $\mathcal {H}_{(\textbf{k})}$ of $\mathcal {H}$ are isomorphic, one to each other. The isometric isomorphism between $\mathcal {H}_{(\varvec{1)}}= W$ and the true-$\textbf{k}$-poly-$\mathbf {\mathfrak {a}}$ space $\mathcal {H}_{(\textbf{k})}$ is given by

$$\begin{aligned} \frac{1}{\sqrt{(\textbf{k}-\varvec{1})!}}\, (\mathbf {\mathfrak {a}}^\dag )^{\textbf{k}-\varvec{1}}|_{W}: W \longrightarrow \mathcal {H}_{(\textbf{k})}, \end{aligned}$$

and the operator

$$\begin{aligned} \frac{1}{\sqrt{(\textbf{k}-\varvec{1})!}}\,{\mathbf {\mathfrak {a}}}^{\textbf{k}-\varvec{1}}|_{\mathcal {H}_{(\textbf{k})}}: \mathcal {H}_{(\textbf{k})} \longrightarrow W \end{aligned}$$

gives the inverse isomorphism.

The space $\mathcal {H}$ is recovered as the orthogonal sum of all its true-$\textbf{k}$-poly-$\mathbf {\mathfrak {a}}$ subspaces,

$$\begin{aligned} \mathcal {H} = \bigoplus _{\textbf{k}\in \mathbb {N}^n} \mathcal {H}_{(\textbf{k})} \end{aligned}$$

The $\textbf{k}$-poly-$\mathbf {\mathfrak {a}}$ subspaces of $\mathcal {H}$, i.e., those that consist of all elements $h \in \mathcal {H}$ satisfying $\mathbf {\mathfrak {a}}^{\textbf{k}}h = 0$, are the tensor products of the $k_j$-poly-$\textsf {a}_j$ subspaces $\textsf {H}_{j,k_j}$ of $\textsf {H}_j$,

$$\begin{aligned} \mathcal {H}_{\textbf{k}} = \ \bigotimes _{j=1}^n \textsf {H}_{j,k_j}. \end{aligned}$$

4 Complex Space $\mathbb {C}^n$ Case

We consider here the case of $\mathcal {H} = L_2(\mathbb {C}^n, d\mu _{\varvec{\alpha }})$, with the weighted measure $d\mu _{\varvec{\alpha }}$, $\varvec{\alpha } = (\alpha _1, \ldots , \alpha _n)$, where all $\alpha _j > 0$, given by

$$\begin{aligned} d\mu _{\varvec{\alpha }} = \frac{\alpha _1\cdots \alpha _n}{\pi ^n}e^{-(\alpha _j|z_1|^2 + \cdots + \alpha _n|z_n|^2)}\,v(\varvec{z}), \end{aligned}$$

(4.1)

where $d v(\varvec{z})$ is the standard Lebesgue measure in $\mathbb {C}^n$.

Note that the most commonly used cases are $\varvec{\alpha } = \varvec{1}=(1, \ldots , 1)$ and $\varvec{\alpha } = (\alpha , \ldots , \alpha )$.

Observe now that

$$\begin{aligned} L_2(\mathbb {C}^n, d\mu _{\varvec{\alpha }}) = \bigotimes _{j=1}^n L_2(\mathbb {C}, d\mu _{\alpha _j}), \end{aligned}$$

(4.2)

where

$$\begin{aligned} d\mu _{\alpha _j} = \frac{\alpha _j}{\pi }\, e^{-\alpha _j|z_j|^2}\, dv(z_j). \end{aligned}$$

Then, for each $\textbf{p},\, \textbf{q}\in \mathbb {Z}_+^n$,

$$\begin{aligned} \Vert \overline{\varvec{z}}^{\textbf{p}}\varvec{z}^{\textbf{q}}\Vert _{L_2(\mathbb {C}^n, d\mu _{\varvec{\alpha }})} = \prod _{j=1}^n \Vert \overline{z}_j^{p_j}z_j^{q_j}\Vert _{L_2(\mathbb {C}, d\mu _{\alpha _j})}, \end{aligned}$$

where

$$\begin{aligned}{} & {} \Vert \overline{z}_j^{p_j}z_j^{q_j}\Vert ^2_{L_2(\mathbb {C}, d\mu _{\alpha _j})} = \frac{\alpha _j}{\pi } \int _{\mathbb {C}}|z_j|^{2(p_j+q_j)} e^{-\alpha _j|z_j|^2}\, dv(z_j) \\{} & {} \quad = \ \frac{1}{\alpha ^{p_j+q_j}\pi } \int _{\mathbb {C}}|w_j|^{2(p_j+q_j)} e^{-|w_j|^2}\, dv(w_j) \\{} & {} \quad = \ \frac{1}{\alpha ^{p_j+q_j}}\int _{\mathbb {R}_+} r^{p_j+q_j} e^{-r}dr = \frac{(p_j+q_j)!}{\alpha ^{p_j+q_j}}. \end{aligned}$$

4.1 Poly-Fock Spaces

By (4.2), we are in the situation of Subsection 3.3. We start with the Hilbert space $L_2(\mathbb {C}, d\mu _{\alpha })$ and two operators $\varvec{a}_{\alpha } = \frac{1}{\sqrt{\alpha }}\frac{\mathfrak {d}}{\mathfrak {d} \overline{z}_j}$ and $\varvec{a}_{\alpha }^{\dag } = -\frac{1}{\sqrt{\alpha }}\frac{\mathfrak {d}}{\mathfrak {d} z_j} + \sqrt{\alpha }\,\overline{z}_j$, densely defined on the linear set $\mathcal {D}$ of all polynomials on z and $\overline{z}$, and satisfying the commutation relation $[\varvec{a}_{\alpha }, \varvec{a}_{\alpha }^{\dag }] = \textsf {I}$.

Let as show that the operators $\varvec{a}_{\alpha }$ and $\varvec{a}_{\alpha }^{\dag }$ are formally adjoint on $\mathcal {D}$. Indeed, consider

$$\begin{aligned} \langle \varvec{a}_{\alpha }(\overline{z}_j^p z_j^q), \overline{z}_j^{\ell }z_j^m \rangle= & {} \frac{p}{\sqrt{\alpha }}\langle \overline{z}_j^{p-1} z_j^q, \overline{z}_j^{\ell }z_j^m \rangle = \delta _{q+\ell ,m+p-1} \frac{p}{\sqrt{\alpha }} \Vert z^{q+\ell }\Vert ^2 \\= & {} \delta _{q+\ell ,m+p-1} \frac{p}{\sqrt{\alpha }} \frac{(q+\ell )!}{\alpha ^{q+\ell }}. \end{aligned}$$

From the other hand side,

$$\begin{aligned}{} & {} \langle \overline{z}_j^p z_j^q, \varvec{a}_{\alpha }^{\dag }(\overline{z}_j^{\ell }z_j^m) \rangle = -\frac{m}{\sqrt{\alpha }}\langle \overline{z}_j^p z_j^q, \overline{z}_j^{\ell }z_j^{m-1}\rangle + \sqrt{\alpha }\langle \overline{z}_j^p z_j^q, \overline{z}_j^{\ell +1}z_j^{m}\rangle \\{} & {} \quad = \ \delta _{q+\ell ,m+p-1} \left( -\frac{m}{\sqrt{\alpha }}\Vert z^{q+\ell }\Vert ^2 +\sqrt{\alpha }\Vert z^{q+\ell +1}\Vert ^2 \right) \\{} & {} \quad = \ \delta _{q+\ell ,m+p-1} \left( -\frac{m}{\sqrt{\alpha }}\frac{(q+\ell )!}{\alpha ^{q+\ell }} + \sqrt{\alpha }\frac{(q+\ell +1)!}{\alpha ^{q+\ell +1}}\right) \\{} & {} \quad = \ \delta _{q+\ell ,m+p-1} \frac{(q+\ell )!}{\sqrt{\alpha }\alpha ^{q+\ell }} \left( -m +q+ \ell + 1\right) \\{} & {} \quad = \ \delta _{q+\ell ,m+p-1} \frac{p}{\sqrt{\alpha _j}} \frac{(q+\ell )!}{\alpha ^{q+\ell }}, \end{aligned}$$

and the result follows.

Introduce the two tuples of the formally adjoint operators

$$\begin{aligned} \mathbf {\mathfrak {a}}_{\varvec{\alpha }} = (\mathfrak {a}_1, \ \mathfrak {a}_2, \ \ldots , \ \mathfrak {a}_n) \qquad \textrm{and} \qquad \mathbf {\mathfrak {a}}_{\varvec{\alpha }}^{\dag } = (\mathfrak {a}_1^{\dag }, \ \mathfrak {a}_2^{\dag }, \ \ldots , \ \mathfrak {a}_n^{\dag }), \end{aligned}$$

densely defined on the finite linear combinations of the monomials $m_{\textbf{p},\textbf{q}}= \overline{z}^{\textbf{p}}z^{\textbf{q}}$, $\textbf{p},\,\textbf{q}\in \mathbb {Z}_+^n$, by

$$\begin{aligned} \mathfrak {a}_j = \textsf {I}\otimes \ldots \otimes \underset{j^{th}- place}{\varvec{a}_{\alpha _j}}\otimes \ldots \otimes \textsf {I} \qquad \textrm{and} \qquad \mathfrak {a}_j^{\dag } = \textsf {I}\otimes \ldots \otimes \underset{j^{th}- place}{\varvec{a}_{\alpha _j}^{\dag }}\otimes \ldots \otimes \textsf {I}, \end{aligned}$$

and satisfying the commutation relations (3.11).

The space $\textsf {L}_{[\textbf{1}]}= \bigcap _{j=1}^n\ker \mathfrak {a}_j$ coincides with the set of all analytic polynomials (polynomials on $\varvec{z}^{\textbf{q}}$), while its closure $\overline{\textsf {L}_{[\textbf{1}]}}$ coincides with the classical Fock space $F^2_{\varvec{\alpha }}(\mathbb {C}^n)$, being the closed subspace of $L_2(\mathbb {C}^n, d\mu _{\varvec{\alpha }})$ which consist of functions analytic in $\mathbb {C}^n$.

Observe that

$$\begin{aligned} F^2_{\varvec{\alpha }}(\mathbb {C}^n) = \bigotimes _{j=1}^n F^2_{\alpha _j}(\mathbb {C}). \end{aligned}$$

Corollary 3.15 extends now the results of [13, Section 4] to the weighted Fock space case, recovering, in particular,the results of [13, Section 4] for the unweighted case.

We summarise them in the following proposition.

Proposition 4.1

Each true-$\textbf{k}$-poly-Fock subspace $F^2_{\varvec{\alpha }, (\textbf{k})}(\mathbb {C}^n)$ of $L_2(\mathbb {C}^n, d\mu _{\varvec{\alpha }})$ has the form

$$\begin{aligned} F^2_{\varvec{\alpha }, (\textbf{k})}(\mathbb {C}^n) = \ \bigotimes _{j=1}^n (\varvec{a}_{\alpha _j}^{\dag })^{k_j-1}F^2_{\alpha _j}(\mathbb {C}), \end{aligned}$$

being the tensor product of the true-$k_j$-poly-Fock subspaces $F^2_{\alpha _j, (k_j)}(\mathbb {C})$ of $L_2(\mathbb {C},d\mu _{\alpha _j})$.

All true-$\textbf{k}$-poly-Fock subspaces $F^2_{\varvec{\alpha }, (\textbf{k})}(\mathbb {C}^n)$ of $L_2(\mathbb {C}^n, d\mu _{\varvec{\alpha }})$ are isomorphic, one to each other. The isometric isomorphism between the Fock space $F^2_{\varvec{\alpha }}(\mathbb {C}^n)$ and the true-$\textbf{k}$-poly-Fock space $F^2_{\varvec{\alpha }, (\textbf{k})}(\mathbb {C}^n)$ is given by

$$\begin{aligned} \frac{1}{\sqrt{(\textbf{k}-\varvec{1})!}}\, (\mathbf {\mathfrak {a}}_{\varvec{\alpha }}^\dag )^{\textbf{k}-\varvec{1}}|_{F^2_{\varvec{\alpha }}(\mathbb {C}^n)}: F^2_{\varvec{\alpha }}(\mathbb {C}^n) \longrightarrow F^2_{\varvec{\alpha }, (\textbf{k})}(\mathbb {C}^n), \end{aligned}$$

and the operator

$$\begin{aligned} \frac{1}{\sqrt{(\textbf{k}-\varvec{1})!}}\,{\mathbf {\mathfrak {a}}}_{\varvec{\alpha }}^{\textbf{k}-\varvec{1}}|_{F_{\varvec{\alpha }, (\textbf{k})}^2(\mathbb {C}^n)}: F^2_{\varvec{\alpha }, (\textbf{k})}(\mathbb {C}^n) \longrightarrow F^2_{\varvec{\alpha }}(\mathbb {C}^n) \end{aligned}$$

gives the inverse isomorphism.

