Isometries, direct sum decompositions, analytic type function spaces, and radial operators

Abstract

The paper connects rather general statements about the direct sum decomposition of a Hilbert space generated by isometries with the study of analytic type subspaces in weighted $L_2$-spaces over the unit disk ${\mathbb {D}}$ or the complex plane ${\mathbb {C}}$, and with characterization of the so-called radial operators acting on these spaces. We characterize the properties of the so-called (m, n)-analytic functions, those that are annihilated by $\frac{\partial ^m }{\partial z^m}\frac{\partial ^n }{\partial {{\overline{z}}}^n}$, which as particular cases include analytic, anti-analytic, harmonic, and various related to them functions. We characterize also the so-called radial operators, with a particular attention to Toeplitz ones, that act on different subspaces of our weighted $L_2$-spaces.

1 Introduction

The paper is based on an approach that connects rather general statements about the direct sum decomposition of a Hilbert space generated by isometries with the study of analytic type subspaces in weighted $L_2$-spaces over the unit disk ${\mathbb {D}}$ or the complex plane ${\mathbb {C}}$, and with characterization of the so-called radial operators acting on these spaces.

Recall that apart of the analytic functions, annihilated by $\frac{\partial }{\partial {{\overline{z}}}}$, there are many other important classes of functions intensively studied for decades, among which, in particular, anti-analytic, annihilated by $\frac{\partial }{\partial z}$, harmonic, annihilated by $\Delta = 4\frac{\partial }{\partial z}\frac{\partial }{\partial {{\overline{z}}}}$, and their derivatives, polyanalytic, anti-polyanalytic, biharmonic, etc. All of them are specific cases of the so-called (m, n)-analytic functions, those that are annihilated by $\frac{\partial ^m }{\partial z^m}\frac{\partial ^n }{\partial {{\overline{z}}}^n}$.

Our aim is to characterize the structure and properties of spaces of the above functions, considered as the subspaces of various $L_2$-spaces in a way that does not depend, as much as possible, on the particular $L_2$-space under consideration.

We are considering weighted Hilbert spaces $L_2(D,d\nu )$, where D is either the unit disk ${\mathbb {D}}$ or the complex plane $\mathbb {C}$, with a probability measure $d\nu (z) = \omega (|z|)dA(z)$, where $dA(z)= \frac{1}{\pi }\textrm{d}x\textrm{d}y$, $z=x+iy$, and whose radial weight function $\omega : D \rightarrow {\mathbb {R}}_+$ is such that the linear span of the monomials $m_{p,q}:=z^p{{\overline{z}}}^q$, for all $p,\,q \in {\mathbb {Z}}_+$, is dense in $L_2(D,d\nu )$. We mention two classical cases: $\omega (|z|) = (\lambda +1)(1-|z|^2)^{\lambda }$, $\lambda > -1$, in the unit disk ${\mathbb {D}}$, and $\omega (|z|) = \alpha e^{-\alpha |z|^2}$, $\alpha > 0$ in ${\mathbb {C}}$. Other examples are given, e.g., in recent papers [3,4,5,6,7,8], where non-standardly weighted spaces related to analytic functions were considered.

The orthogonal bases in these spaces are given by the polynomials $e_{p,q}$, $p,\, q \in {\mathbb {Z}}_+$, of degree p in z and q in ${{\overline{z}}}$, exact formulas for which are given in (3.3) and (3.4).

Given any pair $(m,n) \in {\mathbb {Z}}_+^2 \setminus \{(0,0)\}$, we introduce the pure isometry $W_{m,n}$ on $L_2(D,d\nu )$, defining it on the basis elements by $W_{m,n}: e_{p,q} \mapsto e_{p+m,q+n}$. That is, in Fig. 1 of Sect. 2, the isometry $W_{m,n}$ moves each node (p, q) $(\, \equiv e_{p,q})$ p steps right and q steps up to the node $(p+m,q+n)$ $(\, \equiv e_{p+m,q+n})$.

Being pure isometry, $W_{m,n}$ generates the direct sum decomposition of $L_2(D,d\nu )$

$$\begin{aligned} L_2(D,d\nu ) = \bigoplus _{\ell \in {\mathbb {Z}}_+} W_{m,n}^{\ell }(\ker W_{m,n}^*). \end{aligned}$$

It turns out (Proposition 3.5) that $\ker W_{m,n}^*$ coincides exactly with the closed subspace ${\mathcal {A}}^{(m,n)}$ of $L_2(D,d\nu )$, which consists of (m, n)-analytic functions. We mention as well that $\ker W_{km,kn}^* = \ker (W_{m,n}^*)^k$ coincides with the closed subspace ${\mathcal {A}}_k^{(m,n)} = {\mathcal {A}}^{(km,kn)}$ of $L_2(D,d\nu )$, which consists of k-(m, n)-polyanalytic or (km, kn)-analytic functions.

By Corollary 3.2, a function is k-(m, n)-polyanalytic if and only if it admits the representation as a polynomial on $z^m{{\overline{z}}}^n$ of degree $k-1$ with (m, n)-analytic coefficients.

The above direct sum decomposition leads (Proposition 3.6) to the following assertion. Given any predefined “analytic quality” of functions, characterized by the pair $(m,n) \in {\mathbb {Z}}_+^2 \setminus \{(0,0)\}$, we have

$$\begin{aligned} L_2(D, d\nu ) = \bigoplus _{k \in {\mathbb {N}}} {\mathcal {A}}_{(k)}^{(m,n)}, \end{aligned}$$

where ${\mathcal {A}}_{(k)}^{(m,n)} = {\mathcal {A}}_k^{(m,n)} \ominus {\mathcal {A}}_{k-1}^{(m,n)} = W_{m,n}^{k-1}(\ker W_{m,n}^*)$ are the so-called true-k-(m, n)-polyanalytic function spaces. This result was already known only for polyanalytic, anti-polyanalytic, and polyharmonic functions.

Apart of the pure isometries $W_{m,n}$, let us introduce the unitary isometry (bilateral shift) U, defined on basis elements as follows:

$$\begin{aligned} U : \ e_{p,q} \ \longmapsto \ {\left\{ \begin{array}{ll} e_{p+1,q}, &{} \textrm{if} \ p \ge q \\ e_{p,q-1}, &{} \textrm{if} \ p < q. \end{array}\right. } \end{aligned}$$

Then

$$\begin{aligned} L_2(D, d\nu ) = \bigoplus _{n \in {\mathbb {Z}}}U^n({\mathcal {V}}_0), \end{aligned}$$

where the wandering subspace is taken as

$$\begin{aligned} {\mathcal {V}}_0 = \overline{\textrm{span}}\,\{e_{p,q} : p-q = 0\}. \end{aligned}$$

Note that for each $n \in {\mathbb {Z}}$

$$\begin{aligned} U^n({\mathcal {V}}_0) = {\mathcal {V}}_n = \overline{\textrm{span}}\,\{e_{p,q} : p-q = n\}. \end{aligned}$$

These spaces turn out to be important in Sect. 4 under the study of the so-called radial operators, those that commute with rotation operators $\varpi (t) : f(z) \in {\mathcal {H}} \mapsto f(t^{-1}z)$, for all $t \in {\mathbb {T}}$.

Let now ${\mathcal {H}}_0$ be an infinite-dimensional Hilbert subspace of ${\mathcal {H}}$, whose basis is a certain subset of the basis $\{e_{p,q}\}_{p,q \in {\mathbb {Z}}_+}$ in ${\mathcal {H}}$. We characterize the radial operators on such ${\mathcal {H}}_0$, paying particular attention to the case of Toeplitz operators.

Recall that the Toeplitz operator $T_a$, with symbol $a \in L_{\infty }(D)$ acting on ${\mathcal {H}}_0$, is the compression onto ${\mathcal {H}}_0$ of the multiplication by a operator, acting on ${\mathcal {H}}$, i.e., $T_a \varphi = {{\textbf{P}}}(a \varphi )$, where ${{\textbf{P}}}$ is the orthogonal projection of ${\mathcal {H}}$ onto ${\mathcal {H}}_0$.

In many already known cases (when, for example, ${\mathcal {H}}_0$ is either the space of analytic, or anti-analytic, or harmonic functions), the Toeplitz operator with $L_{\infty }$-symbol is radial if and only if its symbol a is a radial function, i.e., $a(z) = a(|z|)$. At the same time, the situation for an arbitrary subspace ${\mathcal {H}}_0$ is quite different; the result depends upon the subspace ${\mathcal {H}}_0$ in question.

Proposition 4.4 states that the Toeplitz operator $T_a$, with symbol $a \in L_{\infty }(D)$, is radial if and only if the function a admits the representation

$$\begin{aligned} a = b + h, \qquad \textrm{where} \ \ b \ \ \mathrm {is \ radial} \ \ \textrm{and} \ \ h \in \widehat{{\mathcal {H}}}_0^{\perp }, \end{aligned}$$

where

$$\begin{aligned} \widehat{{\mathcal {H}}}_0 = \overline{\textrm{span}}\,\left\{ \overline{e'}e'' : \ \ \mathrm {for \ all \ basis \ elements} \ \ e', \, e'' \in {\mathcal {H}}_0\right\} . \end{aligned}$$

Depending on ${\mathcal {H}}_0$, the set of symbols, generating radial Toeplitz operators, may be wider that a class of radial functions (Example 4.7), or even coincide with a whole $L_{\infty }(D)$ (Example 4.8).

We finish the paper with an alternative description of the radial operators, including Toeplitz, acting in ${\mathcal {A}}$, based on the spectral theorem. Corollary 4.12 characterizes all of them as functions $\varphi (N)$ of the unbounded self-adjoint operator $N = z\frac{\partial }{\partial z}$.

2 Isometries and direct sum decompositions

We start from recalling some well-known facts.

Let V be an isometry on a Hilbert space ${\mathcal {H}}$, i.e., $V^*V = I$. The image of V is always closed in ${\mathcal {H}}$, and let ${\mathcal {L}} = \ker V^* = {\mathcal {H}} \ominus V({\mathcal {H}}) = (\textrm{Im}\,V)^\perp$. As known (and easily follows from $V^*V = I$), $V^m({\mathcal {L}})$ is orthogonal to $V^n({\mathcal {L}})$ for all integers $m \ne n$ in ${\mathbb {Z}}_+ := \{0, 1, 2, \ldots \}$.

Recall that an isometry V is called a pure isometry, or a unilateral shift if

$$\begin{aligned} M_+({\mathcal {L}}) := \bigoplus _{n \in {\mathbb {Z}}_+} V^n({\mathcal {L}}) = {\mathcal {H}}. \end{aligned}$$

The structure of any isometry is characterized by the following theorem; see, e.g., [20, Theorem 1.1]

Theorem 2.1

(Wold decomposition) Let V be an isometry on the Hilbert space ${\mathcal {H}}$. Then, ${\mathcal {H}}$ decomposes into an orthogonal sum ${\mathcal {H}} = {\mathcal {H}}' \oplus {\mathcal {H}}''$, such that ${\mathcal {H}}'$ and ${\mathcal {H}}''$ are invariant for both V and $V^*$, and the restriction of V onto ${\mathcal {H}}'$ is a unitary operator, while the restriction of V onto ${\mathcal {H}}''$ is a pure isometry.

This decomposition is uniquely determined with

$$\begin{aligned} {\mathcal {H}}' = \bigcap _{n \in {\mathbb {Z}}_+} V^n({\mathcal {H}}) \quad \textrm{and} \quad {\mathcal {H}}'' = M_+({\mathcal {L}}), \quad \textrm{where} \quad {\mathcal {L}} = \ker V^*. \end{aligned}$$

One of the spaces ${\mathcal {H}}'$ or ${\mathcal {H}}''$ may be absent, that is, equal to $\{0\}$.

2.1 Pure isometry

In this subsection, we assume that the isometry V under consideration is pure, i.e., the subspace ${\mathcal {H}}'$ is trivial, or, which is the same, ${\mathcal {H}} = {\mathcal {H}}''$.