The space $L_2(\mathbb {C}^n,d\mu _{\varvec{\alpha }})$ coincides with the orthogonal sum of all its true-$\textbf{k}$-poly-Fock subspaces,

$$\begin{aligned} L_2(\mathbb {C}^n,d\mu _{\varvec{\alpha }}) = \bigoplus _{\textbf{k}\in \mathbb {N}^n} F^2_{\varvec{\alpha }, (\textbf{k})}(\mathbb {C}^n). \end{aligned}$$

The $\textbf{k}$-poly-Fock subspaces of $L_2(\mathbb {C}^n,d\mu _{\varvec{\alpha }})$, i.e., those that consist of all elements $h \in L_2(\mathbb {C}^n,d\mu _{\varvec{\alpha }})$ satisfying $\mathbf {\mathfrak {a}}^{\textbf{k}}h = 0$, are the tensor products of the $k_j$-poly-Fock subspaces $F^2_{\alpha _j, k_j}(\mathbb {C})$ of $L_2(\mathbb {C},d\mu _{\alpha _j})$,

$$\begin{aligned} F^2_{\varvec{\alpha }, \textbf{k}}(\mathbb {C}^n) = \ \bigotimes _{j=1}^n F^2_{\alpha _j, k_j}(\mathbb {C}). \end{aligned}$$

Corollary 4.2

Each function $\psi (z,\overline{z})$ from the true-$\textbf{k}$-Fock space $F_{\varvec{\alpha ,(\textbf{k})}}^2(\mathbb {C}^n)$ is uniquely defined by a function $\varphi (\varvec{z}) \in F^2_{\varvec{\alpha }}(\mathbb {C}^n)$ and has the form

$$\begin{aligned} \psi (\varvec{z})=\psi (\varvec{z},\overline{\varvec{z}})=\sum _{\textbf{p}=0}^{\textbf{k}-\varvec{1}} (-1)^{|\textbf{p}|} \frac{\sqrt{\varvec{\alpha }^{\textbf{k}-\varvec{1}}(\textbf{k}-\varvec{1})!}}{\varvec{\alpha }^{\textbf{p}} {\textbf{p}}!\,(\textbf{k}-\varvec{1}-\textbf{p})!}\, \overline{z}^{\,\textbf{k}-\varvec{1}-\textbf{p}}\,\partial ^{\textbf{p}}\varphi (\varvec{z}), \end{aligned}$$

where

$$\begin{aligned} \partial ^{\textbf{p}}\varphi (z) = \frac{\partial ^{|\textbf{p}|} \varphi (z)}{\partial z_1^{p_1} \ldots \partial z_n^{p_n}}. \end{aligned}$$

and

$$\begin{aligned} \Vert \psi \Vert _{F_{\varvec{\alpha },(\textbf{k})}^2(\mathbb {C}^n)} = \Vert \varphi \Vert _{F^2_{\varvec{\alpha }}(\mathbb {C}^n)}. \end{aligned}$$

The true-$\textbf{k}$-Fock space $F_{\varvec{\alpha ,(\textbf{k})}}^2(\mathbb {C}^n)$ admits an orthonormal basis given by the following polynomials

$$\begin{aligned} e^{(\textbf{k})}_{\textbf{q}}{} & {} = \prod _{j=1}^n \sqrt{\alpha _j^{k_j+q_j-1}(k_j-1)!q_j!} \\{} & {} \quad \times \ \frac{(-1)^{p_j}}{\alpha _j^{p_j}p_j!(k_j-1-p_j)!(q_j-p_j)!}\, \overline{z}_j^{k_j-1-p_j}z_j^{q_j-p_j} \\{} & {} = \ \sqrt{\varvec{\alpha }^{\textbf{k}+\textbf{q}-\varvec{1}}(\textbf{k}-\varvec{1})!\textbf{q}!} \\{} & {} \quad \times \ \sum _{\textbf{p}=\varvec{0}}^{\min \{\textbf{k}-\varvec{1},\textbf{q}\}} \frac{(-1)^{|\textbf{p}|}}{\varvec{\alpha }^{\textbf{p}}\textbf{p}!(\textbf{k}-\varvec{1}-\textbf{p})!(\textbf{q}-\textbf{p})!}\, \overline{\varvec{z}}^{\textbf{k}-\varvec{1}-\textbf{p}} \varvec{z}^{\textbf{q}- \textbf{p}}, \end{aligned}$$

where $\textbf{q}\in \mathbb {Z}_+^n$.

The Hilbert space $L_2(\mathbb {C}^n, d\mu _{\varvec{\alpha }})$ admits an orthonormal basis formed by the union of the above orthonormal bases of the true-$\textbf{k}$-Fock spaces $F_{\varvec{\alpha },(\textbf{k})}^2(\mathbb {C}^n)$, $\textbf{k}\in \mathbb {N}^n$.

Note that the polynomials $e^{(\textbf{k})}_{\textbf{q}}$ can be considered as an extension to the case of several variables of the normalized complex Hermite polynomials for $L_2(\mathbb {C}, d\mu _{\alpha _j})$ (the factors in the above formula) introduced and studied in the case of $\alpha _j=1$ e.g. in [8, §12], [9, Section 2], and [11, Section 7].

4.2 Canonical Representation of Tuples $\mathbf {\mathfrak {a}}$ and $\mathbf {\mathfrak {a}}^{\dag }$

We suggest here a canonical representation of the tuples $\mathbf {\mathfrak {a}}$ and $\mathbf {\mathfrak {a}}^{\dag }$ for any (finite or infinite) dimension $\varvec{d}$ of the $\bigcap _{j=1}^n \ker \mathfrak {a}_j$, given in terms of the first order differential operators.

Let, first, $\varvec{d}= \infty$. We set then, $\mathcal {H} = L_2(\mathbb {C}^n, d\mu _{\varvec{1}})$,

$$\begin{aligned} \mathfrak {a}_j = \frac{\partial }{\partial \overline{z}_j}, \qquad \mathfrak {a}_j^{\dag } = - \frac{\partial }{\partial z_j} + \overline{z}_j, \qquad j = 1,..., n, \end{aligned}$$

and $\bigcap _{j=1}^n \ker \mathfrak {a}_j = F^2(\mathbb {C}^n)$ is the classical Fock space of analytic in $\mathbb {C}^n$ functions.

For a finite $\varvec{d}$, we fix any $\varvec{d}$-dimensional subspace $F_{\varvec{d}}$ of the Fock space $F^2(\mathbb {C}^n)$, consider the same

$$\begin{aligned} \mathfrak {a}_j = \frac{\partial }{\partial \overline{z}_j}, \qquad \mathfrak {a}_j^{\dag } = - \frac{\partial }{\partial z_j} + \overline{z}_j, \qquad j = 1,..., n, \end{aligned}$$

and the corresponding Hilbert space

$$\begin{aligned} \mathcal {H}_{\varvec{d}} = \bigoplus _{\textbf{k}\in \mathbb {N}^n} \frac{1}{\sqrt{\textbf{k}!}} (\mathbf {\mathfrak {a}}^{\dag })^{\textbf{k}-\varvec{1}} F_{\varvec{d}}. \end{aligned}$$

4.3 Poly-Vekua Spaces

We start with one complex variable case of item (3) in Example 2.2.

On the set of smooth functions in a domain $D \subset \mathbb {C}$ introduce the operators

$$\begin{aligned} \mathfrak {a}= \frac{\partial }{\partial \overline{z}} - \omega z \qquad \textsf {and} \qquad \mathfrak {b}= \overline{z}. \end{aligned}$$

The functions $\varphi$, satisfying the equation $\mathfrak {a}\varphi =0$, are the particular case of the so-called generalized analytic functions introduced and deeply studied by I. Vekua, see e.g. [16]. Generically they are the functions that satisfy the equation

$$\begin{aligned} \frac{\partial \varphi }{\partial \overline{z}} + A(z)\varphi + B(z)\overline{\varphi } = 0. \end{aligned}$$

In our case $A(z) = -\omega z$, $B(z) = 0$.

The set $\mathfrak {A}(-\omega z,0;\mathbb {C})$ (in the Vekua notation) of all generalized analytic functions f, satisfying $\mathfrak {a}f = 0$, as it easily seen, consists of all functions of the form

$$\begin{aligned} f(z)= e^{\omega z\overline{z}}\varphi (z), \quad \mathrm {with \ analytic} \ \ \varphi (z). \end{aligned}$$

(4.3)

As $[\mathfrak {a},\mathfrak {b}] = I$, Lemma 2.1 implies that k-poly-generalized analytic functions (those satisfying $\mathfrak {a}^k g = 0$) are of the form

$$\begin{aligned} g = \sum _{m=0}^{k-1} \overline{z}^m f_m, \quad \textrm{with} \quad f_m \in \mathfrak {A}(-\omega z,0;\mathbb {C}), \end{aligned}$$

or

$$\begin{aligned} g = e^{\omega z\overline{z}} \varphi _k(z), \quad \textrm{with} \ \ k\text {-poly-analytic} \ \ \varphi _k(z). \end{aligned}$$

Let now $\mathcal {H} = L_2(\mathbb {C}, d\mu )$, with $d\mu = \frac{1}{\pi }e^{-|z|^2}dv(z)$. We define the Vekua space ($\omega$-Vekua space) by

$$\begin{aligned} \mathcal {V}_{\omega }^2(\mathbb {C}) = L_{2}(\mathbb {C},d\mu ) \cap \mathfrak {A}(-\omega z,0;\mathbb {C}). \end{aligned}$$

Taking f of the form (4.3), calculate

$$\begin{aligned} \Vert f\Vert ^2_{L_{2}(\mathbb {C},d\mu )} = \Vert e^{\omega z\overline{z}}\varphi (z)\Vert ^2_{L_{2}(\mathbb {C},d\mu )} = \frac{1}{\pi }\int _{\mathbb {C}} |\varphi (z)|^2 e^{-(1-2\omega )|z|^2}dv(z). \end{aligned}$$

That is, $\mathcal {V}_{\omega }^2(\mathbb {C})$ is non-trivial if and only if $\omega < \frac{1}{2}$, and $f \in \mathcal {V}_{\omega }^2(\mathbb {C})$ if and only if $\varphi \in F^2(\mathbb {C},d\mu _{1-2\omega })$.

Introduce the operator $U_{\omega }: L_2(\mathbb {C}, d\mu ) \rightarrow L_2(\mathbb {C}, d\mu _{1-2\omega })$ by

$$\begin{aligned} U_{\omega }: \, f \, \longmapsto \, \textstyle {\frac{1}{\sqrt{1-2\omega }}}e^{-\omega |z|^2}f. \end{aligned}$$

An easy calculation shows that the operator $U_{\omega }$ is unitary, and its adjoint and inverse $U_{\omega }^* = U_{\omega }^{-1}: L_2(\mathbb {C}, d\mu _{1-2\omega }) \rightarrow L_2(\mathbb {C}, d\mu )$ is given by

$$\begin{aligned} U_{\omega }^{-1}: \, h \, \longmapsto \, \sqrt{1-2\omega }\,e^{\omega |z|^2}h. \end{aligned}$$

Introduce also the k-poly-Vekua space $\mathcal {V}_{\omega ,k}^2(\mathbb {C})$ as the set of all functions f in $L_2(\mathbb {C},d\mu )$ that satisfy the equation $\mathfrak {a}^k f = 0$.

Lemma 4.3

For each $k \in \mathbb {N}$, the operator $U_{\omega }^{-1}$ gives an isometric isomorphism between the k-polyanalytic type spaces $F^2_{1-2\omega ,k}$ and $\mathcal {V}_{\omega ,k}^2$.

A function $f \in L_2(\mathbb {C}, d\mu )$ is k-poly-Vekua if and only if

$f = \sqrt{1-2\omega }\,e^{\omega |z|^2}h$ with $h \in F^2_{1-2\omega ,k}$, and $\Vert f\Vert _{L_2(\mathbb {C}, d\mu )} = \Vert h\Vert _{L_2(\mathbb {C}, d\mu _{1-2\omega })}$.

Proof

For each $f = U_{\omega }^{-1}h \in L_2(\mathbb {C}, d\mu )$, we have

$$\begin{aligned} \mathfrak {a}^k f = \left( \frac{\partial }{\partial \overline{z}} - \omega z \right) ^k \sqrt{1-2\omega }\,e^{\omega |z|^2}h =\sqrt{1-2\omega }\,e^{\omega |z|^2} \frac{\partial ^k h}{\partial \overline{z}^k} = U_{\omega }^{-1}\frac{\partial ^k (U_{\omega }f)}{\partial \overline{z}^k}. \end{aligned}$$

That is, $\mathfrak {a}^k f = 0$ if and only if $\frac{\partial ^k (U_{\omega }f)}{\partial \overline{z}^k} = \frac{\partial ^k h}{\partial \overline{z}^k} = 0$. And the result follows. $\square$

Introduce also the true-k-poly-Vekua space $\mathcal {V}_{\omega ,(k)}^2(\mathbb {C})$ by

$$\begin{aligned} \mathcal {V}_{\omega ,(k)}^2(\mathbb {C}) =\mathcal {V}_{\omega ,k}^2(\mathbb {C}) \ominus \mathcal {V}_{\omega ,k-1}^2(\mathbb {C}), \qquad \textrm{with} \qquad \mathcal {V}_{\omega ,(1)}^2(\mathbb {C}):=\mathcal {V}_{\omega }^2(\mathbb {C}). \end{aligned}$$

Corollary 4.4

We have the following orthogonal sum decomposition of $L_2(\mathbb {C}, d\mu )$ in terms of true-poly-Vekua spaces

$$\begin{aligned} L_2(\mathbb {C}, d\mu ) = \bigoplus _{k=1}^{\infty }\mathcal {V}_{\omega ,(k)}^2(\mathbb {C}). \end{aligned}$$

Proof

Follows from Proposition 4.1 with $n=1$ and $\alpha =1-2\omega$, and Lemma 4.3. $\square$

Recall that the operators

$$\begin{aligned} \textbf{a}_{1-2\omega } = \frac{1}{\sqrt{1-2\omega }}\frac{\partial }{\partial \overline{z}} \qquad \textrm{and} \qquad \textbf{a}_{1-2\omega }^\dag = - \frac{1}{\sqrt{1-2\omega }}\frac{\partial }{\partial z} + \sqrt{1-2\omega }\,\overline{z} \end{aligned}$$

are formally adjoint in $L_2(\mathbb {C}, d\mu _{1-2\omega })$, being defined on the span of all true-poly-Fock spaces $F_{1-2\omega ,(k)}^2(\mathbb {C})$.

Corollary 4.5

The operators

$$\begin{aligned} \mathfrak {a}_{\omega }= & {} U_{\omega }^{-1}\textbf{a}_{1-2\omega }U_{\omega } = \textstyle {\frac{1}{\sqrt{1-2\omega }}\left( \frac{\partial }{\partial \overline{z}} - \omega \,z\right) }, \\ \mathfrak {a}_{\omega }^\dag= & {} U_{\omega }^{-1}\textbf{a}_{1-2\omega }^\dag U_{\omega } = \textstyle {\frac{1}{\sqrt{1-2\omega }}\left( -\frac{\partial }{\partial z} +(1-\omega )\overline{z}\right) } \end{aligned}$$

are formally adjoint in $L_2(\mathbb {C}, d\mu )$, being defined on the span of true-poly-Vekua spaces $\mathcal {V}_{\omega ,(k)}^2(\mathbb {C})$, and satisfy therein the commutation relation

$[\mathfrak {a}_{\omega }, \mathfrak {a}_{\omega }^{\dag }]=I$.

Each function $\psi (z,\overline{z})$ from the true-k-poly-Vekua space $\mathcal {V}_{\omega ,(k)}^2(\mathbb {C})$ is uniquely defined by a function $\varphi (z) \in \mathcal {V}_{\omega }^2(\mathbb {C})$ and has the form

$$\begin{aligned} \psi (z)=\psi (z,\overline{z}) = (\mathfrak {a}_{\omega }^\dag )^{n-1}\varphi (z) \qquad \textrm{and} \qquad \Vert \psi \Vert _{\mathcal {V}_{\omega ,(k)}^2(\mathbb {C})} = \Vert \varphi \Vert _{\mathcal {V}_{\omega }^2(\mathbb {C})}. \end{aligned}$$

Return now to the several variables case. The Hilbert space $L_2(\mathbb {C}^n, d\mu (\varvec{z}))$ splits into the tensor product of n copies of the spaces over $\mathbb {C}$,

$$\begin{aligned} L_2(\mathbb {C}^n, d\mu (\varvec{z})) = \bigotimes _{j=1}^n L_2(\mathbb {C}, d\mu (z_j)) \end{aligned}$$

Given a tuple $\varvec{\omega }=(\omega _1,\ldots , \omega _n)$ with $\omega _j < \frac{1}{2}$. for all $j=1,\ldots ,n$, we introduce formally adjoint operators

$$\begin{aligned} \textbf{a}_{\omega _j}= & {} \frac{1}{\sqrt{1-2\omega _j}}\left( \frac{\mathfrak {d} }{\mathfrak {d} \overline{z}_j} - \omega _j\,z_j\right) \\&\textrm{and}&\\ \textbf{a}_{\omega _j}^\dag= & {} \frac{1}{\sqrt{1-2\omega _j}}\left( -\frac{\mathfrak {d} }{\mathfrak {d} z_j} +(1-\omega _j)\overline{z}_j\right) , \end{aligned}$$

densely defined on the span of true-$k_j$-poly-Vekua subspaces of $L_2(\mathbb {C}, d\mu (z_j))$.

Define then

$$\begin{aligned} \mathfrak {a}_{\omega _j} = \textsf {I}\otimes \ldots \otimes \underset{j^{th}- place}{\textbf{a}_{\omega _j}}\otimes \ldots \otimes \textsf {I} \qquad \textrm{and} \qquad \mathfrak {a}_{\omega _j}^{\dag } = \textsf {I}\otimes \ldots \otimes \underset{j^{th}- place}{\textbf{a}_{\omega _j}^{\dag }}\otimes \ldots \otimes \textsf {I}, \end{aligned}$$

and two tuples of operators

$$\begin{aligned} \mathbf {\mathfrak {a}}_{\varvec{\omega }} = (\mathfrak {a}_{\omega _1}, \ \mathfrak {a}_{\omega _2}, \ \ldots , \ \mathfrak {a}_{\omega _n}) \qquad \textrm{and} \qquad \mathbf {\mathfrak {a}}_{\varvec{\omega }}^{\dag } = (\mathfrak {a}_{\omega _1}^{\dag }, \ \mathfrak {a}_{\omega _2}^{\dag }, \ \ldots , \ \mathfrak {a}_{\omega _n}^{\dag }), \end{aligned}$$

which satisfy the commutation relations (3.11).

Observe that the Vekua space $\mathcal {V}_{\varvec{\omega }}^2(\mathbb {C}^n)$, which consists of all function $f \in L_2(\mathbb {C}^n, d\mu (z))$ that satisfy $\mathbf {\mathfrak {a}}_{\varvec{\omega }}f = 0$, is nothing but the tensor product of the Vekua subspaces $\mathcal {V}_{\omega _j}^2(\mathbb {C})$ of the corresponding $L_2(\mathbb {C}, d\mu (z_j))$.

$$\begin{aligned} \mathcal {V}_{\varvec{\omega }}^2(\mathbb {C}^n) = \bigotimes _{j=1}^n \mathcal {V}_{\omega _j}^2(\mathbb {C}). \end{aligned}$$

Corrolary 3.15 implies then

Proposition 4.6

Each true-$\textbf{k}$-poly-Vekua subspace $\mathcal {V}^2_{\varvec{\omega },(\varvec{k})}(\mathbb {C}^n)$ of $L_2(\mathbb {C}^n, d\mu )$ has the form

$$\begin{aligned} \mathcal {V}^2_{\varvec{\omega },(\varvec{k})}(\mathbb {C}^n) = \ \bigotimes _{j=1}^n (\textsf {a}_{\omega _j}^{\dag })^{k_j-1}\mathcal {V}_{\omega _j,(k_j)}^2(\mathbb {C}), \end{aligned}$$

being the tensor product of the true-$k_j$-poly-Vekua subspaces $\mathcal {V}_{\omega _j,(k_j)}^2(\mathbb {C})$ of $L_2(\mathbb {C},d\mu (z_j))(\mathbb {C})$.

All true-$\textbf{k}$-poly-Vekua subspaces $\mathcal {V}^2_{\varvec{\omega },(\varvec{k})}(\mathbb {C}^n)$ of $L_2(\mathbb {C}^n, d\mu )$ are isomorphic, one to each other. The isometric isomorphism between the Vekua space $\mathcal {V}_{\varvec{\omega }}^2(\mathbb {C}^n)$ and the true-$\textbf{k}$-poly-Vekua space $\mathcal {V}^2_{\varvec{\omega },(\varvec{k})}(\mathbb {C}^n)$ is given by

$$\begin{aligned} \frac{1}{\sqrt{(\textbf{k}-\varvec{1})!}}\, (\mathbf {\mathfrak {a}}_{\varvec{\omega }}^\dag )^{\textbf{k}-\varvec{1}}|_{\mathcal {V}_{\varvec{\omega }}^2(\mathbb {C}^n)}: \, \mathcal {V}_{\varvec{\omega }}^2(\mathbb {C}^n) \longrightarrow \mathcal {V}^2_{\varvec{\omega },(\varvec{k})}(\mathbb {C}^n), \end{aligned}$$

and the operator

$$\begin{aligned} \frac{1}{\sqrt{(\textbf{k}-\varvec{1})!}}\,{\mathbf {\mathfrak {a}}_{\varvec{\omega }}^{\textbf{k}-\varvec{1}}|_{\mathcal {V}^2_{\varvec{\omega },(\varvec{k})}(\mathbb {C}^n)}:\, \mathcal {V}^2_{\varvec{\omega }},(\varvec{k})}(\mathbb {C}^n) \longrightarrow \mathcal {V}_{\varvec{\omega }}^2(\mathbb {C}^n) \end{aligned}$$

gives the inverse isomorphism.

The space $L_2(\mathbb {C}^n,d\mu )$ splits into the orthogonal sum of all its true-$\textbf{k}$-poly-Vekua subspaces,

$$\begin{aligned} L_2(\mathbb {C}^n,d\mu ) = \bigoplus _{\textbf{k}\in \mathbb {N}^n} \mathcal {V}^2_{\varvec{\omega },(\varvec{k})}(\mathbb {C}^n). \end{aligned}$$

The $\textbf{k}$-poly-Vekua subspaces of $L_2(\mathbb {C}^n,d\mu )$, i.e., those that consist of all elements $f \in L_2(\mathbb {C}^n,d\mu )$ satisfying $\mathbf {\mathfrak {a}}_{\varvec{\omega }}^{\textbf{k}}f = 0$, are the tensor products of the $k_j$-poly-Vekua subspaces $\mathcal {V}^2_{\omega _j,k_j}(\mathbb {C})$ of $L_2(\mathbb {C},d\mu (z_j)))$,

$$\begin{aligned} \mathcal {V}^2_{\varvec{\omega },\varvec{k}}(\mathbb {C}^n) = \ \bigotimes _{j=1}^n \mathcal {V}^2_{\omega _j,k_j}(\mathbb {C}). \end{aligned}$$

The next a bit long remark is devoted to a comparison and analysis of two considered in this section realizations in $\mathbb {C}^n$ of the extended Fock constructions leading to the poly-Fock and poly-Vekua spaces.

Remark 4.7

There are two families of objects that depend on n-dimensional parameter tuples: the family of weighted spaces $L_2(\mathbb {C}^n,d\mu _{\varvec{\alpha }})$, where the weighted measure $d\mu _{\varvec{\alpha }}$ (4.1) is defined via the tuple $\varvec{\alpha } =(\alpha _1,\ldots ,\alpha _n)$, all $\alpha _j > 0$, and the family of generalised analytic (and poly-analytic) functions in the Vekua sense, defined via the tuple $\varvec{\omega }= (\omega _1,\ldots ,\omega _n)$, all $\omega _j < \frac{1}{2}$.

For $\varvec{\alpha } = \varvec{1}$ we have the standard unweighted $L_2(\mathbb {C}^n,d\mu )$ space with the Gaussian measure, and for $\varvec{\omega }=\varvec{0}$ the corresponding class of generalised analytic functions coincides with the class of the classical analytic functions of n complex variables.

Further, in the poly-Fock related case, all spaces $L_2(\mathbb {C}^n,d\mu _{\varvec{\alpha }})$ (parameter $\varvec{\alpha }$ varies) split into the orthogonal sum of true-polyanalytic spaces (parameter $\varvec{\omega }=\varvec{0}$ is fixed). While the poly-Vekua related case is in a sense opposite: we are dealing with the single space $L_2(\mathbb {C}^n,d\mu )$ (parameter $\varvec{\alpha }= \varvec{1}$ is fixed), which admits a family of different representations as the orthogonal sum decomposition of differently defined true generalised polyanalytic functions (parameter $\varvec{\omega }$ varies).

At the same time these two cases are in a sense dual. Namely, for each $L_2(\mathbb {C}^n,d\mu _{\varvec{\alpha }})$ there is a unitary operator that maps the orthogonal sum decomposition of $L_2(\mathbb {C}^n,d\mu _{\varvec{\alpha }})$ into true-poly-Fock spaces onto the orthogonal sum decomposition of $L_2(\mathbb {C}^n,d\mu )$ into true-poly-Vekua spaces with $\varvec{\omega } = \frac{1}{2}(\varvec{1}-\varvec{\alpha })$. And visa versa, for each orthogonal sum decomposition of $L_2(\mathbb {C}^n,d\mu )$ into true-poly-Vekua spaces defined by $\varvec{\omega }$, there is a unitary operator (inverse to the above) that maps this decomposition of $L_2(\mathbb {C}^n,d\mu )$ onto the orthogonal sum decomposition of $L_2(\mathbb {C}^n,d\mu _{\varvec{\alpha }})$, with $\varvec{\alpha }= \varvec{1} - 2\varvec{\omega }$, into true-poly-Fock spaces.

Note that, for $\varvec{\alpha } = \varvec{1}$, or equivalently for $\varvec{\omega }=\varvec{0}$, these two cases reduce to a single one: the orthogonal sum decomposition of $L_2(\mathbb {C}^n,d\mu )$ into true-poly-Fock spaces.

5 Unit Ball $\mathbb {B}^n$ Case

We consider here the case of $\mathcal {H} = L_2(\mathbb {B}^n, d\nu _{\lambda })$, with the weighted measure $dv_{\lambda }$, $\lambda > -1$, given by

$$\begin{aligned} dv_{\lambda }(\varvec{z})=c_{\lambda }^{(n)} (1-|\varvec{z}|^2)^{\lambda } dv(\varvec{z}), \qquad \text { where} \qquad c_{\lambda }^{(n)}:= \frac{\Gamma (n+\lambda +1)}{\pi ^n \Gamma (\lambda +1)}. \end{aligned}$$

We mention that, for all $\textbf{p},\, \textbf{q}\in \mathbb {Z}_+^n$, we have that $\Vert \overline{\varvec{z}}^{\textbf{p}}\varvec{z}^{\textbf{q}}\Vert ^2 = \Vert \varvec{z}^{\textbf{p}+ \textbf{q}}\Vert ^2$ in $L_2(\mathbb {B}^n, d\nu _{ \lambda })$ and is given by

$$\begin{aligned}{} & {} \Vert \overline{\varvec{z}}^{\textbf{p}}\varvec{z}^{\textbf{q}}\Vert ^2 = c_{\lambda }^{(n)} \int _{\mathbb {B}^n}\overline{\varvec{z}}^{\textbf{p}}\varvec{z}^{\textbf{q}} \overline{\overline{\varvec{z}}^{\textbf{p}}\varvec{z}^{\textbf{q}}}(1-|\varvec{z}|^2)^{\lambda }dv(\varvec{z}) \\{} & {} \quad = \ c_{\lambda }^{(n)} \int _{\mathbb {B}^n} |\varvec{z}|^{2(\textbf{p}+ \lambda )}(1-|\varvec{z}|^2)^{\lambda }dv(\varvec{z}) \\{} & {} \quad = \ \frac{2^n\Gamma (n+\lambda +1)}{\Gamma (\lambda +1)} \int _{\tau (\mathbb {B}^n)} (1-|z|^2)^{\lambda } \prod _{j=1}^n r_j^{2(p_j+q_j)}r_jdr_j \\{} & {} \quad = \ \frac{\Gamma (n+\lambda +1)}{\Gamma (\lambda +1)} \int _{\Delta _n}(1-(s_1+\ldots +s_n))^{\lambda }\prod _{j=1}^n s_j^{p_j+q_j}ds_j \\{} & {} \quad = \ \frac{\Gamma (n+\lambda +1)}{\Gamma (\lambda +1)} \frac{\prod _{j=1}^n \Gamma (p_j+q_j+1)\Gamma (\lambda +1)}{\Gamma (|\textbf{p}+\textbf{q}|+\lambda + n+1)} \\{} & {} \quad = \ \frac{\Gamma (n+\lambda +1)\prod _{j=1}^n \Gamma (p_j+q_j+1)}{\Gamma (n+|\textbf{p}+\textbf{q}|+\lambda + 1)} \\{} & {} \quad = \ \frac{\Gamma (n+\lambda +1)(\textbf{p}+\textbf{q})!}{\Gamma (n+|\textbf{p}+\textbf{q}|+\lambda + 1)}:= (d^{[\textbf{p}+\varvec{1}]}_{\textbf{q}})^{-2}. \end{aligned}$$

In the above calculation $\tau (\mathbb {B}^n)$ is the base of the unit ball $\mathbb {B}^n$ considered as the Reindhard domain, $\Delta _n$ is the standard n-dimensional simplex, $s_j=r_j^2$, $r_j=|z_j|$, and we use the formula for n-dimensional Beta function [1].

5.1 Poly-Bergman Spaces

As was already mentioned, the operators $\mathfrak {a}_j = \frac{\partial }{\partial \overline{z}_{j}}$ and $\mathfrak {b}_j = \overline{z}_j$, $j=1,\ldots ,n$, act invariantly on the dense in $L_2(\mathbb {\mathbb {B}}^n, d\nu _{\lambda })$ subspace $\mathcal {D} = \mathcal {D}_0$ which consists of finite linear combinations of the monomials $m_{\textbf{p},\textbf{q}}= \overline{\varvec{z}}^{\textbf{p}}\varvec{z}^{\textbf{q}}$, $\textbf{p},\,\textbf{q}\in \mathbb {Z}_+^n$, and $\textsf {L}_{[\textbf{1}]}= \bigcap _{j=1}^n\ker \mathfrak {a}_j$ coincides with the set of all analytic polynomials (polynomials on $\varvec{z}^{\textbf{p}}$).