The next result is a slight modification of [23, Proposition 4.1]. It states that the three objects of the theorem, which always exist in any separable Hilbert space, are tightly connected to each other; any one of them defines another two.

Theorem 2.2

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

(i):

there is a pure isometry V in ${\mathcal {H}}$;

(ii):

the Hilbert space ${\mathcal {H}}$ admits the direct sum decomposition

$$\begin{aligned} {\mathcal {H}} = \bigoplus _{k=1}^{\infty } {\mathcal {H}}_{(k)}, \end{aligned}$$

(2.1)

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

(iii):

there are two mutually adjoint lowering and raising (unbounded) operators ${\mathfrak {a}}$ and ${\mathfrak {a}}^\dag$ having common domain dense in ${\mathcal {H}}$, such that the following commutation relation holds:

$$\begin{aligned}{}[{\mathfrak {a}}, {\mathfrak {a}}^\dag ] = I, \quad \textrm{where} \quad [{\mathfrak {a}}, {\mathfrak {a}}^\dag ] = {\mathfrak {a}}{\mathfrak {a}}^\dag -{\mathfrak {a}}^\dag {\mathfrak {a}}, \end{aligned}$$

(2.2)

and the set of finite linear combinations of elements from all spaces $L_n := ({\mathfrak {a}}^\dag )^{n-1} L_1$, with $L_1 = \ker {\mathfrak {a}}$, is dense in ${\mathfrak {H}}$.

Moreover, the subspaces ${\mathcal {H}}_{(k)}$ in (2.1) are related to the operators V, ${\mathfrak {a}}$ and ${\mathfrak {a}}^\dag$ as follows:

$$\begin{aligned} {\mathcal {H}}_{(1)} = \ker V^* = \ker {\mathfrak {a}} \quad \textrm{and}\quad {\mathcal {H}}_{(k)} = V^{k-1}(\ker V^*) = ({\mathfrak {a}}^\dag )^{k-1}(\ker {\mathfrak {a}}), \quad \textrm{for} \ \ k > 1. \end{aligned}$$

For completeness, we sketch the proof of the theorem, but before we make an observation.

Observation 2.3

Having in mind further application of the results of this section to the spaces ${\mathcal {H}}$ being weighted $L_2$ spaces in the domains $D \subseteq {\mathbb {C}}$, where orthonormal bases are naturally indexed by pairs (p, q) of the highest degrees of variables z and ${{\overline{z}}}$, we will denote by $\{e_{p,k-1}\}_{p \in N}$, with $N =\{0,1,\ldots ,n\}$, the basis of the subspace ${\mathcal {H}}_{(k)}$ of decomposition (2.1). As all spaces ${\mathcal {H}}_{(k)}$ have the same dimension, the index set N does not depend on k.

Proof

(i) $\Rightarrow$ (ii) Let V be a pure isometry in ${\mathcal {H}}$, then we have decomposition (2.1) with ${\mathcal {H}}_{(k)} = V^{k-1}(\ker V^*)$.

(ii) $\Rightarrow$ (i) We define the pure isometry V by the following action on basis elements of ${\mathcal {H}}$:

$$\begin{aligned} V : \, e_{p,q} \ \longmapsto \ e_{p,q+1}, \ \ p \in N, \ q \in {\mathbb {Z}}_+. \end{aligned}$$

(ii) $\Rightarrow$ (iii) This part has already been proven in [23, Proposition 4.1]. We repeat this proof here for completeness. Define ${\mathfrak {a}}$ and ${\mathfrak {a}}^\dag$ by their action on basis elements of ${\mathcal {H}}$ as the weighted versions of the operators $V^*$ and V

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

which satisfy (2.2) on all elements $e_{p,q}$

$$\begin{aligned}{}[{\mathfrak {a}}, {\mathfrak {a}}^\dag ]e_{p,q} = {\left\{ \begin{array}{ll} [q - (q-1)]e_{p,q} = e_{p,q}, &{} p \in N, \ q>0 \\ (1 - 0)e_{p,q} = e_{p,q}, &{} p \in N, \ q=0 \end{array}\right. }. \end{aligned}$$

It is easy to figure out that thus defined operators admit the extension to the common dense in ${\mathcal {H}}$ domain $D_0$

$$\begin{aligned} D_0 = \left\{ f = \sum _{k=1}^{\infty } f_k e_{p_k,q_k} \in {\mathcal {H}} : \ \sum _{k=1}^{\infty } q_k |f_k|^2 < \infty \right\} , \end{aligned}$$

by

$$\begin{aligned} {\mathfrak {a}}f= & {} \sum _{k=1}^{\infty } \sqrt{q_k-1}\,f_k e_{p_k,q_k-1}, \quad \textrm{with} \quad q_k-1 =0 \ \ \textrm{if} \ \ q_k =0, \\ {\mathfrak {a}}^\dag f= & {} \sum _{k=1}^{\infty }\sqrt{q_k}\,f_k e_{p_k,q_k+1}. \end{aligned}$$

Further, it is easy to figure out that the operators ${\mathfrak {a}}$ and ${\mathfrak {a}}^\dag$ are mutually adjoint to each other on $D_0$, satisfy (2.2), and the set of finite linear combinations of elements from all spaces $L_n := ({\mathfrak {a}}^\dag )^{n-1} \ker {\mathfrak {a}}$ is dense in ${\mathfrak {H}}$.

(iii) $\Rightarrow$ (iii) The operator ${\mathfrak {a}}$, being adjoint to ${\mathfrak {a}}^\dag$, is closed, which implies that its kernel $L_1 = \ker {\mathfrak {a}}$ is a closed subspace in ${\mathcal {H}}$. By [22, Proposition 2.2, Corollary 2.3], different subspaces $L_n$ and $L_m$ are orthogonal, and for each $k = 2,3, \ldots$, the weighted raising operator

$$\begin{aligned} \textstyle {\frac{1}{\sqrt{k-1}}}\,{\mathfrak {a}}^\dag |_{L_{k-1}} : L_{k-1} \ \longrightarrow \ L_k \end{aligned}$$

is an isometric isomorphism, while the weighted lowering operator

$$\begin{aligned} \textstyle {\frac{1}{\sqrt{k-1}}}\,{\mathfrak {a}}|_{L_k} : L_k \ \longrightarrow \ L_{k-1} \end{aligned}$$

is its inverse. Then, by [22, Proposition 2.4],

$$\begin{aligned} {\mathcal {H}} = \bigoplus _{k=1}^{\infty } L_k, \end{aligned}$$

giving (2.1) with ${\mathcal {H}}_{(k)} = L_k$. $\square$

Guided by the terminology used in the theory of polyanalytic function spaces, we adopt the following. The subspaces ${\mathcal {H}}_{(k)}$ in the direct sum decomposition (2.1) are defined in terms of the pure isometry V and its adjoint $V^*$, or in terms of the operator ${\mathfrak {a}}$ and its adjoint ${\mathfrak {a}}^\dag$. Note that

$$\begin{aligned} \ker (V^*)^k = \ker {\mathfrak {a}}^k = {\mathcal {H}}_{(1)} \oplus {\mathcal {H}}_{(2)} \oplus \ldots \oplus {\mathcal {H}}_{(k)}. \end{aligned}$$

We denote this space by ${\mathcal {H}}_k$ and call it k-poly-V-space or k-poly-${\mathfrak {a}}$-space, then ${\mathcal {H}}_{(k)} = {\mathcal {H}}_k \ominus {\mathcal {H}}_{k-1}$ is naturally to call true-k-poly-V-space or true-k-poly-${\mathfrak {a}}$-space, with the convention that ${\mathcal {H}}_{(1)} = {\mathcal {H}}_1$.

Having the lowering and raising operators ${\mathfrak {a}}$ and ${\mathfrak {a}}^\dag$ satisfying (2.2), we can reveal a Lie-algebraic nature of the k-poly-V-space ${\mathcal {H}}_k$; see [22, Section 2] for more details. Indeed, for all $k \in {\mathbb {N}}$, we consider three unbounded, densely defined on finite linear combinations of basis elements in ${\mathcal {H}}$, operators; see [21, Formulas (A.1.5)], or [22, Formula (2.11)]

$$\begin{aligned} J^+_k= & {} ({\mathfrak {a}}^\dag )^2 {\mathfrak {a}} - (k-1)\, {\mathfrak {a}}^\dag , \\ J^0_k= & {} {\mathfrak {a}}^\dag {\mathfrak {a}} - \textstyle {\frac{k-1}{2}}I , \\ J^-_k= & {} {\mathfrak {a}} . \end{aligned}$$

These operators obey the ${{\mathfrak {s}}}{{\mathfrak {l}}}(2)$-algebra commutation relations: $[J^-_k, J^+_k]=2 J^0_k$ and $[J^{\pm }_n, J^0_k]= \mp J^{\pm }_k$, and as it is easily seen, the space ${\mathcal {H}}_k$ is invariant under the action of these operators.

Remark 2.4

That is, the k-poly-V-space ${\mathcal {H}}_k$ can be defined alternatively in Lie-algebraic way as the maximal by inclusion (see [22, Observation 2.9]) invariant subspaces for the action the operators $J^+_k$, $J^0_k$, $J^-_k$ in ${\mathcal {H}}$, obeying the ${{\mathfrak {s}}}{{\mathfrak {l}}}(2)$-algebra commutation relations.

An alternative form of decomposition (2.1) is as follows. The set of the k-poly-V-spaces ${\mathcal {H}}_k$, $k\in {\mathbb {N}}$, forms an infinite flag in ${\mathcal {H}}$

$$\begin{aligned} {\mathcal {H}}_1 \ \subset \ {\mathcal {H}}_2\ \subset \ \cdots \ \subset \ {\mathcal {H}}_k \subset \ \cdots \subset \ {\mathcal {H}}, \end{aligned}$$

and

$$\begin{aligned} {\mathcal {H}} = \overline{\bigcup _{k \in {\mathbb {N}}} {\mathcal {H}}_{k}}. \end{aligned}$$

In what follows we assume that all spaces ${\mathcal {H}}_{(k)}$ in decomposition (2.1) are infinite-dimensional, and thus, basis elements $e_{p,q}$ of ${\mathcal {H}}$ are indexed by pairs $(p,q) \in {\mathbb {Z}}_+^2$.

In Fig. 1, we identify each basis element $e_{p,q}$ with the node (p, q) of the lattice ${\mathbb {Z}}_+^2$, which may serve for a visualization of spaces and operators, defined in terms of basis elements.

Fig. 1

Lattice of basis elements

Besides the isometry V, which moves each note one-step up in the above figure, we introduce [23, Formula (3.7)] another pure isometry ${\widetilde{V}}$, which moves each node one-step right

$$\begin{aligned} {\widetilde{V}} : \, e_{p,q} \ \longmapsto \ e_{p+1,q}, \end{aligned}$$

their corresponding subspaces $\widetilde{{\mathcal {H}}}_{(k)}$ for the decomposition

$$\begin{aligned} {\mathcal {H}} = \bigoplus _{k=1}^{\infty } \widetilde{{\mathcal {H}}}_{(k)}, \end{aligned}$$

as well as the corresponding operators $\widetilde{{\mathfrak {a}}}$ and $\widetilde{{\mathfrak {a}}}^\dag$, obeying the relation $[\widetilde{{\mathfrak {a}}}, \widetilde{{\mathfrak {a}}}^\dag ] = I$.

The subspaces ${\mathcal {H}}_{(k)}$ and $\widetilde{{\mathcal {H}}}_{(k)}$, being the closure of the linear span of $\{e_{p,q}\}_{p \in {\mathbb {Z}}_+,\, q = k-1}$ and $\{e_{p,q}\}_{p = k-1,\, q \in {\mathbb {Z}}_+}$, respectively, can be visualized in Fig. 1 as horizontal half-line $q=k-1$ and vertical half-line $p=k-1$, respectively.