The closure $\overline{\textsf {L}_{[\textbf{1}]}}$ coincides obviously with the classical Bergman space $\mathcal {A}^2_{\lambda }(\mathbb {B}^n)$, being the closed subspace of $L_2(\mathbb {B}^n, d\nu _{\lambda })$ which consist of functions analytic in $\mathbb {B}^n$, and whose orthonormal basis formed by the elements

$$\begin{aligned} e^{[\textbf{1}]}_{\textbf{q}} = d^{[\textbf{1}]}_{\textbf{q}}\varvec{z}^{\textbf{q}}, \qquad \textbf{q}= (q_1,\ldots , q_n) \in \mathbb {Z}_+^n. \end{aligned}$$

For the consequent spaces $\textsf {L}_{[\textbf{k}]} = \mathbf {\mathfrak {b}}^{\textbf{k}-\textbf{1}}\textsf {L}_{[\textbf{1}]}$, with $\textbf{k}\in \mathbb {N}^n$, we have

$$\begin{aligned} \overline{\textsf {L}_{[\textbf{k}]}} = \left\{ \overline{\varvec{z}}^{\textbf{k}-\textbf{1}} f(z): \ f \in \mathcal {A}^2_{\lambda }(\mathbb {B}^n)\right\} , \end{aligned}$$

and their bases formed by the elements

$$\begin{aligned} e^{[\mathbf {\textbf{k}}]}_{\textbf{q}} = d^{[\mathbf {\textbf{k}}]}_{\textbf{q}}\overline{\varvec{z}}^{\textbf{k}-\textbf{1}}\varvec{z}^{\textbf{q}}, \qquad \textbf{q}= (q_1,\ldots , q_n) \in \mathbb {Z}_+^n. \end{aligned}$$

In our unit ball setting, the space

$$\begin{aligned} \mathcal {H}_{\textbf{k}} = \textrm{clos}\,\left( \mathop {\dot{\sum }}\limits _{\textbf{p}= \varvec{1}}^{\textbf{k}} \textsf {L}_{[\textbf{p}]}\right) \end{aligned}$$

is called $\textbf{k}$-poly-Bergman space, and is denoted by $\mathcal {A}^2_{\lambda , \textbf{k}}(\mathbb {B}^n)$. The space

$$\begin{aligned} \mathcal {H}_{(\textbf{k})} = \mathcal {H}_{\textbf{k}} \cap \left( \mathop {\dot{\sum }}\limits _{j=1}^n \mathcal {H}_{\textbf{k}-\textbf{e}_j}\right) ^{\perp } = \mathcal {A}^2_{\lambda , \textbf{k}}(\mathbb {B}^n) \cap \left( \mathop {\dot{\sum }}\limits _{j=1}^n \mathcal {A}^2_{\lambda , \textbf{k}-\textbf{e}_j}(\mathbb {B}^n)\right) ^{\perp } \end{aligned}$$

is denoted by $\mathcal {A}^2_{\lambda , (\textbf{k})}(\mathbb {B}^n)$ and is called true-$\textbf{k}$-poly-Bergman space, with the convention that

$$\begin{aligned} \mathcal {A}^2_{\lambda }(\mathbb {B}^n) = \mathcal {A}^2_{\lambda ,\varvec{1}}(\mathbb {B}^n) =\mathcal {A}^2_{\lambda ,(\varvec{1})}(\mathbb {B}^n). \end{aligned}$$

In what follows we will use the basis elements of $\mathcal {A}^2_{\lambda , (\varvec{1}+\varvec{e}_j)}(\mathbb {B}^n)= \mathcal {A}^2_{\lambda , \varvec{1}+\varvec{e}_j}(\mathbb {B}^n) \cap \mathcal {A}^2_{\lambda }(\mathbb {B}^n)^{\perp }$.

Direct calculation shows that the basis is formed by the elements

$$\begin{aligned} e^{(\varvec{1}+\varvec{e}_j)}_{\textbf{q}} = d^{(\varvec{1}+\varvec{e}_j)}_{\textbf{q}} \left( \overline{z}_j \varvec{z}^{\textbf{q}} - \frac{q_j}{n+|\textbf{q}|+\lambda }\varvec{z}^{\varvec{q} -\varvec{e}_j}\right) ,\qquad \textbf{q}= (q_1,\ldots , q_n) \in \mathbb {Z}_+^n, \end{aligned}$$

where

$$\begin{aligned} d^{(\varvec{1}+\varvec{e}_j)}_{\textbf{q}} = \sqrt{\frac{\Gamma (n+|\textbf{q}|+\lambda +2)(n+|\textbf{q}|+\lambda )}{\textbf{q}!\,\Gamma (n+\lambda +1)(n+|\textbf{q}|-q_j+\lambda )}}. \end{aligned}$$

Proposition 5.1

The minimal angle between any two different subspaces $\overline{\textsf {L}_{[\textbf{k}']}}$ and $\overline{\textsf {L}_{[\textbf{k}'']}}$ is equal to zero.

Proof

For a tuple $\textbf{q}$ with $q_j > \max \{k'_j,k''_j\}$, for all $j=1,\ldots ,n$, let us calculate

$$\begin{aligned}{} & {} \langle e^{[\textbf{k}']}_{\textbf{q}-\textbf{k}''+\textbf{1}}, e^{[\textbf{k}'']}_{\textbf{q}-\textbf{k}'+\textbf{1}} \rangle =d^{[\textbf{k}']}_{\textbf{q}-\textbf{k}''+\textbf{1}}d^{[\textbf{k}'']}_{\textbf{q}-\textbf{k}'+\textbf{1}} \langle \overline{\varvec{z}}^{\textbf{k}' - \textbf{1}}\varvec{z}^{\textbf{q}-\textbf{k}''+\textbf{1}}, \overline{\varvec{z}}^{\textbf{k}'' - \textbf{1}}\varvec{z}^{\textbf{q}-\textbf{k}'+\textbf{1}}\rangle \\{} & {} \quad = \ d^{[\textbf{k}']}_{\textbf{q}-\textbf{k}''+\textbf{1}}d^{[\textbf{k}'']}_{\textbf{q}-\textbf{k}'+\textbf{1}} \langle \varvec{z}^{\textbf{q}}, \varvec{z}^{\textbf{q}} \rangle \\{} & {} \quad = \ \sqrt{\frac{\Gamma (n+|\textbf{k}' -\textbf{k}'' + \textbf{q}|+\lambda +1)}{\Gamma (n+\lambda +1) (\textbf{k}'-\textbf{k}''+\textbf{q})!}} \sqrt{\frac{\Gamma (n+|\textbf{k}'' -\textbf{k}' + \textbf{q}|+\lambda +1)}{\Gamma (n+\lambda +1) (\textbf{k}''-\textbf{k}'+\textbf{q})!}} \\{} & {} \qquad \times \ \frac{\Gamma (n+\lambda +1)\textbf{q}!}{\Gamma (n+|\textbf{q}|+\lambda + 1)} \\{} & {} \quad = \ \sqrt{\frac{\Gamma (n+|\textbf{k}'' -\textbf{k}'| + | \textbf{q}|+\lambda +1)}{\Gamma (n+|\textbf{q}|+\lambda + 1)}}\sqrt{\frac{\Gamma (n+|\textbf{k}' -\textbf{k}''| + | \textbf{q}|+\lambda +1)}{\Gamma (n+|\textbf{q}|+\lambda + 1)}} \\{} & {} \qquad \times \ \prod _{j=1}^{n}\sqrt{\frac{\Gamma (q_j+1)}{\Gamma (k'_j - k''_j + q_j+1)}} \sqrt{\frac{\Gamma (q_j+1)}{\Gamma (k''_j - k'_j + q_j+1)}}. \end{aligned}$$

Using several times [6, Formula 8.328.2], we calculate then limit of $\langle e^{[\textbf{k}']}_{\textbf{q}-\textbf{k}''+\textbf{1}}, e^{[\textbf{k}'']}_{\textbf{q}-\textbf{k}'+\textbf{1}} \rangle$, when all $q_j \rightarrow \infty$ (symbolically $\textbf{q}\rightarrow \infty$):

$$\begin{aligned} \lim _{\textbf{q}\rightarrow \infty } \langle e^{[\textbf{k}']}_{\textbf{q}-\textbf{k}''+\textbf{1}}, e^{[\textbf{k}'']}_{\textbf{q}-\textbf{k}'+\textbf{1}} \rangle = 1. \end{aligned}$$

Thus

$$\begin{aligned}{} & {} \cos \varphi ^{(m)}(\overline{\textsf {L}_{[\textbf{k}']}},\overline{\textsf {L}_{[\textbf{k}'']}}) \\{} & {} \quad = \ \sup \left\{ |\langle x, y \rangle |: \ x \in \overline{\textsf {L}_{[\textbf{k}']}}, \ y \in \overline{\textsf {L}_{[\textbf{k}'']}} \ \ \textrm{and} \ \ \Vert x\Vert =\Vert y\Vert = 1 \right\} \\{} & {} \quad = \ \lim _{\textbf{q}\rightarrow \infty } \langle e^{[\textbf{k}']}_{\textbf{q}-\textbf{k}''+\textbf{1}}, e^{[\textbf{k}'']}_{\textbf{q}-\textbf{k}'+\textbf{1}} \rangle = 1, \end{aligned}$$

and the result follows. $\square$

Corollary 5.2

For each $\textbf{k}\in \mathbb {N}^n \setminus \{\varvec{1}\}$, the direct sum of the closed subspaces $\displaystyle {\dot{\sum }_{\textbf{p}= \varvec{1}}^{\textbf{k}} \overline{\textsf {L}_{[\textbf{p}]}}}$ is not closed, and

$$\begin{aligned} \mathcal {A}^2_{\lambda , \textbf{k}}(\mathbb {B}^n) = \textrm{clos}\,\left( \mathop {\dot{\sum }}\limits _{\textbf{p}= \varvec{1}}^{\textbf{k}} \textsf {L}_{[\textbf{p}]}\right) \ \supsetneq \ \mathop {\dot{\sum }}\limits _{\textbf{p}= \varvec{1}}^{\textbf{k}} \overline{\textsf {L}_{[\textbf{p}]}}. \end{aligned}$$

It is instructive to consider an illustrative example.

Example 5.3

We show that the following inclusion is strict

$$\begin{aligned} \mathcal {A}^2_{\lambda }(\mathbb {B}^n) \dotplus \overline{z}_j\mathcal {A}^2_{\lambda }(\mathbb {B}^n) = \overline{\textsf {L}_{[\varvec{1}]}}\dotplus \overline{\textsf {L}_{[\varvec{1}+\varvec{e}_j]}} \varsubsetneq \mathcal {A}^2_{\lambda ,\varvec{1}+\varvec{e}_j}(\mathbb {B}^n). \end{aligned}$$

Consider the function

$$\begin{aligned} f(\varvec{z}) = \sum _{m=1}^{\infty } \frac{1}{m}\,e^{(\varvec{1}+\varvec{e}_j)}_{m\varvec{1}} \in \mathcal {A}^2_{\lambda ,(\varvec{1}+\varvec{e}_j)}(\mathbb {B}^n) \subset \mathcal {A}^2_{\lambda ,\varvec{1}+\varvec{e}_j}(\mathbb {B}^n). \end{aligned}$$

We have that

$$\begin{aligned}{} & {} \ e^{(\varvec{1}+\varvec{e}_j)}_{m\varvec{1}} = \sqrt{\frac{(m+1)(n(m+1)+\lambda )}{n(m+1)-m+\lambda }}e^{[\varvec{1}+\varvec{e}_j]}_{m\varvec{1}} \\{} & {} \quad - \ \sqrt{\frac{m(n(m+1)+\lambda +1)}{n(m+1)-m+\lambda }}e^{[\varvec{1}]}_{m\varvec{1}-\varvec{e}_j} \in \overline{z}_j\mathcal {A}^2_{\lambda }(\mathbb {B}^n) \dotplus \mathcal {A}^2_{\lambda }(\mathbb {B}^n), \end{aligned}$$

implying that

$$\begin{aligned}{} & {} \ f(z) = \sum _{m=1}^{\infty } \sqrt{\frac{(m+1)(n(m+1)+\lambda )}{m^2(n(m+1)-m+\lambda )}}e^{[\varvec{1}+\varvec{e}_j]}_{m\varvec{1}} \\{} & {} \quad - \ \sum _{m=1}^{\infty } \sqrt{\frac{n(m+1)+\lambda +1}{m(n(m+1)-m+\lambda )}}e^{[\varvec{1}]}_{m\varvec{1}-\varvec{e}_j} =f_1(\varvec{z}) +f_2(\varvec{z}), \end{aligned}$$

where none of $f_1(\varvec{z})$ and $f_2(\varvec{z})$ belongs to $L_2(\mathbb {B}^n, d\nu _{\lambda })$. At the same time, f(z) approximates in norm by its partial sums each one of which does belong to $\overline{z}_j\mathcal {A}^2_{\lambda }(\mathbb {B}^n) \dotplus \mathcal {A}^2_{\lambda }(\mathbb {B}^n)$.

The following lemma shows that different true-$\textbf{k}$-poly-Bergman spaces are not necessarily orthogonal.

Lemma 5.4

For any two indices $1\le j' < j'' \le n$, the minimal angle between $\mathcal {A}^2_{\lambda , (\varvec{1}+\varvec{e}_{j'})}(\mathbb {B}^n)$ and $\mathcal {A}^2_{\lambda , (\varvec{1}+\varvec{e}_{j''})}(\mathbb {B}^n)$ is zero.