The above two isometries V and ${\widetilde{V}}$ permit us to define an infinite family of other pure isometries, each one of which, according to Theorem 2.2, leads to the corresponding direct sum decomposition of ${\mathcal {H}}$ onto k-poly-spaces and to specific to this isometry operators ${\mathfrak {a}}$ and ${\mathfrak {a}}^\dag$.

Indeed, any pair $(m,n) \in {\mathbb {Z}}_+^2 \setminus \{(0,0)\}$ defines the pure isometry $W_{m,n} = {\widetilde{V}}^mV^n = V^n {\widetilde{V}}^m$.

In Fig. 1, the isometry $W_{m,n}$ moves each node (p, q) $(\, \equiv e_{p,q})$ p steps right and q steps up to the node $(p+m,q+n)$ $(\, \equiv e_{p+m,q+n})$.

In the next proposition, we collect some obvious properties of $W_{m,n}$ and related to $W_{m,n}$ objects.

Proposition 2.5

We have

(i):

$W_{0,1} = V$ and $W_{1,0} = {\widetilde{V}}$;

(ii):

${\mathcal {H}}^{(m,n)} = \ker W_{m,n}^* = \widetilde{{\mathcal {H}}}_m + {\mathcal {H}}_n = (\widetilde{{\mathcal {H}}}_m \ominus (\widetilde{{\mathcal {H}}}_m \cap {\mathcal {H}}_n)) \oplus ({\mathcal {H}}_n \ominus (\widetilde{{\mathcal {H}}}_m \cap {\mathcal {H}}_n)) \oplus (\widetilde{{\mathcal {H}}}_m \cap {\mathcal {H}}_n)$;

(iii):

${\mathcal {H}}_{(k)}^{(m,n)} = W_{m,n}^{k-1}(\ker W_{m,n}^*)$;

(iv):

${\mathcal {H}}_{k}^{(m,n)} = \ker W_{km,kn}^*$;

(v):

${\mathcal {H}} = \bigoplus _{k \in {\mathbb {N}}} {\mathcal {H}}_{(k)}^{(m,n)} = \overline{\bigcup _{k \in {\mathbb {N}}} \ker W_{km,kn}^*}$.

Remark 2.6

The last item of the proposition says that, for any pair $(m,n) \in {\mathbb {Z}}_+^2 \setminus \{(0,0)\}$, the Hilbert space ${\mathcal {H}}$ can be either represented as a direct sum of its true-k-poly-$W_{m,n}$-subspaces ${\mathcal {H}}_{(k)}^{(m,n)}$, or via the flag if its k-poly-$W_{m,n}$-subspaces ${\mathcal {H}}_{k}^{(m,n)}$ ( $\equiv$ flag of its $W_{km,kn}$-subspaces ${\mathcal {H}}^{(km,kn)}\,$).

The above two isometries V and ${\widetilde{V}}$ permit us to define also two mutually adjoint partial isometries

$$\begin{aligned} S = V{\widetilde{V}}^*= {\widetilde{V}}^*V \quad \textrm{and} \quad S^* = {\widetilde{V}}V^* = V^*{\widetilde{V}}, \end{aligned}$$

which act on the basis elements [23, Formulas (3.3) and (3.4)] as follows:

$$\begin{aligned} S e_{p,q} = {\left\{ \begin{array}{ll} e_{p-1,q+1}, &{} \textrm{if} \quad p \ge 1 \\ 0, &{} \textrm{if} \quad p = 0 \end{array}\right. } \qquad \textrm{and} \qquad S^* e_{p,q} = {\left\{ \begin{array}{ll} e_{p+1,q-1}, &{} \textrm{if} \quad q \ge 1 \\ 0, &{} \textrm{if} \quad q = 0 \end{array}\right. }. \end{aligned}$$

In Fig. 1, the operator S moves each node to the nearest up-left node, while the operator $S^*$ moves each node to the nearest down-right node, with the convention that the result is 0 when the destination node lies outside of ${\mathbb {Z}}_+^2$.

That is, the operator S is a partial isometry with $\ker S = \widetilde{{\mathcal {H}}}_1$ and $(\textrm{Im}\, S)^{\perp } = {\mathcal {H}}_1$, while its adjoint operator $S^*$ is a partial isometry with $\ker S^* ={\mathcal {H}}_1$ and $(\textrm{Im}\, S^*)^{\perp } = \widetilde{{\mathcal {H}}}_1$.

Some benefit from the introduction of operators S and $S^*$ is shown in the following proposition.

Proposition 2.7

For any pair $(m,n) \in {\mathbb {Z}}_+^2 \setminus \{(0,0)\}$

(i):

the space ${\mathcal {H}}^{(m,n)} = \ker W_{m,n}^*$ admits the representation

$$\begin{aligned} {\mathcal {H}}^{(m,n)} = \ker S^{m+n} (S^*)^n = \ker \, (S^*)^{m+n}S^m; \end{aligned}$$

(ii):

the orthogonal projection $P_{m,n} : \, {\mathcal {H}} \longrightarrow {\mathcal {H}}^{(m,n)}$ has the form

$$\begin{aligned} P_{m,n} = I - S^n(S^*)^{m+n}S^m = I - (S^*)^m S^{m+n}(S^*)^n. \end{aligned}$$

Proof

(i): By item (2) of Proposition 2.5

$$\begin{aligned} {\mathcal {H}}^{(m,n)} = \overline{\textrm{span}}\,\{ e_{p,q} : \ \mathrm {such \ that} \ \ p< m, \ \ \textrm{or} \ \ q < n\}. \end{aligned}$$

Given any $(p,q) \in {\mathbb {Z}}_+^2$, calculate

$$\begin{aligned} S^{m+n} (S^*)^n e_{p,q} = {\left\{ \begin{array}{ll} S^{m+n}e_{p+n,q-n}, &{} q \ge n\\ 0, &{} q< n \end{array}\right. } \ = \ {\left\{ \begin{array}{ll} e_{p-m,q+m}, &{} q \ge n \ \textrm{and} \ p \ge m \\ 0, &{} p< m \\ 0, &{} q < n \end{array}\right. }, \end{aligned}$$

and the result follows.

Analogously

$$\begin{aligned} (S^*)^{m+n}S^m e_{p,q} = {\left\{ \begin{array}{ll} (S^*)^{m+n}e_{p+n,q-n}, &{} p \ge m\\ 0, &{} p< m \end{array}\right. } \ = \ {\left\{ \begin{array}{ll} e_{p-m,q+m}, &{} q \ge n \ \textrm{and} \ p \ge m \\ 0, &{} p< m \\ 0, &{} q < n \end{array}\right. } , \end{aligned}$$

which finishes the proof of (1).

(ii): By the standard property of isometries, the orthogonal projection onto $\ker W_{m,n}^*$ is given by $P_{m,n} = I - W_{m,n}W_{m,n}^*$. Thus

$$\begin{aligned} P_{m,n}= & {} I - V^n{\widetilde{V}}^m (V^*)^n({\widetilde{V}}^*)^m = I - V^n [({\widetilde{V}}^*)^n{\widetilde{V}}^n]{\widetilde{V}}^m (V^*)^n [(V^*)^m V^m]({\widetilde{V}}^*)^m \\= & {} I - V^n({\widetilde{V}}^*)^n{\widetilde{V}}^{m+n} (V^*)^{m+n} V^m({\widetilde{V}}^*)^m = I - S^n(S^*)^{m+n}S^m. \end{aligned}$$

The operator $P_{m,n}$ is self-adjoint, and thus

$$\begin{aligned} P_{m,n} = P_{m,n}^* =(I - S^n(S^*)^{m+n}S^m)^* = I - (S^*)^m S^{m+n}(S^*)^n. \end{aligned}$$

$\square$

Corollary 2.8

For all pairs $(m,n) \in {\mathbb {Z}}_+^2 \setminus \{(0,0)\}$ and each $k \in {\mathbb {N}}$:

(i):

the k-poly-$W_{m,n}$-space ${\mathcal {H}}_{k}^{(m,n)}$ admits the representation

$$\begin{aligned} {\mathcal {H}}_{k}^{(m,n)} = \ker S^{k(m+n)} (S^*)^{kn} = \ker \, (S^*)^{k(m+n)}S^{km}; \end{aligned}$$

(ii):

the orthogonal projection $P_{m,n;k} : \, {\mathcal {H}} \longrightarrow {\mathcal {H}}_k^{(m,n)}$ has the form

$$\begin{aligned} P_{m,n;k} = I - S^{kn}(S^*)^{k(m+n)}S^{km} = I - (S^*)^{km} S^{k(m+n)}(S^*)^{kn}. \end{aligned}$$

Proof

It follows from item (4) of Proposition 2.5. $\square$

2.2 Unitary isometry

In this subsection, we consider the opposite situation to that of Sect. 2.1 in the Wold decomposition, i.e., the subspace ${\mathcal {H}}''$ is trivial, or, which is the same, ${\mathcal {H}} = {\mathcal {H}}'$. That is, the isometry U under consideration is a unitary operator.

Recall, see, e.g., [20, Chapter I, Section 2], that a unitary operator U is called a bilateral shift if there is a (not uniquely defined) subspace ${\mathcal {L}}$, called wandering, such that $U^m({\mathcal {L}})$ is orthogonal to $U^n({\mathcal {L}})$ for each integers $m\ne n$ and

$$\begin{aligned} {\mathcal {H}} = \bigoplus _{n \in {\mathbb {Z}}}U^n({\mathcal {L}}). \end{aligned}$$

Let the Hilbert space ${\mathcal {H}}$ be as in Theorem 2.2, having the direct sum decomposition (2.1) with infinite-dimensional summands ${\mathcal {H}}_{(k)}$, and whose basis elements $e_{p,q}$, $p,q \in {\mathbb {Z}}_+$, are rearranged in the lattice as in Fig. 1. Apart of the pure isometries V, ${\widetilde{V}}$, and $W_{m,n}$ of the previous subsection, we introduce the unitary isometry U defined on the basis elements as follows:

$$\begin{aligned} U : \ e_{p,q} \ \longmapsto \ {\left\{ \begin{array}{ll} e_{p+1,q}, &{} \textrm{if} \ p \ge q \\ e_{p,q-1}, &{} \textrm{if} \ p< q \end{array}\right. } \ = \ {\left\{ \begin{array}{ll} {\widetilde{V}}e_{p,q}, &{} \textrm{if} \ p \ge q \\ V^*e_{p,q}, &{} \textrm{if} \ p < q \end{array}\right. }. \end{aligned}$$

(2.3)

As it easily seen, U is a bilateral shift, whose wandering subspace can be taken as

$$\begin{aligned} {\mathcal {V}}_0 = \overline{\textrm{span}}\,\{e_{p,q} : p-q = 0\}. \end{aligned}$$

Then

$$\begin{aligned} {\mathcal {H}} = \bigoplus _{n \in {\mathbb {Z}}}U^n({\mathcal {V}}_0), \end{aligned}$$

(2.4)

where for each $n \in {\mathbb {Z}}$

$$\begin{aligned} U^n({\mathcal {V}}_0) = {\mathcal {V}}_n = \overline{\textrm{span}}\,\{e_{p,q} : p-q = n\}. \end{aligned}$$

Such spaces will be important in considerations of Sect. 4.

3 Analytic type function spaces

3.1 (m, n)-Analytic functions

The powers of differential operators $\frac{\partial }{\partial {{\overline{z}}}}$ and $\frac{\partial }{\partial z}$ permit us to single out various important subclasses of smooth in a domain $D \subset {\mathbb {C}}$ functions. Recall that a function f is called

• analytic if it satisfies the equation $\frac{\partial }{\partial {{\overline{z}}}}f = 0$,

• anti-analytic if it satisfies the equation $\frac{\partial }{\partial z}f = 0$,

• k-polyanalytic if it satisfies the equation $\frac{\partial ^k }{\partial {{\overline{z}}}^k}f = 0$,

• k-anti-polyanalytic if it satisfies the equation $\frac{\partial ^k }{\partial z^k}f = 0$,

• harmonic if it satisfies the equation $\Delta f = 4\frac{\partial }{\partial z}\frac{\partial }{\partial {{\overline{z}}}}f = 0$,

• biharmonic if it satisfies the equation $\Delta ^2 f = 16\frac{\partial ^2 }{\partial z^2}\frac{\partial ^2 }{\partial {{\overline{z}}}^2}f = 0$.

More general, we call a function f (m, n)-analytic if it satisfies the equation $\frac{\partial ^m }{\partial z^m}\frac{\partial ^n }{\partial {{\overline{z}}}^n}f = 0$, and k-(m, n)-polyanalytic if it satisfies the equation $\left( \frac{\partial ^m }{\partial z^m}\frac{\partial ^n }{\partial {{\overline{z}}}^n}\right) ^kf = 0$.

Evidently, a function is k-(m, n)-polyanalytic if and only if it is (km, kn)-analytic, and, in this notation, analytic functions are (0, 1)-analytic, anti-analytic functions are (1, 0)-analytic, harmonic functions are (1, 1)-analytic, etc.

In a sense, the pair (m, n), via the above definition, characterizes the “analytic quality” of differentiable functions from ${\mathcal {H}}$.

Note that, to the best of our knowledge, the class of (m, n)-analytic functions was introduced first in [16].

We denote by ${\mathcal {O}}^{(m,n)}_k = {\mathcal {O}}^{(m,n)}_k(D)$ the linear space of all k-(m, n)-polyanalytic in a domain D functions, with the convention that ${\mathcal {O}}_n := {\mathcal {O}}^{(0,n)} = {\mathcal {O}}^{(0,1)}_n$ is the space of n-polyanalytic functions and $\widetilde{{\mathcal {O}}}_m := {\mathcal {O}}^{(m,0)} = {\mathcal {O}}^{(1,0)}_m$ is the space of m-anti-polyanalytic functions.

As known, see, e.g., [1, Section 1.1], each n-polyanalytic function $\varphi$ and each m-anti-polyanalytic function $\psi$ admit the representation

$$\begin{aligned} \varphi = \sum _{q=0}^{n-1} {{\overline{z}}}^q \varphi _q \qquad \textrm{and} \qquad \psi = \sum _{p=0}^{m-1} z^p \psi _p, \end{aligned}$$

where all $\varphi _q$ and $\psi _p$ are analytic and anti-analytic functions, respectively.

Note that the sum $\varphi + \psi$ of any n-polyanalytic function $\varphi$ and any m-anti-polyanalytic function $\psi$ is obviously (m, n)-analytic. The converse statement was proven in [16, Proposition 2.1], resulting that a function f is (m, n)-analytic if and only if it is a sum of n-polyanalytic and m-anti-polyanalytic functions.

The interrelations between k-(m, n)-polyanalytic and $(k-1)$-(m, n)-polyanalytic functions are described the following theorem.

Theorem 3.1

Given any $k \ge 2$, for each function $f \in {\mathcal {O}}^{(m,n)}_k$, the function $g = \frac{\partial ^m }{\partial z^m}\frac{\partial ^n }{\partial {{\overline{z}}}^n}f$ belongs to ${\mathcal {O}}^{(m,n)}_{k-1}$.

A function f belongs to ${\mathcal {O}}^{(m,n)}_k$ if and only if it admits the representation $f = z^m{{\overline{z}}}^n g_1 + g_2$, where $g_1,g_2 \in {\mathcal {O}}^{(m,n)}_{k-1}$.

Proof

The first statement is trivial. The condition $f \in {\mathcal {O}}^{(m,n)}_k$ is equivalent to $0 = \left( \frac{\partial ^m }{\partial z^m}\frac{\partial ^n }{\partial {{\overline{z}}}^n}\right) ^k f = \left( \frac{\partial ^m }{\partial z^m}\frac{\partial ^n }{\partial {{\overline{z}}}^n}\right) ^{k-1} g$, implying $g \in {\mathcal {O}}^{(m,n)}_{k-1}$.

Part $\Rightarrow$ of the second statement: We represent $f = \varphi + \psi$, with $\varphi \in {\mathcal {O}}_{kn}$ and $\psi \in \widetilde{{\mathcal {O}}}_{km}$.

Then

$$\begin{aligned} \varphi \ = \sum _{q=0}^{kn-1} {{\overline{z}}}^q \varphi _q \ = \sum _{q=(k-1)n}^{kn-1} {{\overline{z}}}^q \varphi _q \ \ + \sum _{q=0}^{(k-1)n-1} {{\overline{z}}}^q \varphi _q := \varphi ^{(1)} + \varphi ^{(2)}; \end{aligned}$$

here, all functions $\varphi _q$ are analytic. This, in particular, implies that $\varphi ^{(2)} \in {\mathcal {O}}_{(k-1)n}$.

Representing each $\varphi _q$, $q = (k-1)n, (k-1)n+1, \ldots , kn-1$, as a power series $\varphi _q = \sum _{s \in {\mathbb {Z}}_+} a_{q,s}z^s$, we have

$$\begin{aligned} \varphi _q = \sum _{s=0}^{m-1} a_{q,s}z^s + z^m\sum _{s=0}^{\infty } a_{q,s+m}z^s, \end{aligned}$$

and thus

$$\begin{aligned} \varphi ^{(1)}= & {} {{\overline{z}}}^n\sum _{q=(k-2)n}^{(k-1)n-1} {{\overline{z}}}^q \left( \sum _{s=0}^{m-1} a_{q,s}z^s + z^m\sum _{s=0}^{\infty } a_{q,s+m}z^s \right) \\= & {} z^m{{\overline{z}}}^n\sum _{q=(k-2)n}^{(k-1)n-1} {{\overline{z}}}^q \sum _{s=0}^{\infty } a_{q,s+m}z^s + \sum _{s=0}^{m-1} z^s\sum _{q=(k-1)n}^{kn-1} a_{q,s}{{\overline{z}}}^q := z^m{{\overline{z}}}^n\varphi ^{(3)} + \varphi ^{(4)}. \end{aligned}$$

That is, $\varphi = z^m{{\overline{z}}}^n\varphi ^{(3)} + \varphi ^{(4)} + \varphi ^{(2)}$, where $\varphi ^{(2)},\varphi ^{(3)} \in {\mathcal {O}}_{(k-1)n}$ and $\varphi ^{(4)} \in \widetilde{{\mathcal {O}}}_{m}$.

Analogously, $\psi = z^m{{\overline{z}}}^n\psi ^{(3)} + \psi ^{(4)} + \psi ^{(2)}$, where $\psi ^{(2)},\psi ^{(3)} \in \widetilde{{\mathcal {O}}}_{(k-1)m}$ and $\psi ^{(4)} \in {\mathcal {O}}_{n}$.

Thus, $f = z^m{{\overline{z}}}^ng_1 + g_2$, where $g_1 = \varphi ^{(3)} + \psi ^{(3)}$ and $g_2 = (\psi ^{(4)} + \varphi ^{(4)}) + (\varphi ^{(2)} + \psi ^{(2)})$ belong to ${\mathcal {O}}^{(m,n)}_{k-1}$.

Part $\Leftarrow$ of the second statement: Let $f = z^m{{\overline{z}}}^n g_1 + g_2$, with $g_1,g_2 \in {\mathcal {O}}^{(m,n)}_{k-1}$. Inclusion ${\mathcal {O}}^{(m,n)}_{k-1} \subset {\mathcal {O}}^{(m,n)}_k$ implies that $g_2 \in {\mathcal {O}}^{(m,n)}_k$. Then, $g_1 = \varphi + \psi$, with $\varphi = \sum _{q=0}^{(k-1)n-1} {{\overline{z}}}^q \varphi _q \in {\mathcal {O}}_{(k-1)n}$ and $\psi = \sum _{p=0}^{(k-1)m-1} z^p \psi _p\in \widetilde{{\mathcal {O}}}_{(k-1)m}$, and thus

$$\begin{aligned} z^m{{\overline{z}}}^n g_1 = \sum _{q=n}^{kn-1} {{\overline{z}}}^q (z^m\varphi _q) + \sum _{p=m}^{km-1} z^p ({{\overline{z}}}^n\psi _p) \in {\mathcal {O}}^{(m,n)}_k. \end{aligned}$$

$\square$

An equivalent assertion to the second statement of the above theorem, giving the characterization of the k-(m, n)-polyanalytic functions, is as follows.

Corollary 3.2

A function f belongs to ${\mathcal {O}}^{(m,n)}_k$ if and only if it admits the representation

$$\begin{aligned} f = (z^m{{\overline{z}}}^n)^{k-1}g_{k-1} + (z^m{{\overline{z}}}^n)^{k-2}g_{k-2} + \ldots +z^m{{\overline{z}}}^n g_1 + g_0, \end{aligned}$$

where all functions $g_{\ell }$ belong to ${\mathcal {O}}^{(m,n)}$.

The functions $g_{\ell }$ can be naturally called (m, n)-analytic components of the k-(m, n)-polyanalytic function f.

Another form of the above corollary reads as follows:

Corollary 3.3

A function is (km, kn)-analytic if and only if it admits the representation as a polynomial of degree $k-1$ in $z^m{{\overline{z}}}^n$ with coefficients being (m, n)-analytic.

3.2 Spaces and bases

In what follows we will systematically use the results of Sect. 2 for a specific class of Hilbert function spaces.

Let D be either ${\mathbb {D}}$ or ${\mathbb {C}}$, and let J be either [0, 1) or ${\mathbb {R}}_+$, so that $D = J \times {\mathbb {T}}$ (with $\{0\} \times {\mathbb {T}}$ glued with the origin), where ${\mathbb {T}}$ is the unit circle in ${\mathbb {C}}$. Following [23], our Hilbert space ${\mathcal {H}}$ in question can be any weighted Hilbert space $L_2(D, d\nu )$, with a probability measure $d\nu (z) = \omega (|z|)dA(z)$, where $dA(z)= \frac{1}{\pi }dxdy$, $z=x+iy$, and whose radial weight function $\omega : D \rightarrow {\mathbb {R}}_+$ is such that the linear span of the monomials $m_{p,q}:=z^p{{\overline{z}}}^q$, for all $p,\,q \in {\mathbb {Z}}_+$, is dense in ${\mathcal {H}}$.

There are many of such spaces have been considered in the literature. However, we give an example of only two classical ones, in which we will illustrate later on some of the results.

Example 3.4

Two basic classical spaces are: $L_2({\mathbb {D}}, d\nu _{\lambda })$, where the measure $d\nu _{\lambda }$ is given by

$$\begin{aligned} d\nu _{\lambda }(z) = (\lambda +1)(1-|z|^2)^{\lambda }dA(z), \quad \textrm{for} \quad \lambda > -1, \end{aligned}$$

(3.1)

and $L_2({\mathbb {C}}, d\mu _{\alpha })$, where the Gaussian measure $d\mu _{\alpha }$ is given by

$$\begin{aligned} d\mu _{\alpha }(z) = \alpha e^{-\alpha |z|^2}dA(z), \quad \textrm{for} \quad \alpha >0. \end{aligned}$$

(3.2)

Recall briefly the construction of an orthonormal basis in ${\mathcal {H}}$; see, e.g., [2, 13, 23] for details.