Proof

To simplify the notation and without loss of generality, we assume that $j'=1$ and $j''=2$. In calculations to follow we set $\textbf{q}_m = (m,m,0,\ldots ,0)$, and consider two norm one elements

$$\begin{aligned} x_m = e^{(\varvec{1}+\varvec{e}_1)}_{\textbf{q}_m-\varvec{e}_2} = d_m \left( \overline{z}_1\varvec{z}^{\textbf{q}_m-\varvec{e}_2} - \frac{m}{n+2m+\lambda } \varvec{z}^{\textbf{q}_m-\varvec{e}_1-\varvec{e}_2}\right) \in \mathcal {A}^2_{\lambda , (\varvec{1}+\varvec{e}_1)}(\mathbb {B}^n), \end{aligned}$$

where

$$\begin{aligned} d_m = \sqrt{\frac{\Gamma (n+2m+\lambda +1)(n+2m+\lambda -1)}{m!(m-1)!\Gamma (n+\lambda +1)(n+m+\lambda -1)}}, \end{aligned}$$

and

$$\begin{aligned} y_m = e^{(\varvec{1}+\varvec{e}_2)}_{\textbf{q}_m-\varvec{e}_1} = d_m \left( \overline{z}_2\varvec{z}^{\textbf{q}_m-\varvec{e}_1} - \frac{m}{n+2m+\lambda } \varvec{z}^{\textbf{q}_m-\varvec{e}_1-\varvec{e}_2}\right) \in \mathcal {A}^2_{\lambda , (\varvec{1}+\varvec{e}_2)}(\mathbb {B}^n). \end{aligned}$$

Calculate now

$$\begin{aligned}{} & {} \langle x_m, y_m\rangle = d_m^2 \left\langle \overline{z}_1\varvec{z}^{\textbf{q}_m-\varvec{e}_2} \right. \\{} & {} \qquad - \ \left. \frac{m}{n+2m+\lambda -1} \varvec{z}^{\textbf{q}_m-\varvec{e}_1-\varvec{e}_2},\, \overline{z}_2\varvec{z}^{\textbf{q}_m-\varvec{e}_1} - \frac{m}{n+2m+\lambda -1} \varvec{z}^{\textbf{q}_m-\varvec{e}_1-\varvec{e}_2}\right\rangle \\{} & {} \quad = \ d_m^2 \left( \Vert \varvec{z}^{\textbf{q}_m}\Vert ^2 - \frac{m}{n+2m+\lambda -1}\Vert \varvec{z}^{\textbf{q}_m - \varvec{e}_2}\Vert ^2 \right. \\{} & {} \qquad \ - \left. \frac{m}{n+2m+\lambda -1}\Vert \varvec{z}^{\textbf{q}_m - \varvec{e}_1}\Vert ^2 + \frac{m^2}{(n+2m+\lambda -1)^2}\Vert \varvec{z}^{\textbf{q}_m - \varvec{e}_1- \varvec{e}_2}\Vert ^2\right) \\{} & {} \quad = \ d_m^2 \left( \frac{\Gamma (n+\lambda +1)(m!)^2}{\Gamma (n+2m+\lambda +1)} - \frac{2m}{n+2m+\lambda -1}\frac{\Gamma (n+\lambda +1)m!(m-1)!}{\Gamma (n+2m+\lambda )} \right. \\{} & {} \qquad + \left. \frac{m^2}{(n+2m+\lambda -1)^2}\frac{\Gamma (n+\lambda +1)(m-1)!(m-1)!}{\Gamma (n+2m+\lambda -1)}\right) \\{} & {} \quad = \ d_m^2 \frac{\Gamma (n+\lambda +1)(m!)^2}{\Gamma (n+2m+\lambda )} \left( \frac{1}{n+2m+\lambda } - \frac{1}{n+2m+\lambda -1}\right) \\{} & {} \quad = \ - \frac{m}{n+m+\lambda -1}. \end{aligned}$$

Then $\lim _{m \rightarrow \infty } |\langle x_m, y_m\rangle | = 1$ implies the statement of the lemma. $\square$

Corollary 5.5

The direct sum of all “the same level” (closed) true-poly-Bergman spaces

$$\begin{aligned} \mathcal {A}^2_{\lambda , (\varvec{1}+\varvec{e}_{1})}(\mathbb {B}^n) \dotplus \mathcal {A}^2_{\lambda , (\varvec{1}+\varvec{e}_{2})}(\mathbb {B}^n) \dotplus \ldots \dotplus \mathcal {A}^2_{\lambda , (\varvec{1}+\varvec{e}_{n})}(\mathbb {B}^n) \end{aligned}$$

is not closed.

The above lemma also implies that, contrary to the one-dimensional case of $\mathbb {D}$ and $\mathbb {C}$, as well as the several variables cases of the previous sections, for the unit ball $\mathbb {B}^n$ we have the following “no-go” assertion.

Corollary 5.6

There does not exist any orthonormal basis of $L_2(\mathbb {\mathbb {B}}^n, d\nu _{\lambda })$ formed by the union of orthonormal bases of the true-poly-Bergman spaces.

This in turn implies

Corollary 5.7

The operators $\mathfrak {a}_j= \frac{\partial }{\partial \overline{z}_j}$ can not have their formally adjoint $\mathfrak {b}_j=\mathfrak {a}_j^{\dag }$, $j=1,\ldots ,n$, which satisfy Definition 3.1 of the extended Fock space construction.

Moreover, there do not exist any two tuples of formally adjoint operators $\mathfrak {a}_j$ and $\mathfrak {a}_j^{\dag }$, $j=1,2,\ldots ,n$, densely defined in $L_2(\mathbb {\mathbb {B}}^n, d\nu _{\lambda })$, such that

$$\begin{aligned} \left[ \mathfrak {a}_j, \mathfrak {a}^{\dag }_{\ell }\right] = \delta _{j,\ell }I, \quad \left[ \mathfrak {a}_j, \mathfrak {a}_{\ell }\right] = 0, \quad \left[ \mathfrak {a}^{\dag }_j, \mathfrak {a}^{\dag }_{\ell }\right] = 0, \quad j,\ell = 1,2,\ldots ,n; \end{aligned}$$

and

$$\begin{aligned} \textrm{clos}\left( \bigcap _{j=1}^n\ker \mathfrak {a}_j\right) = \mathcal {A}^2(\mathbb {B}^n) \qquad \textrm{and} \qquad \mathcal {A}^2_{(\textbf{k})}(\mathbb {B}^n) = (\mathbf {\mathfrak {a}}^{\dag })^{\textbf{k}-\varvec{1}}\mathcal {A}^2(\mathbb {B}^n). \end{aligned}$$

Corollary 5.8

The Hilbert space $L_2(\mathbb {\mathbb {B}}^n, d\nu _{\lambda })$ can not be represented as the orthogonal sum of all its true-poly-Bergman spaces. We have just the following representations of $L_2(\mathbb {\mathbb {B}}^n, d\nu _{\lambda })$ in terms of poly-Bergman and true-poly-Bergman spaces

$$\begin{aligned} L_2(\mathbb {\mathbb {B}}^n, d\nu _{\lambda }) = \textrm{clos}\, \left( \bigcup _{\textbf{k}\in \mathbb {N}^n} \mathcal {A}^2_{\lambda , \textbf{k}}(\mathbb {B}^n)\right) = \textrm{clos}\,\left( \underset{\textbf{k}\in \mathbb {N}^n}{\mathop {\dot{\sum }}\limits } \mathcal {A}^2_{\lambda , (\textbf{k})}(\mathbb {B}^n)\right) . \end{aligned}$$

5.2 Poly-Vekua Spaces

We study here the case of item (4) of Example 2.2, i.e.,

$$\begin{aligned} \mathfrak {a}_j = \frac{\partial }{\partial \overline{z}_j} + \mu \frac{z_j}{1-|z|^2} \qquad \textrm{and} \qquad \mathfrak {b}_j = \overline{z}_j, \quad \textrm{with} \ \ \mu > \textstyle {-\frac{1}{2}}, \ \ j = 1,2,\ldots , n. \end{aligned}$$

These operators act invariantly on the set of smooth functions in $\mathbb {B}^n$, and the set of functions f satisfying the equations $\mathfrak {a}_jf = 0$, for all $j= 1,2,\ldots , n$, forms a class of n-variable generalised analytic functions $\mathfrak {A}(\mu \frac{z_j}{1-|z|^2}, 0; \mathbb {B}^n)$ in the Vekua sense.

As it easily seen, $f \in \mathfrak {A}(\mu \frac{z_j}{1-|z|^2}, 0; \mathbb {B}^n)$ if and only if $f = (1- |z|^2)^{\mu }\varphi (z)$, where $\varphi$ is analytic. Then, by Lemma 2.1, a function f is $\textbf{k}$-poly-generalised analytic, i.e. satisfies the equations $\mathfrak {a}_j^{k_j}f=0$, for all $j= 1,2,\ldots , n$, if and only if if admits the representation

$$\begin{aligned} f = \sum _{\textbf{q}=\textbf{0}}^{\textbf{k}-\textbf{1}} \overline{\varvec{z}}^{\textbf{q}} (1- |z|^2)^{\mu }\varphi _{\textbf{q}}, \ \ \mathrm {with \ analytic} \ \ \varphi _{\textbf{q}}, \end{aligned}$$

or $f = (1- |z|^2)^{\mu }\varphi$, where $\varphi = {\sum _{\textbf{q}=\textbf{0}}^{\textbf{k}-\textbf{1}}} \overline{\varvec{z}}^{\textbf{q}}\varphi _{\textbf{q}}$ is $\textbf{k}$-polyanalytic.

We introduce then unweighted space $L_2(\mathbb {B}^n, d\nu )$, where the measure $d\nu$ given by

$$\begin{aligned} d\nu (\varvec{z})= \frac{\Gamma (n+1)}{\pi ^n }dv(\varvec{z}), \end{aligned}$$

and define the Vekua space $\mathcal {V}^2_{\mu }(\mathbb {B}^n)$ as the intersection $L_2(\mathbb {B}^n, d\nu ) \cap \mathfrak {A}(\mu \frac{z_j}{1-|z|^2}, 0; \mathbb {B}^n)$.

Similarly, for a multi-index $\textbf{k}\in \mathbb {N}^n$, we define the $\textbf{k}$-poly-Vekua space $\mathcal {V}^2_{\mu , \textbf{k}}(\mathbb {B}^n)$ as the intersection of $L_2(\mathbb {B}^n, d\nu )$ with the space of the $\textbf{k}$-poly-generalised analytic functions, with the convention that $\mathcal {V}^2_{\mu , \varvec{1}}(\mathbb {B}^n) = \mathcal {V}^2_{\mu }(\mathbb {B}^n)$.

It is evident that $f=(1- |z|^2)^{\mu }\varphi$ belongs to $L_2(\mathbb {B}^n, d\nu )$ if and only if the function $\varphi$ belongs to $L_2(\mathbb {B}^n, d\nu _{2\mu })$. We introduce thus the isometric isomorphism

$$\begin{aligned} U_{\mu }: \, L_2(\mathbb {B}^n, d\nu ) \longrightarrow L_2(\mathbb {B}^n, d\nu _{2\mu }) \end{aligned}$$

by

$$\begin{aligned} U_{\mu }: \, f \ \longmapsto \ \sqrt{\frac{\Gamma (n+1)\Gamma (2\mu +1)}{\Gamma (n+2\mu +1)}} (1- |z|^2)^{-\mu }f. \end{aligned}$$

This operator is obviously unitary and

$$\begin{aligned} U_{\mu }^*: \, \varphi \ \longmapsto \ \sqrt{\frac{\Gamma (n+2\mu +1)}{\Gamma (n+1)\Gamma (2\mu +1)}} (1- |z|^2)^{\mu } \varphi . \end{aligned}$$

Lemma 5.9

The operator $U_{\mu }$ maps isomorphically and isometrically $\textbf{k}$-poly-Vekua spaces $\mathcal {V}^2_{\mu , \textbf{k}}(\mathbb {B}^n)$ in $L_2(\mathbb {B}^n, d\nu )$ onto the $\textbf{k}$-poly-Bergman spaces $\mathcal {A}^2_{2\mu , \textbf{k}}(\mathbb {B}^n)$ in $L_2(\mathbb {B}^n, d\nu _{2\mu })$.

Proof

Follows from

$$\begin{aligned} U_{\mu }\left( \frac{\partial }{\partial \overline{z}_j} + \mu \frac{z_j}{1-|z|^2}\right) U_{\mu }^* = \frac{\partial }{\partial \overline{z}_j}, \quad \mathrm {for \ all} \quad j=1,\ldots ,n. \end{aligned}$$

$\square$

As a corollary we can now characterize the properties of the poly-Vekua spaces based on the corresponding properties of the poly-Bergman spaces that are given in Lemma 5.5 and Corollaries 5.5–5.8.

6 Generalized Gaussian Measure Spaces in $\mathbb {C}^2$

We consider here Hilbert spaces with a generalized Gaussian measure in several variable case. To simplify the presentation we consider the functions of just two complex variables, i.e., the spaces over $\mathbb {C}^2$.

We start with the one-dimensional case, and introduce, following e.g., [4], the space $L_2(\mathbb {C},d\mu _2)$, where the probability generalized Gaussian measure $d\mu _2$ is given by

$$\begin{aligned} d\mu _2(z) = \frac{2}{\pi ^{\frac{3}{2}}}e^{-|z|^4}dv(z). \end{aligned}$$

A straightforward calculation, with the use of [6, Formula 3.326.2], shows that

$$\begin{aligned} \Vert \overline{z}^p z^q\Vert ^2 = \Vert z^{p+q}\Vert ^2 = \frac{1}{\sqrt{\pi }}\Gamma \left( \frac{p+q+1}{2}\right) . \end{aligned}$$

A simple calculation implies then that

$$\begin{aligned}{} & {} \ \left\langle \frac{\partial }{\partial \overline{z}}\overline{z}^pz^q, \overline{z}^kz^q \right\rangle = \delta _{q+k,p+\ell -1} \frac{p}{\sqrt{\pi }}\Gamma \left( \frac{q+k+1}{2}\right) \\{} & {} \quad = \ \left\langle \overline{z}^pz^q, \left( -\frac{\partial }{\partial z} + 2z\overline{z}^2\right) \overline{z}^kz^q \right\rangle , \end{aligned}$$

which means that the operators

$$\begin{aligned} \mathfrak {a}= \frac{\partial }{\partial \overline{z}} \qquad \textrm{and} \qquad \mathfrak {a}^{\dag } = -\frac{\partial }{\partial z} + 2z\overline{z}^2 \end{aligned}$$

(6.1)

are formally adjoint being defined on the linear span of monomials $m_{p,q}=\overline{z}^pz^q$, with $p,\,q \in ~\mathbb {Z}_+$. But unfortunately this does not lead to the extended Fock construction, since, in particular,

$$\begin{aligned} {[}\mathfrak {a}, \mathfrak {a}^{\dag }] = 4z\overline{z} \ne I. \end{aligned}$$

At the same time, the pair of operators

$$\begin{aligned} \mathfrak {a}= \frac{\partial }{\partial \overline{z}} \qquad \textrm{and} \qquad \mathfrak {b}= \overline{z}, \end{aligned}$$

densely defined on the linear span $\mathcal {D}_0$ of all monomials $m_{p,q}=\overline{z}^pz^q$, does lead to the extended Fock construction. We have that $\textsf {L}_{\mathbf {[1]}} = \ker \textbf{a}|_{\mathcal {D}_0}$ consists of all analytic polynomials, its closure $\overline{\textsf {L}_{\mathbf {[1]}}}$ coincides with the weighted Fock space $\mathcal {F}^2_2(\mathbb {C})$, in the notation of [4], the k-poly-Fock spaces $\mathcal {F}^2_{2,k}(\mathbb {C})$ are given then by

$$\begin{aligned} \mathcal {F}^2_{2,k}(\mathbb {C})= & {} \textrm{clos}\left( \textsf {L}_{\mathbf {[1]}} \dotplus \overline{z}\textsf {L}_{\mathbf {[1]}} \dotplus \cdots \dotplus \overline{z}^{k-1}\textsf {L}_{\mathbf {[1]}}\right) \\= & {} \textrm{clos}\left( \mathcal {F}^2_2(\mathbb {C}) \dotplus \overline{z}\mathcal {F}^2_2(\mathbb {C}) \dotplus \cdots \dotplus \overline{z}^{k-1}\mathcal {F}^2_2(\mathbb {C})\right) . \end{aligned}$$

Note in this connection that the direct sum of the closed spaces $\overline{z}^{j}\mathcal {F}^2_2(\mathbb {C})$ is not closed. This follows from the fact that the minimal angle between any two different spaces $\overline{z}^{j_1}\mathcal {F}^2_2(\mathbb {C})$ and $\overline{z}^{j_2}\mathcal {F}^2_2(\mathbb {C})$ is zero. It can be easily verified by calculating $\lim _{\ell \rightarrow \infty } |\langle x_{\ell },y_{\ell }\rangle |$, where $x_{\ell }$ and $y_{\ell }$ are normalised elements $\overline{z}^{j_1}z^{\ell } \in \overline{z}^{j_1}\mathcal {F}^2_2(\mathbb {C})$ and $\overline{z}^{j_2}z^{\ell +j_2-j_1} \in \overline{z}^{j_2}\mathcal {F}^2_2(\mathbb {C})$, respectively.