For all $p,\,q, \, k, \, \ell \in {\mathbb {Z}}_+$

$$\begin{aligned} \langle m_{p,q}, m_{k,\ell } \rangle= & {} \frac{1}{\pi } \int _0^{2\pi } e^{\theta (p-q+\ell -k)}d\theta \int _J r^{p+q+k+\ell }\omega (r)r\textrm{d}r \\= & {} \delta _{p-q,k-\ell } \int _J s^{1/2(p+q+k+\ell )} \omega (\sqrt{s}) \textrm{d}s, \end{aligned}$$

implying that the monomials $m_{p,q}$, having different values of $p-q$, are orthogonal to each other. For $\xi = |p-q|$, we introduce the space $L_2(J,d\eta _{\xi })$, where $d\eta _{\xi }(s) = s^{\xi }\omega (\sqrt{s}) \textrm{d}s$. Then, see, e.g., [19, Chapter 2] for exact formulas and details, there exists a sequence $\{P^{(\xi )}_k\}_{k \in {\mathbb {Z}}_+}$ of the orthonormal polynomials with real coefficients

$$\begin{aligned} \langle P^{(\xi )}_k, P^{(\xi )}_{\ell } \rangle = \int _J P^{(\xi )}_k (s)P^{(\xi )}_{\ell }(s) s^{\xi }\omega (\sqrt{s}) \textrm{d}s = \delta _{k,\ell }, \end{aligned}$$

giving a basis of $L_2(J,d\eta _{\xi })$.

Further, the collection of functions

$$\begin{aligned} e_{p,q} = e_{p,q}(rt) = t^{p-q}r^{|p-q|} P^{(|p-q|)}_{\min \{p,q\}}(r^2), \quad z = rt \quad \textrm{and} \quad p,q \in {\mathbb {Z}}_+ \end{aligned}$$

(3.3)

forms an orthonormal basis on ${\mathcal {H}}$.

Note that the basis elements $e_{p,q}$ can be represented with some real constants $a_0,\ldots , a_{\min \{p,q\}}$ as

$$\begin{aligned} e_{p,q}(z,{{\overline{z}}}) =\sum _{k = 0}^{\min \{p,q\}}a_k z^{p-k}{{\overline{z}}}^{q-k}. \end{aligned}$$

(3.4)

Again, it is convenient to identity each basis element $e_{p,q}$ with the node (p, q) of the lattice ${\mathbb {Z}}_+^2$, visualizing them in Fig. 1.

We introduce then the pure isometries V, ${\widetilde{V}}$, and $W_{m,n}$, with $(m,n) \in {\mathbb {Z}}_+^2 \setminus \{(0,0)\}$, as well as the mutually adjoint partial isometries S and $S^*$, defining all of them on the basis elements as in Sect. 2.1.

3.3 (m, n)-Analytic function subspaces of ${\mathcal {H}}$

We introduce now the corresponding subspaces of our generic ${\mathcal {H}} = L_2(D, d\nu )$. We denote by ${\mathcal {A}}^{(m,n)}$ the subspace of ${\mathcal {H}}$, which consists of all (m, n)-analytic functions; with the convention that ${\mathcal {A}} = {\mathcal {A}}_1 = {\mathcal {A}}_{(1)} = {\mathcal {A}}^{(0,1)}$, $\widetilde{{\mathcal {A}}} = \widetilde{{\mathcal {A}}}_1 = \widetilde{{\mathcal {A}}}_{(1)} = {\mathcal {A}}^{(1,0)}$, ${\textsf {H}} = {\textsf {H}}_1 = {\textsf {H}}_{(1)} = {\mathcal {A}}^{(1,1)}$, being, respectively, the subspaces of analytic, anti-analytic, and harmonic functions.

We introduce as well the spaces of

• k-polyanalytic functions ${\mathcal {A}}_k = {\mathcal {A}}^{(0,k)}$,

• true-k-polyanalytic functions ${\mathcal {A}}_{(k)} = {\mathcal {A}}_k \ominus {\mathcal {A}}_{k-1} = {\mathcal {A}}^{(0,k)} \ominus {\mathcal {A}}^{(0,k-1)}$,

• k-anti-polyanalytic functions $\widetilde{{\mathcal {A}}}_k = {\mathcal {A}}^{(k,0)}$,

• true-k-anti-polyanalytic functions $\widetilde{{\mathcal {A}}}_{(k)} = \widetilde{{\mathcal {A}}}_k \ominus \widetilde{{\mathcal {A}}}_{k-1} = {\mathcal {A}}^{(k,0)} \ominus {\mathcal {A}}^{(k-1,0)}$,

• k-polyharmonic functions ${{\textsf {H}}}_k = {\mathcal {A}}^{(k,k)}$,

• true-k-polyharmonic functions ${{\textsf {H}}}_{(k)} = {{\textsf {H}}}_k \ominus {{\textsf {H}}}_{k-1} = {\mathcal {A}}^{(k,k)} \ominus {\mathcal {A}}^{(k-1,k-1)}$.

Note that all these subspaces are closed in ${\mathcal {H}}$. For ${\mathcal {A}}_{(k)}$ and ${\mathcal {A}}_{(k)}$, with $k \in {\mathbb {N}}$, it follows, see, e.g., [23, Corollary 3.6], from

$$\begin{aligned} {\mathcal {A}}_k= & {} \overline{\textrm{span}}\,\left\{ e_{p,q} : \, p \in \mathbb {Z_+}, \ \ q = 0,1,\ldots ,k-1\right\} , \nonumber \\ \widetilde{{\mathcal {A}}}_k= & {} \overline{\textrm{span}}\,\left\{ e_{p,q} : \, q \in \mathbb {Z_+}, \ \ p = 0,1,\ldots ,k-1\right\} , \end{aligned}$$

(3.5)

for ${\mathcal {A}}^{(m,n)}$ it follows from, see, e.g., [16, Proposition 2.1], implying that

$$\begin{aligned} {\mathcal {A}}^{(m,n)} = \widetilde{{\mathcal {A}}}_m \oplus [{\mathcal {A}}_n \ominus (\widetilde{{\mathcal {A}}}_m \cap {\mathcal {A}}_n)] = [\widetilde{{\mathcal {A}}}_m \ominus (\widetilde{{\mathcal {A}}}_m \cap {\mathcal {A}}_n)] \oplus {\mathcal {A}}_n, \end{aligned}$$

or in symmetric form

$$\begin{aligned} {\mathcal {A}}^{(m,n)} = [\widetilde{{\mathcal {A}}}_m \ominus (\widetilde{{\mathcal {A}}}_m \cap {\mathcal {A}}_n)] \oplus [{\mathcal {A}}_n \ominus (\widetilde{{\mathcal {A}}}_m \cap {\mathcal {A}}_n)] \oplus (\widetilde{{\mathcal {A}}}_m \cap {\mathcal {A}}_n). \end{aligned}$$

(3.6)

An alternative characterization of the (m, n)-analytic spaces can be done based on the isometries and subspaces ${\mathcal {H}}^{(m,n)}$ of Sect. 2. Indeed, formulas (3.5) and (3.6) together with Proposition 2.5 imply

Proposition 3.5

We have

$$\begin{aligned} {\mathcal {A}}_k= & {} {\mathcal {H}}^{(0,k)} = \ker W_{0,k}^* = \ker (V^*)^k, \qquad \widetilde{{\mathcal {A}}}_k = {\mathcal {H}}^{(k,0)} = \ker W_{k,0}^* = \ker ({\widetilde{V}}^*)^k, \\ \textsf {H}_k= & {} {\mathcal {H}}^{(k,k)} = \ker W_{k,k}^* = \ker (V^*)^k({\widetilde{V}}^*)^k, \\ {\mathcal {A}}^{(m,n)}= & {} \ker W_{m,n}^* = (\widetilde{{\mathcal {H}}}_m \ominus (\widetilde{{\mathcal {H}}}_m \cap {\mathcal {H}}_n)) \oplus ({\mathcal {H}}_n \ominus (\widetilde{{\mathcal {H}}}_m \cap {\mathcal {H}}_n)) \oplus (\widetilde{{\mathcal {H}}}_m \cap {\mathcal {H}}_n). \end{aligned}$$

Moreover, by item (5) of Proposition 2.5, the Hilbert space $L_2(D, d\nu )$ can be decomposed into a direct sum of functions having any predefined “analytic quality”; the result which was previously known only for analytic, anti-analytic, and harmonic cases.

Proposition 3.6

Given any pair $(m,n) \in {\mathbb {Z}}_+^2 \setminus \{(0,0)\}$, we have the following orthogonal decomposition of $L_2(D, d\nu )$ into the true-k-(m, n)-polyanalytic function spaces:

$$\begin{aligned} L_2(D, d\nu ) = \bigoplus _{k \in {\mathbb {N}}} {\mathcal {A}}_{(k)}^{(m,n)}, \end{aligned}$$

(3.7)

alternatively an infinite flag of k-(m, n)-polyanalytic function spaces

$$\begin{aligned} {\mathcal {A}}^{(m,n)} \ \subset \ {\mathcal {A}}^{(m,n)}_2 \ \subset \ \cdots \ \subset \ {\mathcal {A}}^{(m,n)}_k \subset \ \cdots \subset \ L_2(D, d\nu ) \end{aligned}$$

leads to the following description of $L_2(D, d\nu )$:

$$\begin{aligned} L_2(D, d\nu ) = \overline{\bigcup _{k \in {\mathbb {N}}} {\mathcal {A}}^{(m,n)}_k }. \end{aligned}$$

Given a pair (m, n), let $\ell$ be their greatest common divisor, so that $m = \ell m_0$ and $n = \ell n_0$, with relatively prime $m_0$ and $n_0$. Then, ${\mathcal {A}}^{(m,n)} = {\mathcal {A}}^{(m_0,n_0)}_{\ell } = \bigoplus _{s = 1}^{\ell }{\mathcal {A}}^{(m_0,n_0)}_{(s)}$, and thus, (3.7) can be regrouped onto the direct sum of the smaller, not decomposable more, true-$\ell$-$(m_0,n_0)$ polyanalytic spaces ${\mathcal {A}}^{(m_0,n_0)}_{(s)}$, with $s \in {\mathbb {N}}$.

That is, all essentially different decompositions (3.7) of the Hilbert space $L_2(D, d\nu )$ are classified by the pairs (m, n) of relatively prime integers.

Proposition 2.7 permits us to characterize the spaces ${\mathcal {A}}^{(m,n)}$ and the corresponding projections in terms of the operators S and $S^*$.

Proposition 3.7

For each pair $(m,n) \in {\mathbb {Z}}_+^2 \setminus \{(0,0)\}$, the space ${\mathcal {A}}^{(m,n)}$ admits the representation ${\mathcal {A}}^{(m,n)} = \ker S^{m+n} (S^*)^n = \ker \, (S^*)^{m+n}S^m$, and the orthogonal projection

$P_{m,n} : L_2(D, d\nu ) \longrightarrow {\mathcal {A}}^{(m,n)}$ is given by $P_{m,n} = I - S^n(S^*)^{m+n}S^m = I - (S^*)^m S^{m+n}(S^*)^n$.

Example 3.8

Let ${\mathcal {H}}$ be the first space of Example 3.4 with $\lambda = 0$, i.e., ${\mathcal {H}} = L_2({\mathbb {D}},dA(z))$. As shown in [23, Proposition 5.1], in this case, the operators S and $S^*$ coincide with the mutually adjoint two-dimensional singular integral operators

$$\begin{aligned} (S_{{\mathbb {D}}}f)(z) = - \int _{{\mathbb {D}}} \frac{f(w)\, dA(w)}{(w-z)^2} \quad \textrm{and} \quad (S_{{\mathbb {D}}}^*f)(z) = - \int _{{\mathbb {D}}} \frac{f(w)\, dA(w)}{({{\overline{w}}}-{{\overline{z}}})^2}. \end{aligned}$$

Then, by Proposition 3.7

$$\begin{aligned} {\mathcal {A}}^{(m,n)} = \ker S_{{\mathbb {D}}}^{m+n} (S_{{\mathbb {D}}}^*)^n = \ker \, (S_{{\mathbb {D}}}^*)^{m+n}S_{{\mathbb {D}}}^m, \end{aligned}$$

and the orthogonal projection $P_{m,n} : L_2({\mathbb {D}},dA(z)) \longrightarrow {\mathcal {A}}^{(m,n)}$ is given by

$$\begin{aligned} P_{m,n} = I - S_{{\mathbb {D}}}^n(S_{{\mathbb {D}}}^*)^{m+n}S_{{\mathbb {D}}}^m = I - (S_{{\mathbb {D}}}^*)^m S_{{\mathbb {D}}}^{m+n}(S_{{\mathbb {D}}}^*)^n, \end{aligned}$$

implying that the orthogonal projections $P_{m,n;k}$ onto k-(m, n)-polyanalytic space and $P_{m,n;(k)}$ onto true-k-(m, n)-polyanalytic space are given, respectively, by

$$\begin{aligned} P_{m,n;k}= & {} I - S_{{\mathbb {D}}}^{kn}(S_{{\mathbb {D}}}^*)^{k(m+n)}S_{{\mathbb {D}}}^{km} = I - (S_{{\mathbb {D}}}^*)^{km} S_{{\mathbb {D}}}^{k(m+n)}(S_{{\mathbb {D}}}^*)^{kn}, \\ P_{m,n;(k)}= & {} S_{{\mathbb {D}}}^{(k-1)n}(S_{{\mathbb {D}}}^*)^{(k-1)(m+n)}S_{{\mathbb {D}}}^{(k-1)m} - S_{{\mathbb {D}}}^{kn}(S_{{\mathbb {D}}}^*)^{k(m+n)}S_{{\mathbb {D}}}^{km} \\= & {} (S_{{\mathbb {D}}}^*)^{(k-1)m} S_{{\mathbb {D}}}^{(k-1)(m+n)}(S_{{\mathbb {D}}}^*)^{(k-1)n} - (S_{{\mathbb {D}}}^*)^{km} S_{{\mathbb {D}}}^{k(m+n)}(S_{{\mathbb {D}}}^*)^{kn}. \end{aligned}$$

In particular, this recovers [16, Formula (9)] formula for the projection onto the (m, n)-analytic space, the consequent formulas for harmonic and polyharmonic projections, as well as Dzuraev formulas [9, Formulas (5.31) and (5.32)] for projections onto k-polyanalytic and k-anti-polyanalytic functions.

Let us observe that each basis element $e_{p,q}$ of the space $L_2(D, d\nu )$ is related to the basis element $e_{0,0}$ (the function identically equals to 1, as the measure $\omega$ is probability) by

$$\begin{aligned} e_{p,q}= & {} W_{p,q}e_{0,0} = {\widetilde{V}}^pV^qe_{0,0}= V^q{\widetilde{V}}^p e_{0,0} = \frac{1}{\sqrt{p!q!}} (\widetilde{{\mathfrak {a}}}^\dag )^p ({\mathfrak {a}}^\dag )^q e_{0,0} \\ {}= & {} \frac{1}{\sqrt{p!q!}}({\mathfrak {a}}^\dag )^q(\widetilde{{\mathfrak {a}}}^\dag )^p e_{0,0}, \end{aligned}$$

where the raising operators ${\mathfrak {a}}^\dag$ and $\widetilde{{\mathfrak {a}}}^\dag$ are defined (Theorem 2.2) by isometries V and ${\widetilde{V}}$.

In particular

$$\begin{aligned} e_{p+m,q+n}= & {} W_{m,n} e_{p,q} = \sqrt{\frac{p!q!}{(p+m)! (q+n)!}}(\widetilde{{\mathfrak {a}}}^\dag )^m ({\mathfrak {a}}^\dag )^n e_{p,q} \\= & {} \sqrt{\frac{p!q!}{(p+m)! (q+n)!}}({\mathfrak {a}}^\dag )^n(\widetilde{{\mathfrak {a}}}^\dag )^m e_{p,q}. \end{aligned}$$

We denote by ${\mathcal {E}}^{(m,n)}$, ${\mathcal {E}}_k^{(m,n)}$, and ${\mathcal {E}}_{(k)}^{(m,n)}$ the sets of all basis elements of the spaces ${\mathcal {A}}^{(m,n)}$, ${\mathcal {A}}_k^{(m,n)}$, and ${\mathcal {A}}_{(k)}^{(m,n)}$, respectively. We have then

$$\begin{aligned} {\mathcal {E}}^{(m,n)}= & {} \left\{ e_{p,q} : \ p< m,\ q \in {\mathbb {Z}}_+ \quad \textrm{or} \quad p \in {\mathbb {Z}}_+, \ q< n \right\} , \nonumber \\ {\mathcal {E}}_k^{(m,n)}= & {} \left\{ e_{p,q} : \ p< km,\ q \in {\mathbb {Z}}_+ \quad \textrm{or} \quad p \in {\mathbb {Z}}_+, \ q< kn \right\} , \nonumber \\ {\mathcal {E}}_{(k)}^{(m,n)}= & {} \left\{ e_{p,q} : \ (k-1)m \le p< km,\ q \in {\mathbb {Z}}_+ \quad \textrm{or} \quad p \in {\mathbb {Z}}_+, \ (k-1)n \le q < kn \right\} .\nonumber \\ \end{aligned}$$

(3.8)

That is, the operator $W_{m,n}$ gives the one-to-one correspondence between ${\mathcal {E}}_{(k)}^{(m,n)}$ and ${\mathcal {E}}_{(k+1)}^{(m,n)}$, which extends to the isometrical isomorphism between ${\mathcal {A}}_{(k)}^{(m,n)}$ and ${\mathcal {A}}_{(k+1)}^{(m,n)}$, while the operators $(\widetilde{{\mathfrak {a}}}^\dag )^m ({\mathfrak {a}}^\dag )^n$ maps bijectively ${\mathcal {E}}_{(k)}^{(m,n)}$ to a complete orthonormal system of elements in ${\mathcal {A}}_{(k+1)}^{(m,n)}$.

The normalizing coefficient for $(\widetilde{{\mathfrak {a}}}^\dag )^m ({\mathfrak {a}}^\dag )^ne_{p,q}$, with $e_{p,q} \in {\mathcal {E}}_{(k)}^{(m,n)}$, is $\sqrt{\frac{p!q!}{(p+m)! (q+n)!}}$.

Example 3.9

Let ${\mathcal {H}}$ be the second space of Example 3.4, i.e., ${\mathcal {H}} = L_2({\mathbb {C}},d\mu _{\alpha }(z))$, $\alpha > 0$.

In this case, see [23, Example 4.5] for details

$$\begin{aligned} {\mathfrak {a}}^\dag = - \frac{1}{\sqrt{\alpha }}\frac{\partial }{\partial {z}} + \sqrt{\alpha }{{\overline{z}}} \qquad \textrm{and} \qquad \widetilde{{\mathfrak {a}}}^\dag = - \frac{1}{\sqrt{\alpha }}\frac{\partial }{\partial {{{\overline{z}}}}} + \sqrt{\alpha } z, \end{aligned}$$

implying that

$$\begin{aligned} e_{p,q}= & {} \frac{1}{\sqrt{p!q!}}({\mathfrak {a}}^\dag )^q (\widetilde{{\mathfrak {a}}}^\dag )^p e_{0,0} \nonumber \\= & {} \sqrt{\alpha ^{p+q}p!q!} \sum _{k=0}^{\min \{p,q\}} \frac{(-1)^k}{\alpha ^k k!(p-k)!(q-k)!}\, z^{p-k} {{\overline{z}}}^{q-k}, \quad p,q \in {\mathbb {Z}}_+. \end{aligned}$$

(3.9)

Thus, (3.8), together with (3.9), gives us the exact description of bases for (m, n)-analytic space ${\mathcal {A}}^{(m,n)}$, k-(m, n)-polyanalytic space ${\mathcal {A}}_k^{(m,n)}$, and true-k-(m, n)-polyanalytic space ${\mathcal {A}}_{(k)}^{(m,n)}$; in addition, the operator $\left( - \frac{1}{\sqrt{\alpha }}\frac{\partial }{\partial {z}} + \sqrt{\alpha }{{\overline{z}}}\right) ^n \left( - \frac{1}{\sqrt{\alpha }}\frac{\partial }{\partial {{{\overline{z}}}}} + \sqrt{\alpha } z\right) ^m$ maps bijectively ${\mathcal {E}}_{(k)}^{(m,n)}$ to a complete orthonormal system of elements in ${\mathcal {A}}_{(k+1)}^{(m,n)}$.

4 Radial operators

4.1 Radial operators

We extend here the results on radial operators [2, 14, 15] to the case of our spaces ${\mathcal {H}} = L_2(D, d\nu )$ and various their subspaces. In what follows, we adopt the approach used in [2, Sections 7, 8] and [15, Sections 5, 6].

For each $t \in {\mathbb {T}}$, the rotation operator $\varpi (t) : f(z) \in {\mathcal {H}} \mapsto f(t^{-1}z) \in {\mathcal {H}}$ is obviously unitary. We denote by $\varPi$ the set of all operators $\varpi (t)$. Note that $\varPi$ is actually a group, giving unitary representation of ${\mathbb {T}}$ in ${\mathcal {B}}({\mathcal {H}})$.

Recall that the operator $S \in {\mathcal {B}}({\mathcal {H}})$ is called radial if it commutes with $\varpi (t)$, for all $t \in {\mathbb {T}}$. We denote by ${\mathcal {R}} = {\mathcal {R}}({\mathcal {H}})$ the set of all radial operators in ${\mathcal {H}}$, that is

$$\begin{aligned} {\mathcal {R}} = \left\{ S \in {\mathcal {B}}({\mathcal {H}}) : \ \varpi (t)S =S\varpi (t) \ \ \ \mathrm {for \ all} \ \ t \in {\mathbb {T}} \right\} . \end{aligned}$$

In other words, ${\mathcal {R}}$ is the commutant (or centralizer) of $\varPi$, ${\mathcal {R}} = \varPi '$, being a von Neumann subalgebra of the von Neumann algebra ${\mathcal {B}}({\mathcal {H}})$; see, e.g., [18].

Observe that each $\varpi (t)$ is diagonal in ${\mathcal {H}}$ with respect to its orthogonal basis ${\mathcal {E}} = \{e_{p,q}\}_{p,q \in {\mathbb {Z}}_+}$. Indeed, (3.3) implies

$$\begin{aligned} \varpi (t)e_{p,q} = t^{-(p-q)}e_{p,q}. \end{aligned}$$

(4.1)

As in [2, Definition 1] and [15, Definition 5.1], we introduce the following general concept.

Let a Hilbert space ${\mathcal {H}}$ be represented as a direct sum ${\mathcal {H}} = \bigoplus _{j \in J} {\mathcal {L}}_j$ of a finite or countable number of its mutually orthogonal subspaces. And let ${\mathcal {Q}}$ be a self-adjoint subset of bounded linear operators in ${\mathcal {H}}$. We say then that the family of subspaces $\{{\mathcal {L}}_j\}_{j \in J}$ diagonalizes ${\mathcal {Q}}$ if

• 1. each operator $Q \in {\mathcal {Q}}$, being restricted to any ${\mathcal {L}}_j$, is a scalar operator, $Q|_{{\mathcal {L}}_j} = \lambda _{Q,j}I$;

• 2. for every j, k in J with $j \ne k$, there exists $Q \in {\mathcal {Q}}$, such that $\lambda _{Q,j} \ne \lambda _{Q,k}$.