The true-k-poly-Fock spaces $\mathcal {F}^2_{2,(k)}(\mathbb {C})$ are defined then by

$$\begin{aligned} \mathcal {F}^2_{2,(k)}(\mathbb {C}) = \mathcal {F}^2_{2,k}(\mathbb {C}) \cap \mathcal {F}^2_{2,k-1}(\mathbb {C})^{\perp }, \end{aligned}$$

leading to

$$\begin{aligned} L_2(\mathbb {C},d\mu _2) = \textrm{clos}\left( \bigcup _{k \in \mathbb {N}}\mathcal {F}^2_{2,k}(\mathbb {C})\right) = \bigoplus _{k \in \mathbb {N}}\mathcal {F}^2_{2,(k)}(\mathbb {C}). \end{aligned}$$

Let us recall briefly (for details we refer to [15]) another characterization of the true-poly-Fock spaces. The set of polynomials (see e.g., [15, Formula 2.9]),

$$\begin{aligned} e_{p,q}(z,\overline{z}) = \sum _{k = 0}^{\min \{p,q\}}c_{p,q}(k) \overline{z}^{p-k}z^{q-k}, \qquad p,\, q \in \mathbb {Z}_+, \end{aligned}$$

(6.2)

where $c_{p,q}(k)$ are explicitly calculable constants, forms an orthonormal basis of the space $L_2(\mathbb {C},d\mu _2)$. Example 2.4 in [15] gives the first six these basis elements.

Formulas (4.4) in [15]:

$$\begin{aligned} \textbf{a}&:&\, e_{p,q} \ \longmapsto {\left\{ \begin{array}{ll} \sqrt{p} e_{p-1,q}, &{} q \in \mathbb {Z}_+, \ p>0 \\ 0, &{} q \in \mathbb {Z}_+, \ p=0 \end{array}\right. } \\&\textrm{and}&\\ \textbf{a}^\dag&:&\, e_{p,q} \ \longmapsto \ \sqrt{p+1} e_{p+1,q}, \ \ p,q \in \mathbb {Z}_+ \end{aligned}$$

defines then the formally adjoint operators $\textbf{a}$ and $\textbf{a}^{\dag }$, densely defined on the span of all monomials $m_{p,q}$ (and even on the span of the true-poly-Fock spaces), that satisfy the commutation relation $[\textbf{a},\textbf{a}^{\dag }] = I$.

Contrary to the operators (6.1), the operators $\textbf{a}$ and $\textbf{a}^{\dag }$ lead to the extended Fock construction, and obey the properties

$$\begin{aligned} \ker \textbf{a}= \mathcal {F}^2_2(\mathbb {C}):=\mathcal {F}^2_{2,(1)}(\mathbb {C}) \qquad \textrm{and} \qquad (\textbf{a}^{\dag })^k(\mathcal {F}^2_2(\mathbb {C})) = \mathcal {F}^2_{2,(k+1)}(\mathbb {C}),\ \ k \in \mathbb {N}. \end{aligned}$$

We note as well that, for each $k \in \mathbb {N}$,

$$\begin{aligned} \mathcal {F}^2_{2,(k)}(\mathbb {C}) = \textrm{clos}\left( \textrm{span} \{e_{k-1,q}:\, q \in \mathbb {Z}_+\}\right) , \end{aligned}$$

leading again to

$$\begin{aligned} L_2(\mathbb {C},d\mu _2) = \bigoplus _{k \in \mathbb {N}}\mathcal {F}^2_{2,(k)}(\mathbb {C}). \end{aligned}$$

Passing to the two complex variables setting, we mansion that there are two different natural generalizations of the weight factor $e^{|z|^4}$, namely $e^{|z_1|^4+|z_2|^4}$ and $e^{|\varvec{z}|^4}=e^{(|z_1|^2+|z_2|^2)^2}$.

We consider these cases separately.

6.1 Weight Factor $e^{|z_1|^4+|z_2|^4}$ Case

This is a simple case, as the Hilbert space $L_2(\mathbb {C}^2, d\mu _{\varvec{2}})$, with the probability measure $d\mu _{\varvec{2}}(\varvec{z}) = \frac{4}{\pi ^3}e^{|z_1|^4+|z_2|^4}dv(\varvec{z})$ splits into the tensor product

$$\begin{aligned} L_2(\mathbb {C}^2, d\mu _{\varvec{2}}) = L_2(\mathbb {C},d\mu _2(z_1)) \otimes L_2(\mathbb {C},d\mu _2(z_2)). \end{aligned}$$

We introduce then two pairs of of formally adjoint operators

$$\begin{aligned} \mathbf {\mathfrak {a}}= (\mathfrak {a}_1, \ \mathfrak {a}_2) \qquad \textrm{and} \qquad \mathbf {\mathfrak {a}}^{\dag } = (\mathfrak {a}_1^{\dag }, \ \mathfrak {a}_2^{\dag }), \end{aligned}$$

densely defined on the finite linear combinations of the monomials $m_{\textbf{p},\textbf{q}}= \overline{z}^{\textbf{p}}z^{\textbf{q}}$, $\textbf{p},\,\textbf{q}\in \mathbb {Z}_+^n$, by

$$\begin{aligned} \mathfrak {a}_1 = \textbf{a}\otimes \textsf {I}, \quad \mathfrak {a}_2 = \textsf {I} \otimes \textbf{a}\qquad \textrm{and} \qquad \mathfrak {a}_1^{\dag } = \textbf{a}^{\dag } \otimes \textsf {I}, \quad \mathfrak {a}_2^{\dag } = \textsf {I} \otimes \textbf{a}^{\dag }, \end{aligned}$$

and satisfying the commutation relations (3.11).

The space $\textsf {L}_{[\textbf{1}]}= \ker \mathfrak {a}_1 \cap \ker \mathfrak {a}_2$ coincides with the set of all analytic polynomials (polynomials on $\varvec{z}^{\textbf{q}}$), while its closure $\overline{\textsf {L}_{[\textbf{1}]}}$ coincides with the generalized Fock space $\mathcal {F}^2_{\varvec{2}}(\mathbb {C}^2)$, being the closed subspace of $L_2(\mathbb {C}^2, d\mu _{\varvec{2}})$ which consist of functions analytic in $\mathbb {C}^2$.

Observe that space $\mathcal {F}^2_{\varvec{2}}(\mathbb {C}^2)$ splits into the tensor product of two Fock spaces $\mathcal {F}^2_2(\mathbb {C})$ of analytic functions in $z_1$ and $z_2$, respectively,

$$\begin{aligned} \mathcal {F}^2_{\varvec{2}}(\mathbb {C}^2) =\mathcal {F}^2_2(\mathbb {C}) \otimes \mathcal {F}^2_2(\mathbb {C}). \end{aligned}$$

Corollary 3.15 implies now

Proposition 6.1

Each true-$\textbf{k}$-poly-Fock subspace $F^2_{\varvec{2}, (\textbf{k})}(\mathbb {C}^2)$ of $L_2(\mathbb {C}^2, d\mu _{\varvec{2}})$ has the form

$$\begin{aligned} \mathcal {F}^2_{\varvec{2}, (\textbf{k})}(\mathbb {C}^2) = \mathcal {F}^2_{2,(k_1)}(\mathbb {C}) \otimes \mathcal {F}^2_{2,(k_2)}(\mathbb {C}). \end{aligned}$$

All true-$\textbf{k}$-poly-Fock subspaces $\mathcal {F}^2_{\varvec{2}, (\textbf{k})}(\mathbb {C}^2)$ of $L_2(\mathbb {C}^2, d\mu _{\varvec{2}})$ are orthogonal and are isomorphic to each other. The isometric isomorphism between the Fock space $\mathcal {F}^2_{\varvec{2}}(\mathbb {C}^2)$ and the true-$\textbf{k}$-poly-Fock space $\mathcal {F}^2_{\varvec{2}, (\textbf{k})}(\mathbb {C}^2)$ is given by

$$\begin{aligned} \frac{1}{\sqrt{(\textbf{k}-\varvec{1})!}}\, (\mathbf {\mathfrak {a}}^\dag )^{\textbf{k}-\varvec{1}}|_{\mathcal {F}^2_{\varvec{2}}(\mathbb {C}^2)}: \mathcal {F}^2_{\varvec{2}}(\mathbb {C}^2) \longrightarrow \mathcal {F}^2_{\varvec{2}, (\textbf{k})}(\mathbb {C}^2), \end{aligned}$$

and the operator

$$\begin{aligned} \frac{1}{\sqrt{(\textbf{k}-\varvec{1})!}}\,{\mathbf {\mathfrak {a}}}^{\textbf{k}-\varvec{1}}|_{\mathcal {F}_{\varvec{2}, (\textbf{k})}^2(\mathbb {C}^2)}: \mathcal {F}^2_{\varvec{2}, (\textbf{k})}(\mathbb {C}^2) \longrightarrow \mathcal {F}^2_{\varvec{2}}(\mathbb {C}^2) \end{aligned}$$

gives the inverse isomorphism.

The space $L_2(\mathbb {C}^2, d\mu _{\varvec{2}})$ coincides with the orthogonal sum of all its true-$\textbf{k}$-poly-Fock subspaces,

$$\begin{aligned} L_2(\mathbb {C}^2, d\mu _{\varvec{2}}) = \bigoplus _{\textbf{k}\in \mathbb {N}^2} \mathcal {F}^2_{\varvec{2}, (\textbf{k})}(\mathbb {C}^2). \end{aligned}$$

The $\textbf{k}$-poly-Fock subspaces of $L_2(\mathbb {C}^2, d\mu _{\varvec{2}})$, i.e., those that consist of all elements $h \in L_2(\mathbb {C}^2, d\mu _{\varvec{2}})$ satisfying $\mathbf {\mathfrak {a}}^{\textbf{k}}h = 0$, are the tensor products of the $k_j$-poly-Fock subspaces $\mathcal {F}^2_{2, k_j}(\mathbb {C})$ of $L_2(\mathbb {C},\mu _{2}(z_j))$,

$$\begin{aligned} \mathcal {F}^2_{\varvec{}, \textbf{k}}(\mathbb {C}^2) = \mathcal {F}^2_{2, k_1}(\mathbb {C}) \otimes \mathcal {F}^2_{2, k_2}(\mathbb {C}). \end{aligned}$$

The true-$\textbf{k}$-Fock space $F_{\varvec{2,(\textbf{k})}}^2(\mathbb {C}^n)$ admits an orthonormal basis given by the following polynomials (see (6.2))

$$\begin{aligned} e^{(\textbf{k})}_{\textbf{q}} = e_{k_1-1,q_1} \otimes .e_{k_2-1,q_2}, \qquad q_1,\ q_2 \in \mathbb {Z}_+. \end{aligned}$$

The Hilbert space $L_2(\mathbb {C}^2, d\mu _{\varvec{2}})$ admits an orthonormal basis formed by the union of the above orthonormal bases of the true-$\textbf{k}$-Fock spaces $\mathcal {F}_{\varvec{2},(\textbf{k})}^2(\mathbb {C}^n)$, $\textbf{k}\in \mathbb {N}^2$.

6.2 Weight Factor $e^{(|z_1|^2+|z_2|^2)^2}$ Case

As we will see, this case is very much similar to the case of the unit ball $\mathbb {B}^n$.

We introduce the Hilbert space $L_2(\mathbb {C}^2, d\mu (\varvec{z}))$, where the probability measure $d\mu (\varvec{z})$ is given by

$$\begin{aligned} d\mu (\varvec{z}) = \frac{2}{\pi ^2}e^{(|z_1|^2+|z_2|^2)^2} dv(\varvec{z}). \end{aligned}$$

A straightforward calculation, with the use of [6, Formula 3.621.5], shows that

$$\begin{aligned} \Vert \overline{\varvec{z}}^{\textbf{p}} \varvec{z}^{\textbf{q}}\Vert ^2{} & {} = \Vert \varvec{z}^{\textbf{p}+\textbf{q}}\Vert ^2 = \frac{\Gamma (p_1+q_1+1)\Gamma (p_2+q_2+1)}{\Gamma (|\textbf{p}+\textbf{q}|+2)}\Gamma \left( \frac{|\textbf{p}+\textbf{q}|}{2}+1\right) \nonumber \\{} & {} = \ (d^{[\textbf{p}+\varvec{1}]}_{\textbf{q}})^{-2}. \end{aligned}$$

(6.3)

Having in mind the study of polyanalytic functions, we introduce the operators

$$\begin{aligned} \mathfrak {a}_1= \frac{\partial }{\partial \overline{z}_1} \qquad \textrm{and} \qquad \mathfrak {a}_2= \frac{\partial }{\partial \overline{z}_2}, \end{aligned}$$

and try to use them together with their formally adjoint operators in the generalized Fock space construction. A bit lengthy calculation based on (6.3) shows that

$$\begin{aligned} \mathfrak {a}_1^{\dag } = -\frac{\partial }{\partial z_1} + 2 |\varvec{z}|^2\overline{z}_1 \qquad \textrm{and} \qquad \mathfrak {a}_2^{\dag } = -\frac{\partial }{\partial z_2} + 2 |\varvec{z}|^2\overline{z}_2 \end{aligned}$$

being defined on the dense set $\mathcal {D}_0$ in $L_2(\mathbb {C}^2, d\mu (\varvec{z}))$ formed by the finite linear combinations of the monomials $m_{\textbf{p},\textbf{q}} = \overline{\varvec{z}}^{\textbf{p}}\varvec{z}^{\textbf{q}}$, $\textbf{p},\,\textbf{q}\in \mathbb {Z}_+^2$. Unfortunately such an approach does not work since, in particular,

$$\begin{aligned}{}[\mathfrak {a}_j, \mathfrak {a}_k^{\dag }] \ne \delta _{j,k}I \qquad \textrm{and} \qquad [\mathfrak {a}_j^{\dag }, \mathfrak {a}_k^{\dag }] \ne 0. \end{aligned}$$

At the same time the pairs of operators

$$\begin{aligned} \mathbf {\mathfrak {a}}=(\mathfrak {a}_1,\mathfrak {a}_2) \qquad \textrm{and} \qquad \mathbf {\mathfrak {b}}= (\mathfrak {b}_1,\mathfrak {b}_2), \quad \textrm{with} \quad \mathfrak {a}_j= \frac{\partial }{\partial \overline{z}_j}, \quad \mathfrak {b}_j = \overline{z}_j, \end{aligned}$$

do can be used in the generalized Fock space construction directed to the study of the polyanalytic functions.