In particular, this means that, with respect to the decomposition ${\mathcal {H}} = \bigoplus _{j \in J} {\mathcal {L}}_j$, each operator $Q \in {\mathcal {Q}}$ is nothing but the direct sum of scalar operators, $Q = \bigoplus _{j \in J}\lambda _{Q,j}I$, implying the following representation for the commutant of ${\mathcal {Q}}$:

$$\begin{aligned} {\mathcal {Q}}' = \bigoplus _{j \in J} {\mathcal {B}}({\mathcal {L}}_j). \end{aligned}$$

We introduce now the unitary isometry U defined in our ${\mathcal {H}} = L_2(D, d\nu )$ by (2.3), and the direct sum decomposition (2.4)

$$\begin{aligned} {\mathcal {H}} = \bigoplus _{n \in {\mathbb {Z}}}U^n({\mathcal {V}}_0), \end{aligned}$$

where for each $n \in {\mathbb {Z}}$

$$\begin{aligned} U^n({\mathcal {V}}_0) = {\mathcal {V}}_n = \overline{\textrm{span}}\,\{e_{p,q} : p-q = n\}. \end{aligned}$$

(4.2)

By (4.1), the family of subspaces $\{{\mathcal {V}}_n\}_{n \in {\mathbb {Z}}}$ diagonalizes the set $\varPi$ of rotation operators, and thus, we have

Proposition 4.1

The algebra ${\mathcal {R}}$ of radial operators admits the representation

$$\begin{aligned} {\mathcal {R}} = \varPi ' = \bigoplus _{n \in {\mathbb {Z}}} {\mathcal {B}}({\mathcal {V}}_n). \end{aligned}$$

Let now ${\mathcal {H}}_0$ be an infinite-dimensional closed subspace of ${\mathcal {H}}$, such that its basis can be chosen as a certain subset of the basis ${\mathcal {E}} = \{e_{p,q}\}_{p,q \in {\mathbb {Z}}_+}$ in ${\mathcal {H}}$. Or equivalently, take an infinite subset of ${\mathcal {E}}_0$ of the set ${\mathcal {E}} = \{e_{p,q}\}_{p,q \in {\mathbb {Z}}_+}$ of basis elements in ${\mathcal {H}}$, and denote by ${\mathcal {H}}_0$ the subspace of ${\mathcal {H}}$ with basis ${\mathcal {E}}_0$.

Then, (4.1) ensures that ${\mathcal {H}}_0$ is invariant under the action of family $\varPi$. Let $N \subset {\mathbb {Z}}$ be the set of integers for which ${\mathcal {V}}_n^{0} = {\mathcal {H}}_0 \cap {\mathcal {V}}_n \ne \{0\}$. It is straightforward that

$$\begin{aligned} {\mathcal {H}}_0 = \bigoplus _{n \in N} {\mathcal {V}}_n^0. \end{aligned}$$

Further, for each $n \in N$, the space ${\mathcal {V}}_n^{0}$ is invariant for each $\varpi (t)$, $t \in {\mathbb {T}}$, and $\varpi (t)$, being restricted to ${\mathcal {V}}_n^{0}$ is a scalar operator, $\varpi (t)_0 := \varpi (t)_0|_{{\mathcal {V}}_n^{0}} = t^{-n}I$.

This yields the following:

Lemma 4.2

The family of subspaces $\{{\mathcal {V}}_n^0\}_{n \in N}$ diagonalizes the set $\varPi _0 = \{\varpi (t)_0\}_{t \in {\mathbb {T}}}$, and the algebra ${\mathcal {R}}_0$ of radial in ${\mathcal {H}}_0$ operators is of the form

$$\begin{aligned} {\mathcal {R}}_0 = \varPi _0' = \bigoplus _{n \in N} {\mathcal {B}}({\mathcal {V}}_n^0). \end{aligned}$$

Further, for each $n \in N$, we renumber the basis elements $e_{p,q} \in {\mathcal {V}}_n$ ($p-q = n$) as follows:

$$\begin{aligned} e^{(n)}_{\ell }:= {\left\{ \begin{array}{ll} e_{\ell + n,\ell }, &{} \textrm{if} \ \ n\ge 0 \\ e_{\ell , \ell -n}, &{} \textrm{if} \ \ n < 0 \end{array}\right. }, \quad \ell \in {\mathbb {Z}}_+, \end{aligned}$$

and let then $\{e^{(n)}_j\}_{j = 1,2,\ldots , \dim {\mathcal {V}}_n^0}$ be the basis of ${\mathcal {V}}_n^0 = {\mathcal {H}}_0 \cap {\mathcal {V}}_n$.

This leads to the following description of operators in ${\mathcal {R}}_0$.

Lemma 4.3

According to the direct sum decomposition

$$\begin{aligned} {\mathcal {H}}_0 = \bigoplus _{n \in N} {\mathcal {V}}_n^0, \end{aligned}$$

each operator $S \in {\mathcal {R}}_0$ has the form

$$\begin{aligned} S = \bigoplus _{n \in N} S_n, \end{aligned}$$

where each $S_n$ admits the following matrix representation:

$$\begin{aligned} \left( \left\langle S_ne^{(n)}_j, e^{(n)}_k\right\rangle \right) _{j,k = 1}^{\dim {\mathcal {V}}_n^0}. \end{aligned}$$

(4.3)

4.2 Radial Toeplitz operators

Among all operators in ${\mathcal {R}}_0$, we single out and describe the so-called radial Toeplitz operators.

Recall that, by its standard definition, the Toeplitz operator $T_a$, with symbol $a \in L_{\infty }(D)$ acting on ${\mathcal {H}}_0$, is the compression onto ${\mathcal {H}}_0$ of the multiplication by a operator, acting on ${\mathcal {H}}$, that is

$$\begin{aligned} T_a : \, \varphi \in {\mathcal {H}}_0 \ \longmapsto \ {{\textbf{P}}}(a \varphi ) \in {\mathcal {H}}_0, \end{aligned}$$

where ${{\textbf{P}}}$ is the orthogonal projection of ${\mathcal {H}}$ onto ${\mathcal {H}}_0$.

Recall first that a function $a \in L_{\infty }(D)$ is called radial if there exists a function ${\widetilde{a}} \in L_{\infty }(J)$, such that $a(z) = {\widetilde{a}}(|z|)$ for a.e. $z \in D$. Without overusing the notation, we will simply understand $a(z) = a(|z|)$ to be a radial function. It is well known (and easy to check) that a function $a \in L_{\infty }(D)$ is radial if and only if, for all $t \in {\mathbb {T}}$, $a(tz) = a(z)$ for a.e. $z \in D$.

In many already known cases (when, for example, ${\mathcal {H}}_0$ is either the space of analytic, or anti-analytic, or harmonic functions), the Toeplitz operator with $L_{\infty }$-symbol is radial if and only if its symbol is a radial function. At the same time, the situation for an arbitrary infinite-dimensional closed subspace ${\mathcal {H}}_0$ of ${\mathcal {H}}$ is a bit different, the result may depend of the subspace ${\mathcal {H}}_0$ in question.

Approaching to it, let us remind first, see, e.g., [10, 26], that given a function $a \in L_{\infty }(D)$, its radialization rad(a) is given by

$$\begin{aligned} rad(a)(z) = \frac{1}{2\pi } \int _0^{2\pi } a(e^{i\theta }z) d\theta = \frac{1}{2\pi } \int _{{\mathbb {T}}} a(tz)\frac{dt}{it}. \end{aligned}$$

The radialization rad(a) of a function a is a radial function, and a function a is radial if and only if it coincides with its radialization.

Then, given a subspace ${\mathcal {H}}_0$ of ${\mathcal {H}}$, (3.4) implies that the product $\overline{e'}e''$ of any two its basis elements is a polynomial in z and ${{\overline{z}}}$. We denote by $\widehat{{\mathcal {H}}}_0$ the subspace of ${\mathcal {H}}$ being the closure of the linear span of all such polynomials

$$\begin{aligned} \widehat{{\mathcal {H}}}_0 = \overline{\textrm{span}}\,\left\{ \overline{e'}e'' : \ \ \mathrm {for \ all \ basis \ elements} \ \ e', \, e'' \in {\mathcal {H}}_0\right\} . \end{aligned}$$

Proposition 4.4

Let ${\mathcal {H}}_0$ be an infinite-dimensional closed subspace of ${\mathcal {H}}$. Then, the Toeplitz operator $T_a$, with symbol $a \in L_{\infty }(D)$, is radial if and only if the function a admits the representation

$$\begin{aligned} a = b + h, \quad \textrm{where} \ \ b \ \ \mathrm {is \ radial} \ \ \textrm{and} \ \ h \in \widehat{{\mathcal {H}}}_0^{\perp }. \end{aligned}$$

(4.4)

In particular, for each radial $L_{\infty }$-symbol, the corresponding Toeplitz operator is radial.

Proof

Toeplitz operator $T_a$ is radial if and only if $\varpi (t)T_a = T_a \varpi (t)$, for all $t \in {\mathbb {T}}$, equivalently $T_a = \varpi (t^{-1})T_a \varpi (t)$, for all $t \in {\mathbb {T}}$, or if and only if all matrix elements of these two operator coincide

$$\begin{aligned} \langle T_a e',e''\rangle = \langle \varpi (t^{-1})T_a \varpi (t)e',e''\rangle , \quad \mathrm {for \ all \ basis \ elements} \ \ e', \, e'' \in {\mathcal {H}}_0. \end{aligned}$$

We have

$$\begin{aligned} \langle T_a e',e''\rangle = \langle {{\textbf{P}}}(ae'),e''\rangle = \langle ae', {{\textbf{P}}}e''\rangle = \langle ae', e''\rangle , \end{aligned}$$

and taking into account (4.1)

$$\begin{aligned} \langle \varpi (t^{-1})T_a \varpi (t)e',e''\rangle= & {} \langle T_a \varpi (t)e', \varpi (t)e''\rangle = \langle a\varpi (t)e', {{\textbf{P}}}(\varpi (t)e'')\rangle \\= & {} \langle a\varpi (t)e', \varpi (t)e''\rangle = \langle \varpi (t^{-1})a\varpi (t)e', e''\rangle = \langle a(tz)e', e''\rangle . \end{aligned}$$

Thus, a Toeplitz operator $T_a$ is radial if and only if

$$\begin{aligned} \langle [a(z) - a(tz)] e', e'' \rangle = 0, \quad \mathrm {for \ all} \ \ t \in {\mathbb {T}} \ \ \mathrm {and \ all \ basis \ elements} \ \ e', \, e'' \in {\mathcal {H}}_0, \end{aligned}$$

or

$$\begin{aligned} \langle [a(z) - a(tz)], \overline{e'}e'' \rangle = 0, \quad \mathrm {for \ all} \ \ t \in {\mathbb {T}} \ \ \mathrm {and \ all \ basis \ elements} \ \ e', \, e'' \in {\mathcal {H}}_0, \end{aligned}$$

being equivalent to $a(z) - a(tz) \in \widehat{{\mathcal {H}}}_0^{\perp }$, for all $t \in {\mathbb {T}}$. Integrating on t over ${\mathbb {T}}$, we arrive to $a - rad(a) \in \widehat{{\mathcal {H}}}_0^{\perp }$. Thus, for all $t \in {\mathbb {T}}$, we have $a(tz) - rad(a)(|z|) \in \widehat{{\mathcal {H}}}_0^{\perp }$, implying $a(z) - a(tz) \in \widehat{{\mathcal {H}}}_0^{\perp }$.

That is, we arrive to one more equivalence: a Toeplitz operator $T_a$ is radial if and only if its symbol satisfies the condition

$$\begin{aligned} a - rad(a)\, \in \, \widehat{{\mathcal {H}}}_0^{\perp }. \end{aligned}$$

(4.5)

This, in particular, implies the part “only if” of the proposition. For the part “if”, let us assume that the $L_{\infty }$-function a admits the representation $a = b + h$, with radial b and $h \in \widehat{{\mathcal {H}}}_0^{\perp }$. Then, $rad(a) = b + rad(h)$, and thus, $a - rad(a) = h - rad(h) \in \widehat{{\mathcal {H}}}_0^{\perp }$. $\square$

Corollary 4.5

If a subspace ${\mathcal {H}}_0$ is such that $\widehat{{\mathcal {H}}}_0 = {\mathcal {H}}$, then the Toeplitz operator $T_a$ in ${\mathcal {H}}_0$ is radial if and only if its symbol a is a radial function.