In more details. The set $\textsf {L}_{\mathbf {[\varvec{1}]}} = \ker \textbf{a}_1|_{\mathcal {D}_0}\cap \ker \textbf{a}_2|_{\mathcal {D}_0}$ consists of all analytic polynomials, its closure $\overline{\textsf {L}_{\mathbf {[\varvec{1}]}}}$ coincides with the Fock subspace $\mathcal {F}^2(\mathbb {C}^2)$ of $L_2(\mathbb {C}^2, d\mu (\varvec{z}))$. Introduce then the sets $\textsf {L}_{\mathbf {[\textbf{k}]}} = \mathbf {\mathfrak {b}}^{\textbf{k}-\varvec{1}}\textsf {L}_{\mathbf {[\varvec{1}]}}$, $\textbf{k}\in \mathbb {N}^n$, whose linear span is dense in $L_2(\mathbb {C}^2, d\mu (\varvec{z}))$.

Observe that $\overline{\textsf {L}_{\mathbf {[\textbf{k}]}}} = \overline{\varvec{z}}^{\textbf{k}-\varvec{1}}\mathcal {F}^2(\mathbb {C}^2)$, and that the basis of is formed by

$$\begin{aligned} e^{[\textbf{k}]}_{\textbf{q}} = d^{[\textbf{k}]}_{\textbf{q}} \overline{\varvec{z}}^{\textbf{k}-\varvec{1}}\varvec{z}^{\textbf{q}}, \qquad \textbf{q}\in \mathbb {Z}_+^2. \end{aligned}$$

Lemma 6.2

The minimal angle between any two different spaces $\overline{\textsf {L}_{\mathbf {[\textbf{k}']}}}$ and $\overline{\textsf {L}_{\mathbf {[\textbf{k}'']}}}$ is equal to zero.

Proof

For $\textbf{q}_m=(m,m)$ with $m > \max \{k'_j,k''_j\}$, $j=1,2$, we calculate

$$\begin{aligned}{} & {} \langle e^{[\textbf{k}']}_{\textbf{q}_m-\textbf{k}''+\textbf{1}}, e^{[\textbf{k}'']}_{\textbf{q}_m-\textbf{k}'+\textbf{1}} \rangle \\{} & {} \quad = \ d^{[\textbf{k}']}_{\textbf{q}_m-\textbf{k}''+\textbf{1}}d^{[\textbf{k}'']}_{\textbf{q}_m-\textbf{k}'+\textbf{1}} \langle \overline{\varvec{z}}^{\textbf{k}' - \textbf{1}}\varvec{z}^{\textbf{q}_m-\textbf{k}''+\textbf{1}}, \overline{\varvec{z}}^{\textbf{k}'' - \textbf{1}}\varvec{z}^{\textbf{q}_m-\textbf{k}'+\textbf{1}}\rangle \\{} & {} \quad = \ d^{[\textbf{k}']}_{\textbf{q}_m-\textbf{k}''+\textbf{1}}d^{[\textbf{k}'']}_{\textbf{q}_m-\textbf{k}'+\textbf{1}} \langle \varvec{z}^{\textbf{q}_m}, \varvec{z}^{\textbf{q}_m} \rangle \\{} & {} \quad \ = \ \sqrt{\frac{\Gamma (|\textbf{k}'-\textbf{k}''| + 2m+2)}{\Gamma (k'_1-k''_1+m+1)\Gamma (k'_2-k''_2+m+1)} \frac{1}{\Gamma (\frac{|\textbf{k}'-\textbf{k}''|}{2}+m+1)}} \\{} & {} \qquad \ \times \ \sqrt{\frac{\Gamma (|\textbf{k}''-\textbf{k}'| + 2m+2)}{\Gamma (k''_1-k'_1+m+1)\Gamma (k''_2-k'_2+m+1)} \frac{1}{\Gamma (\frac{|\textbf{k}''-\textbf{k}'|}{2}+m+1)}} \\{} & {} \qquad \times \ \frac{\Gamma (m+1)^3}{\Gamma (2m+2)} \\{} & {} \quad \ = \ \sqrt{\frac{\Gamma (|\textbf{k}'-\textbf{k}''| + 2m+2)}{\Gamma (2m+2)}} \sqrt{\frac{\Gamma (|\textbf{k}''-\textbf{k}'| + 2m+2)}{\Gamma (2m+2)}} \\{} & {} \qquad \times \ \sqrt{\frac{\Gamma (m+1)}{\Gamma (k'_1-k''_1+m+1)}}\\{} & {} \qquad \ \times \ \sqrt{\frac{\Gamma (m+1)}{\Gamma (k''_1-k'_1+m+1)}}\sqrt{\frac{\Gamma (m+1)}{\Gamma (k'_2-k''_2+m+1)}} \sqrt{\frac{\Gamma (m+1)}{\Gamma (k''_2-k'_2+m+1)}} \\{} & {} \qquad \ \times \ \sqrt{\frac{\Gamma (m+1)}{\Gamma (\frac{|\textbf{k}'-\textbf{k}''|}{2}+m+1)}} \sqrt{\frac{\Gamma (m+1)}{\Gamma (\frac{|\textbf{k}''-\textbf{k}'|}{2}+m+1)}}. \end{aligned}$$

Using [6, Formula 8.328.2] in each square root, we have

$$\begin{aligned} \lim _{m \rightarrow \infty } \langle e^{[\textbf{k}']}_{\textbf{q}_m-\textbf{k}''+\textbf{1}}, e^{[\textbf{k}'']}_{\textbf{q}_m-\textbf{k}'+\textbf{1}} \rangle = 1, \end{aligned}$$

and the result follows by the definition of the minimal angle between the subspaces. $\square$

The lemma implies that the direct sum of closed subspaces $\overline{\textsf {L}_{\mathbf {[\textbf{k}]}}} = \overline{\varvec{z}}^{\textbf{k}-\varvec{1}}\mathcal {F}^2(\mathbb {C}^2)$

$$\begin{aligned} \mathop {\dot{\sum }}\limits _{\textbf{p}= \varvec{1}}^{\textbf{k}} \overline{\varvec{z}}^{\textbf{k}-\varvec{1}}\mathcal {F}^2(\mathbb {C}^2) \end{aligned}$$

is not closed. That is, the space

$$\begin{aligned} \mathcal {H}_{\textbf{k}} = \textrm{clos}\,\left( \mathop {\dot{\sum }}\limits _{\textbf{p}= \varvec{1}}^{\textbf{k}} \overline{\varvec{z}}^{\textbf{k}-\varvec{1}}\mathcal {F}^2(\mathbb {C}^2) \right) = \textrm{clos}\,\left( \mathop {\dot{\sum }}\limits _{\textbf{p}= \varvec{1}}^{\textbf{k}} \textsf {L}_{[\textbf{p}]}\right) \end{aligned}$$

is the $\textbf{k}$-poly-Fock space $\mathcal {F}^2_{\textbf{k}}(\mathbb {C}^2)$, which consists of all functions $f \in L_2(\mathbb {C}^2, d\mu (\varvec{z}))$ satisfying $\mathbf {\mathfrak {a}}^{\textbf{k}}f = 0$.

Then the space

$$\begin{aligned} \mathcal {H}_{(\textbf{k})} = \mathcal {H}_{\textbf{k}} \cap \left( \mathop {\dot{\sum }}\limits _{j=1}^2 \mathcal {H}_{\textbf{k}-\textbf{e}_j}\right) ^{\perp } = \mathcal {F}^2_{\textbf{k}}(\mathbb {C}^2) \cap \left( \mathcal {F}^2_{\textbf{k}-\textbf{e}_1}(\mathbb {C}^2) \dotplus \mathcal {F}^2_{\textbf{k}-\textbf{e}_2}(\mathbb {C}^2)\right) ^{\perp } \end{aligned}$$

is denoted by $\mathcal {F}^2_{(\textbf{k})}(\mathbb {C}^2)$ and is called true-$\textbf{k}$-poly-Fock space, with the convention that

$$\begin{aligned} \mathcal {F}^2(\mathbb {C}^2) = \mathcal {F}^2_{\varvec{1}}(\mathbb {C}^2) =\mathcal {F}^2_{(\varvec{1})}(\mathbb {C}^2). \end{aligned}$$

Note that we meet here with a third possible situation different from the previous two (Proposition 6.1 and Corollary 5.5):

Lemma 6.3

The true-poly-Fock spaces $\mathcal {F}^2_{(\varvec{1}+ \varvec{e}_1)}(\mathbb {C}^2)$ and $\mathcal {F}^2_{(\varvec{1}+ \varvec{e}_2)}(\mathbb {C}^2)$ (recall that $\varvec{1}+ \varvec{e}_1=(2,1)$ and $\varvec{1}+ \varvec{e}_2=(1,2)$) are not orthogonal, but their direct sum

$$\begin{aligned} \mathcal {F}^2_{(\varvec{1}+ \varvec{e}_1)}(\mathbb {C}^2) \dotplus \mathcal {F}^2_{(\varvec{1}+ \varvec{e}_2)}(\mathbb {C}^2) \end{aligned}$$

is closed

We give the proof of the lemma in the Appendix.

Corollary 6.4

There does not exist any orthonormal basis of $L_2(\mathbb {C}^2, d\mu (\varvec{z}))$ formed by the union of orthonormal bases of the true-poly-Fock spaces.

This in turn implies

Corollary 6.5

There does not exist any pair of formally adjoint operators $\mathbf {\mathfrak {a}}=(\mathfrak {a}_1,\mathfrak {a}_2)$ and $\mathbf {\mathfrak {a}}^{\dag }=(\mathfrak {a}_1^{\dag },\mathfrak {a}_2^{\dag })$ that act invariantly on the linear span of the monomials, satisfy therein the commutation relations

$$\begin{aligned} \left[ \mathfrak {a}_j, \mathfrak {a}^{\dag }_{\ell }\right] = \delta _{j,\ell }I, \quad \left[ \mathfrak {a}_j, \mathfrak {a}_{\ell }\right] = 0, \quad \left[ \mathfrak {a}^{\dag }_j, \mathfrak {a}^{\dag }_{\ell }\right] = 0, \quad j,\ell = 1,2; \end{aligned}$$

and such that

$$\begin{aligned} \textrm{clos}(\ker \mathfrak {a}_1 \cap \ker \mathfrak {a}_2) = \mathcal {F}^2(\mathbb {C}^2) \quad \textrm{and} \quad \mathcal {F}^2_{(\textbf{k})}(\mathbb {C}^2) = (\mathfrak {a}^{\dag }_1)^{k_1-1}(\mathfrak {a}^{\dag }_2)^{k_2-1}\mathcal {F}^2(\mathbb {C}^2). \end{aligned}$$

Corollary 6.6

The Hilbert space $L_2(\mathbb {C}^2, d\mu (\varvec{z}))$ can not be represented as the orthogonal sum of all its true-poly-Fock spaces. We have just the following representations of $L_2(\mathbb {C}^2, d\mu (\varvec{z}))$ in terms of poly-Fock and true-poly-Fock spaces

$$\begin{aligned} L_2(\mathbb {C}^2, d\mu (\varvec{z}))= \textrm{clos}\, \left( \bigcup _{\textbf{k}\in \mathbb {N}^2} \mathcal {F}^2_{ \textbf{k}}(\mathbb {C}^2)\right) = \textrm{clos}\,\left( \underset{\textbf{k}\in \mathbb {N}^2}{\mathop {\dot{\sum }}\limits } \mathcal {F}^2_{(\textbf{k})}(\mathbb {C}^2)\right) . \end{aligned}$$

Appendix

We give here the detailed proof of Lemma 6.3.

To prove the non-orthogonality of $\mathcal {F}^2_{(\varvec{1}+ \varvec{e}_1)}(\mathbb {C}^2)$ and $\mathcal {F}^2_{(\varvec{1}+ \varvec{e}_2)}(\mathbb {C}^2)$, consider $\overline{z}_1z_1 -\frac{\sqrt{\pi }}{4} \in \mathcal {F}^2_{(\varvec{1}+ \varvec{e}_1)}(\mathbb {C}^2)$ and $\overline{z}_2z_2 -\frac{\sqrt{\pi }}{4} \in \mathcal {F}^2_{(\varvec{1}+ \varvec{e}_2)}(\mathbb {C}^2)$, and calculate

$$\begin{aligned} \left\langle \overline{z}_1z_1 -\frac{\sqrt{\pi }}{4}, \overline{z}_2z_2 -\frac{\sqrt{\pi }}{4} \right\rangle = \frac{1}{6} - \frac{\pi }{16} \ne 0. \end{aligned}$$

To prove the second statement of the lemma we need to check that the minimal angle between $\mathcal {F}^2_{(\varvec{1}+ \varvec{e}_1)}(\mathbb {C}^2)$ and $\mathcal {F}^2_{(\varvec{1}+ \varvec{e}_2)}(\mathbb {C}^2)$ is greater than zero (see Lemma 3.7), that is to check that

$$\begin{aligned} \sup \left\{ \frac{|\langle x, y \rangle |}{\Vert x\Vert \Vert y\Vert }: \ x \in \mathcal {F}^2_{(\varvec{1}+ \varvec{e}_1)}(\mathbb {C}^2), \ y \in \mathcal {F}^2_{(\varvec{1}+ \varvec{e}_2)}(\mathbb {C}^2) \right\} < 1. \end{aligned}$$

An easy calculation, based on (6.3), shows that the orthonormal bases of the spaces $\mathcal {F}^2_{(\varvec{1}+ \varvec{e}_1)}(\mathbb {C}^2)$ and $\mathcal {F}^2_{(\varvec{1}+ \varvec{e}_2)}(\mathbb {C}^2)$ are formed by the normalising elements

$$\begin{aligned}{} & {} \overline{z}_1\varvec{z}^{\textbf{q}}- \frac{q_1}{|\textbf{q}|+1}\frac{\Gamma (\frac{|\textbf{q}|}{2}+1)}{\Gamma (\frac{|\textbf{q}|}{2}+\frac{1}{2})} \varvec{z}^{\textbf{q}-\varvec{e}_1}, \ \ \textbf{q}\in \mathbb {N}\times \mathbb {Z}, \\{} & {} \overline{z}_2\varvec{z}^{\textbf{q}}- \frac{q_2}{|\textbf{q}|+1}\frac{\Gamma (\frac{|\textbf{q}|}{2}+1)}{\Gamma (\frac{|\textbf{q}|}{2}+\frac{1}{2})}\varvec{z}^{\textbf{q}-\varvec{e}_2}, \ \ \textbf{q}\in \mathbb {Z}\times \mathbb {N}, \end{aligned}$$

respectively. It is easy to figure out then that the only pairs of basis elements of $\mathcal {F}^2_{(\varvec{1}+ \varvec{e}_1)}(\mathbb {C}^2)$ and $\mathcal {F}^2_{(\varvec{1}+ \varvec{e}_2)}(\mathbb {C}^2)$, which are not orthogonal, have (in a convenient notation) the form

$$\begin{aligned} e_{1,\textbf{q}} = \frac{x_{\textbf{q}}}{\Vert x_{\textbf{q}}\Vert }\qquad \textrm{and} \qquad e_{2,\textbf{q}} = \frac{y_{\textbf{q}}}{\Vert y_{\textbf{q}}\Vert }, \end{aligned}$$

with

$$\begin{aligned} x_{\textbf{q}}= & {} \overline{z}_1\varvec{z}^{\textbf{q}-\varvec{e}_2}- \frac{q_1}{|\textbf{q}|}\frac{\Gamma (\frac{|\textbf{q}|}{2}+\frac{1}{2})}{\Gamma (\frac{|\textbf{q}|}{2})} \varvec{z}^{\textbf{q}-\varvec{e}_1-\varvec{e}_2} \\ y_{\textbf{q}}= & {} \overline{z}_2\varvec{z}^{\textbf{q}-\varvec{e}_1}- \frac{q_2}{|\textbf{q}|}\frac{\Gamma (\frac{|\textbf{q}|}{2}+\frac{1}{2})}{\Gamma (\frac{|\textbf{q}|}{2})} \varvec{z}^{\textbf{q}-\varvec{e}_1-\varvec{e}_2}, \qquad \textbf{q}\in \mathbb {N}^2. \end{aligned}$$

That is the only pair of elements x, y, that are not orthogonal, are of the form

$$\begin{aligned} x= & {} \sum _{\textbf{q}\in \mathbb {N}^2} c_{\textbf{q}}e_{1,\textbf{q}}, \ \ \Vert x\Vert = \left( \sum _{\textbf{q}\in \mathbb {N}^2} |c_{\textbf{q}}|^2\right) ^{\frac{1}{2}}, \\ y= & {} \sum _{\textbf{q}\in \mathbb {N}^2} d_{\textbf{q}}e_{2,\textbf{q}}, \ \ \Vert y\Vert = \left( \sum _{\textbf{q}\in \mathbb {N}^2} |d_{\textbf{q}}|^2\right) ^{\frac{1}{2}}. \end{aligned}$$

Then

$$\begin{aligned}{} & {} \frac{|\langle x, y \rangle |}{\Vert x\Vert \Vert y\Vert } = \frac{1}{\Vert x\Vert \Vert y\Vert } \left| \left\langle \sum _{\textbf{q}\in \mathbb {N}^2} c_{\textbf{q}}e_{1,\textbf{q}}, \sum _{\textbf{p}\in \mathbb {N}^2} d_{\textbf{p}}e_{2,\textbf{p}}\right\rangle \right| \\{} & {} \quad = \ \frac{1}{\Vert x\Vert \Vert y\Vert } \left| \sum _{\textbf{q}\in \mathbb {N}^2} c_{\textbf{q}}\overline{d_{\textbf{q}}} \langle e_{1,\textbf{q}}, e_{2,\textbf{q}}\rangle \right| \\{} & {} \quad \le \ \frac{\sup \langle e_{1,\textbf{q}}, e_{2,\textbf{q}}\rangle }{\Vert x\Vert \Vert y\Vert } \sum _{\textbf{q}\in \mathbb {N}^2} |c_{\textbf{q}}\overline{d_{\textbf{q}}}| \le \sup _{\textbf{q}\in \mathbb {N}^2} \langle e_{1,\textbf{q}}, e_{2,\textbf{q}}\rangle \end{aligned}$$

Note that this inequality becomes equality on the family of pairs $e_{1,\textbf{q}}$, $e_{2,\textbf{q}}$, where $\textbf{q}\in \mathbb {N}^2$. That is, it is sufficient to estimate

$$\begin{aligned} \sup \left\{ \frac{|\langle x_{\textbf{q}}, y_{\textbf{q}} \rangle |}{\Vert x_{\textbf{q}}\Vert \Vert y_{\textbf{q}}\Vert }: \ \textbf{q}\in \mathbb {N}^2 \right\} . \end{aligned}$$

A bit lengthy calculation, again based on (6.3), shows that

$$\begin{aligned}{} & {} \langle x_{\textbf{q}}, y_{\textbf{q}} \rangle = \frac{\Gamma (q_1+1)\Gamma (q_2+1)}{\Gamma \left( \frac{|\textbf{q}|}{2} +1\right) \Gamma \left( |\textbf{q}| +2\right) } \\{} & {} \quad \times \ \left[ \Gamma \left( \frac{|\textbf{q}|}{2} +1\right) ^2 - \Gamma \left( \frac{|\textbf{q}|}{2} +\frac{1}{2}\right) \Gamma \left( \frac{|\textbf{q}|}{2} +\frac{3}{2}\right) \right] , \\{} & {} \ \Vert x_{\textbf{q}}\Vert ^2 = \frac{q_1\Gamma (q_1+1)\Gamma (q_2)}{\Gamma \left( \frac{|\textbf{q}|}{2} +1\right) \Gamma \left( |\textbf{q}| +2\right) } \\{} & {} \quad \times \ \left[ \frac{q_1+1}{q_1}\Gamma \left( \frac{|\textbf{q}|}{2} +1\right) ^2 - \Gamma \left( \frac{|\textbf{q}|}{2} +\frac{1}{2}\right) \Gamma \left( \frac{|\textbf{q}|}{2} +\frac{3}{2}\right) \right] , \\{} & {} \ \Vert y_{\textbf{q}}\Vert ^2 = \frac{q_2\Gamma (q_1)\Gamma (q_2+1)}{\Gamma \left( \frac{|\textbf{q}|}{2} +1\right) \Gamma \left( |\textbf{q}| +2\right) } \\{} & {} \quad \times \ \left[ \frac{q_2+1}{q_2}\Gamma \left( \frac{|\textbf{q}|}{2} +1\right) ^2 - \Gamma \left( \frac{|\textbf{q}|}{2} +\frac{1}{2}\right) \Gamma \left( \frac{|\textbf{q}|}{2} +\frac{3}{2}\right) \right] . \end{aligned}$$

Thus

$$\begin{aligned} \frac{|\langle x_{\textbf{q}}, y_{\textbf{q}} \rangle |}{\Vert x_{\textbf{q}}\Vert \Vert y_{\textbf{q}}\Vert } = \frac{|1 -G\left( \frac{|\textbf{q}|}{2}\right) |}{\left[ \frac{1}{q_1} +1 -G\left( \frac{|\textbf{q}|}{2}\right) \right] ^{\frac{1}{2}}\left[ \frac{1}{q_2} +1 -G\left( \frac{|\textbf{q}|}{2}\right) \right] ^{\frac{1}{2}}}, \end{aligned}$$

(6.4)

where

$$\begin{aligned} G(s):= \frac{\Gamma \left( s +\frac{1}{2}\right) \Gamma \left( s +\frac{3}{2}\right) }{\Gamma \left( s +1\right) ^2}. \end{aligned}$$

Let us analyse now the behavior of G(s), where half-integer positive number $s \in S:= \left\{ m + \frac{k}{2}: \, m \in \mathbb {N}, \ k =0,1 \right\}$ varies. Observe that

$$\begin{aligned} G(s+1) = \frac{\left( s+\frac{1}{2}\right) \left( s +\frac{3}{2}\right) }{(s+1)^2}G(s)\, < \, G(s), \end{aligned}$$

and, by [6, Formula 8.328.2], $\displaystyle {\lim _{s \rightarrow \infty } G(s) = 1}$, implying that $G(s) > 1$, for all $s \in S$. Then

$$\begin{aligned} G(s + \textstyle {\frac{1}{2}}) = \displaystyle {\frac{s+1}{s +\frac{1}{2}} \frac{1}{G(s)} \,> \, \frac{1}{G(s)}} \qquad \textrm{or} \qquad G(s + \textstyle {\frac{1}{2}})\displaystyle {G(s) >1}, \end{aligned}$$

implying that $G(s + \frac{1}{2}) < G(s)$, for all $s \in S$.

Thus the sequence $\{G(s)\}_{s \in S}$ strictly decreasing with $\displaystyle {\lim _{s \rightarrow \infty } G(s) = 1}$. This fact may have an independent interest.

In particular, this implies that $|1 -G\left( \frac{|\textbf{q}|}{2}\right) | = G\left( \frac{|\textbf{q}|}{2}\right) -1$.

Further, continuing to analyze formula (6.4), observe that for a fixed $|\textbf{q}| = q_1+q_2$, the function

$$\begin{aligned}{} & {} \left[ \textstyle {\frac{1}{q_1} +1 -G\left( \frac{|\textbf{q}|}{2}\right) }\right] \left[ \textstyle {\frac{1}{q_2} +1 -G\left( \frac{|\textbf{q}|}{2}\right) }\right] \\{} & {} \quad = \ \left[ \textstyle {\frac{1}{q_1} +1 -G\left( \frac{|\textbf{q}|}{2}\right) }\right] \left[ \textstyle {\frac{1}{|\textbf{q}|-q_1} +1 -G\left( \frac{|\textbf{q}|}{2}\right) }\right] \\{} & {} \quad = \ \textstyle {\frac{C'(|\textbf{q}|)}{q_1(|\textbf{q}|-q_1)}} +C''(|\textbf{q}|), \end{aligned}$$

where $q_1 \in [1,|\textbf{q}|]$ and $C',\, C''$ are constants (depending on $|\textbf{q}|$), attains the minimum at $q_1 = \frac{|\textbf{q}|}{2}$. Thus

$$\begin{aligned} \frac{|\langle x_{\textbf{q}}, y_{\textbf{q}} \rangle |}{\Vert x_{\textbf{q}}\Vert \Vert y_{\textbf{q}}\Vert }{} & {} = \frac{G\left( \frac{|\textbf{q}|}{2}\right) -1}{\left[ \frac{1}{q_1} +1 -G\left( \frac{|\textbf{q}|}{2}\right) \right] ^{\frac{1}{2}}\left[ \frac{1}{q_2} +1 -G\left( \frac{|\textbf{q}|}{2}\right) \right] ^{\frac{1}{2}}} \\{} & {} \le \ \frac{G\left( \frac{|\textbf{q}|}{2}\right) -1}{\frac{2}{|\textbf{q}|} +1 -G\left( \frac{|\textbf{q}|}{2}\right) } =\frac{\frac{|\textbf{q}|}{2}\left( G\left( \frac{|\textbf{q}|}{2}\right) -1\right) }{1 -\frac{|\textbf{q}|}{2}\left( G\left( \frac{|\textbf{q}|}{2}\right) -1\right) }. \end{aligned}$$

Our aim now is to calculate $\displaystyle {\lim _{s\rightarrow \infty }} s(G(s)-1)$. Recall (see e.g. [6, Formula 8.327.1]) the asymptotics of $\Gamma (s)$ for large values of s,

$$\begin{aligned} \Gamma (s) = s^{s-\frac{1}{2}}e^{-s}\sqrt{2\pi }\left( 1 +\frac{1}{12s} + O(s^{-2})\right) . \end{aligned}$$

Thus

$$\begin{aligned} G(s){} & {} = \frac{\Gamma \left( s +\frac{1}{2}\right) \Gamma \left( s +\frac{3}{2}\right) }{\Gamma \left( s +1\right) ^2} = \ \frac{(s+\frac{1}{2})^{s}(s+\frac{3}{2})^{s+1}}{(s+1)^{2s+1}} \\{} & {} \quad \times \ \frac{\left( 1 + \frac{1}{12(s+\frac{1}{2})} +O(s^{-2})\right) \left( 1 + \frac{1}{12(s+\frac{3}{2})} +O(s^{-2})\right) }{\left( 1 + \frac{1}{12(s+1)} +O(s^{-2})\right) ^2} \\{} & {} = \ \left( \frac{s+\frac{1}{2}}{s+1}\right) ^s \left( \frac{s+\frac{3}{2}}{s+1}\right) ^{s+1}\, \frac{1 + \frac{1}{6s} +O(s^{-2})}{1 + \frac{1}{6s} +O(s^{-2})} \\{} & {} = \ \exp \left[ s\ln \left( 1-\frac{1}{2(s+1)}\right) \right] \\{} & {} \quad \times \ \exp \left[ (s+1)\ln \left( 1+\frac{1}{2(s+1)}\right) \right] \left( 1+O(s^{-2})\right) \\{} & {} = \ \exp \left[ -\frac{s}{2(s+1)} -\frac{s}{4(s+1)^2}+O(s^{-2})\right] \\{} & {} \quad \times \ \exp \left[ \frac{s+1}{2(s+1)} -\frac{s+1}{4(s+1)^2}+O(s^{-2})\right] \left( 1 +O(s^{-2})\right) \\{} & {} = \ \exp \left[ -\frac{1}{2} + \frac{1}{2(s+1)} - \frac{1}{4(s+1)}+O(s^{-2})\right] \\{} & {} \quad \times \ \exp \left[ \frac{1}{2} - \frac{1}{4(s+1)}+O(s^{-2})\right] \left( 1 +O(s^{-2})\right) \\{} & {} = \ \exp \left[ O(s^{-2})\right] \left( 1 +O(s^{-2})\right) = 1+O(s^{-2}). \end{aligned}$$

Thus $\displaystyle {\lim _{|\textbf{q}|\rightarrow \infty }} \textstyle {\frac{2}{|\textbf{q}|}}\left( G\left( \frac{|\textbf{q}|}{2}\right) -1\right) = 0$, implying

$$\begin{aligned} \lim _{|\textbf{q}|\rightarrow \infty } \frac{|\langle x_{\textbf{q}}, y_{\textbf{q}} \rangle |}{\Vert x_{\textbf{q}}\Vert \Vert y_{\textbf{q}}\Vert } = 0. \end{aligned}$$

That is

$$\begin{aligned} \sup _{\textbf{q}\in \mathbb {N}^2} \frac{|\langle x_{\textbf{q}}, y_{\textbf{q}} \rangle |}{\Vert x_{\textbf{q}}\Vert \Vert y_{\textbf{q}}\Vert } \end{aligned}$$

is reached at some value $\textbf{q}'=(q_1',q_2')$, but

$$\begin{aligned} \frac{|\langle x_{\textbf{q}'}, y_{\textbf{q}'} \rangle |}{\Vert x_{\textbf{q}'}\Vert \Vert y_{\textbf{q}'}\Vert } \ne 1 \end{aligned}$$

since (see (6.4) and [6, Formula 8.339.2]) it is an irrational number (contains $\pi$).

This finishes the proof of Lemma 6.3.