Example 4.6

Let ${\mathcal {H}}_0$ be a space, such that, for all $n \in {\mathbb {Z}}_+$, either $e_{n,0} = \frac{z^n}{\Vert z^n\Vert }$ or $e_{0,n} = \frac{{{\overline{z}}}^n}{\Vert {{\overline{z}}}^n\Vert }$ belongs to ${\mathcal {H}}_0$. Then, all polynomials $z^n{{\overline{z}}}^m$ belong to $\widehat{{\mathcal {H}}}_0$, whose linear span is dense in both $\widehat{{\mathcal {H}}}_0$ and ${\mathcal {H}}$. Thus, such ${\mathcal {H}}_0$ satisfies the condition of Corollary 4.5, and thus, all radial Toeplitz operators in ${\mathcal {H}}_0$ are generated by radial symbols only.

All spaces ${\mathcal {A}}$, ${\mathcal {A}}_k$, $\widetilde{{\mathcal {A}}}$, $\widetilde{{\mathcal {A}}}_k$, H, $H_k$, as well as more general spaces ${\mathcal {A}}^{(m,n)}$, with $(m,n) \in {\mathbb {Z}}_+^2 \setminus \{(0,0)\}$ are of this sort.

Example 4.7

Let

$$\begin{aligned} {\mathcal {H}}_0 = {\mathcal {A}}_{even} = \overline{\textrm{span}}\,\{e_{2p,0} : \, p \in {\mathbb {Z}}_+ \} = \overline{\textrm{span}}\,\{ 1, \, z^2, \,z^4,\, \ldots \}, \end{aligned}$$

then $\widehat{{\mathcal {H}}}_0$ coincides with the subspace of ${\mathcal {H}}$ with consisting of all even functions in D. This implies that a function a generates the radial Toeplitz operator $T_a$ if and only if the difference $a - rad(a)$ is an odd function.

Example 4.8

Given $n \in {\mathbb {Z}}$, let ${\mathcal {H}}_0 = {\mathcal {V}}_n$, see (4.2). By (3.3), for all basis elements $e'$ and $e''$ in ${\mathcal {V}}_n$, the product $\overline{e'}e''$ is a radial function. Thus, for all $t \in {\mathbb {T}}$

$$\begin{aligned}{} & {} \langle \varpi (t^{-1})T_a \varpi (t)e',e''\rangle = \langle T_a \varpi (t)e', \varpi (t)e''\rangle = \langle a\varpi (t)e', {{\textbf{P}}}(\varpi (t)e'')\rangle \\ {}{} & {} \quad = \langle a\varpi (t)e', \varpi (t)e''\rangle \\{} & {} \quad = \langle a, \overline{\varpi (t)e'}\varpi (t)e''\rangle = \langle a,\varpi (t)(\overline{e'}e'')\rangle = \langle a,\overline{e'}e''\rangle = \langle a e', e''\rangle = \langle T_a e', e''\rangle , \end{aligned}$$

which implies that, for each $a \in L_{\infty }(D)$ the Toeplitz operator $T_a$ is radial.

Observation 4.9

Condition (4.5) implies

$$\begin{aligned} \langle [a - rad(a)] e', e'' \rangle = 0, \quad \mathrm {for \ all \ basis \ elements} \ \ e', \, e'' \in {\mathcal {H}}_0, \end{aligned}$$

which is equivalent to $T_{a - rad(a)} = 0$, or $T_a = T_{rad(a)}$. That is, although the set of symbols, which generate radial Toeplitz operators, can be essentially wider then the set of radial symbols, characterizing the operator $T_a$ we can replace it with $T_{rad(a)}$. Note that for a symbol a represented in the form (4.4), its radialization is $rad(a) = b + rad(h)$.

We describe now radial Toeplitz operators, acting on the above mentioned spaces.

4.3 Radial Toeplitz operators in ${\mathcal {A}}^{(m,n)}$

In this case, for each $s \in {\mathbb {Z}}$, the spaces

${\mathcal {V}}^0_s = {\mathcal {A}}^{(m,n)} \cap {\mathcal {V}}_s$ are not trivial and admit the following isomorphic characterization:

$$\begin{aligned} {\mathcal {V}}^0_s \ \cong \ {\left\{ \begin{array}{ll} {\mathbb {C}}^m, &{} \textrm{for} \ \ s \le 0 \\ {\mathbb {C}}^{m-s}, &{} \textrm{for} \ \ 0< s < m-n \\ {\mathbb {C}}^n, &{} \textrm{for} \ \ s \ge m-n \end{array}\right. }, \qquad \textrm{if} \quad m \ge n \end{aligned}$$

(4.6)

and

$$\begin{aligned} {\mathcal {V}}^0_s \ \cong \ {\left\{ \begin{array}{ll} {\mathbb {C}}^n, &{} \textrm{for} \ \ s \ge 0 \\ {\mathbb {C}}^{n+s}, &{} \textrm{for} \ \ m-n< s< 0 \\ {\mathbb {C}}^m, &{} \textrm{for} \ \ s \le m-n \end{array}\right. }, \qquad \textrm{if} \quad m < n. \end{aligned}$$

(4.7)

Note that a simple visualization of these isomorphisms can be done by drawing lines, which correspond to the spaces in question, in Fig. 1.

By Example 4.6, radial Toeplitz operators in ${\mathcal {A}}^{(m,n)}$ are only those whose symbols are radial functions. Then, according to Lemma 4.3, in the direct sum decomposition

$$\begin{aligned} {\mathcal {A}}^{(m,n)} = \bigoplus _{s \in {\mathbb {Z}}} {\mathcal {V}}_s^0, \end{aligned}$$

each radial Toeplitz operator $T_a^{(s)}$ has the form

$$\begin{aligned} T_a = \bigoplus _{s \in {\mathbb {Z}}} T_a^{(s)}. \end{aligned}$$

Lemma 4.10

Under the isomorphisms (4.6) and (4.7), the operator $T_a^{(s)}$ is identified with the following symmetric matrix:

$$\begin{aligned} T_a^{(s)} = \left( \gamma _a^{(s)}(j,k)\right) _{j,k = 1}^{\dim {\mathcal {V}}_s^0}, \end{aligned}$$

where

$$\begin{aligned} \gamma _a^{(s)}(j,k) = \gamma _a^{(s)}(k,j) = \int _J a(\sqrt{r})r^{|s|}P^{(|s|)}_{j-1}(r)P^{(|s|)}_{k-1}(r)\omega (\sqrt{r})dr. \end{aligned}$$

Proof

The basis elements of each ${\mathcal {V}}_s^0$ are of the form

$$\begin{aligned} e^{(s)}_j \ = \ {\left\{ \begin{array}{ll} e_{j-1+s,j-1}, &{} \textrm{if} \ \ s \ge 0 \\ e_{j-1, j-1-s}, &{} \textrm{if} \ \ s < 0 \end{array}\right. }, \qquad j = 1,2,\ldots . \dim {\mathcal {V}}_s^0. \end{aligned}$$

Thus, by (4.3)

$$\begin{aligned} T_a^{(s)} = \left( \left\langle T_a^{(s)}e^{(s)}_j, e^{(s)}_k \right\rangle \right) _{j,k = 1}^{\dim {\mathcal {V}}_s^0}, \end{aligned}$$

where by (3.3) for $s \ge 0$

$$\begin{aligned}{} & {} \left\langle T_a^{(\ell )}e^{(s)}_j, e^{(s)}_k\right\rangle = \int _D a(r) e_{j-1+s,j-1}\,\overline{e_{k-1+s,k-1}}\omega (r)dA(z) \\{} & {} \quad = 2\int _J a(r)r^{2s}P^{(s)}_{j-1}(r^2)P^{(s)}_{k-1}(r^2)\omega (r)rdr = \int _J a(\sqrt{r})r^{s}P^{(s)}_{j-1}(r)P^{(s)}_{k-1}(r)\omega (\sqrt{r})dr, \end{aligned}$$

and for $s < 0$

$$\begin{aligned} \left\langle T_a^{(\ell )}e^{(s)}_j, e^{(s)}_k\right\rangle= & {} \int _D a(r) e_{j,j-s}\,\overline{e_{k,k-s}}\omega (r)dA(z) \\= & {} 2\int _J a(r)r^{2|s|}P^{(|s|)}_{j-1}(r^2)P^{(|s|)}_{k-1}(r^2)\omega (r)rdr \\ {}= & {} \int _J a(\sqrt{r})r^{|s|}P^{(|s|)}_{j-1}(r)P^{(|s|)}_{k-1}(r)\omega (\sqrt{r})dr. \end{aligned}$$

$\square$

4.4 Radial Toeplitz operators in ${\mathcal {A}}$

We specify here the results of the previous subsection to the space ${\mathcal {A}} = {\mathcal {A}}^{(0,1)}$ of analytic functions. In this case, for each $p \in {\mathbb {Z}}_+$, we have that the spaces ${\mathcal {V}}^0_k = {\mathcal {A}} \cap {\mathcal {V}}_k = {\mathbb {C}}e_{k,0}$ are one-dimensional. Lemma 4.10 implies now that each Toeplitz operator $T_a$ with bounded radial symbol a can be identified with its spectral sequence $\gamma _a = \{\gamma _a(k)\}_{k \in {\mathbb {Z}}_+}$, where

$$\begin{aligned} \gamma _a(k)= & {} \int _J a(\sqrt{r})r^{k}(P^{(k)}_0)^2 \omega (\sqrt{r})dr = \Vert z^p\Vert ^{-2}\int _J a(\sqrt{r})r^{k}\omega (\sqrt{r})dr \nonumber \\= & {} \left( \int _J r^{k}\omega (\sqrt{r})dr\right) ^{-1}\int _J a(\sqrt{r})r^{k}\omega (\sqrt{r})dr. \end{aligned}$$

(4.8)

That is, each Toeplitz operator $T_a$ with bounded radial symbol a admits the representation

$$\begin{aligned} T_a = \sum _{k \in {\mathbb {Z}}_+}\gamma _a(k)\textrm{P}_k, \end{aligned}$$

where $\textrm{P}_k$ are the one-dimensional orthogonal projections of ${\mathcal {A}}$ onto ${\mathcal {V}}^0_k = {\mathbb {C}}e_{k,0}$, and the equality is understood in the strong sense, i.e., for each $f \in {\mathcal {A}}$, $T_a f= \sum _{k \in {\mathbb {Z}}_+}\gamma _a(k)\textrm{P}_kf$. All similar representations in what follows are to be understood in the same way.

Moreover, the $C^*$-algebra generated by radial Toeplitz operators is commutative.

4.5 Radial Toeplitz operators in ${\mathcal {A}}_{even}$

Consider the space ${\mathcal {A}}_{even}$ of Example 4.7. Then, the symbols a, that generate radial Toeplitz operators, satisfy the condition $a - rad(a)$ is an odd function, and under this condition $T_a = T_{rad(a)}$. Further, the matrix elements of the operator $T_a$ are as follows:

$$\begin{aligned} \langle T_a e_{2j.0}, e_{2k,0} \rangle= & {} \int _D rad(a)(r) \frac{z^{2j}}{\Vert z^{2j}\Vert }\frac{{{\overline{z}}}^{2k}}{\Vert {{\overline{z}}}^{2k}\Vert } \omega (r)dA(z) \\= & {} \delta _{j,k} \frac{2}{\Vert z^{2j}\Vert ^2}\int _J rad(a)(r) r^{4j}\omega (r) rdr \\ {}= & {} \delta _{j,k} \frac{1}{\Vert z^{2j}\Vert }\int _J rad(a)(\sqrt{r}) r^{2j}\omega (\sqrt{r}) dr. \end{aligned}$$

That is, the operator $T_a$ is diagonal with respect to the basis $\{e_{2j.0}\}_{j \in {\mathbb {Z}}_+}$, and the corresponding eigenvalues are

$$\begin{aligned} \gamma _a(j) = \frac{1}{\Vert z^{2j}\Vert }\int _J rad(a)(\sqrt{r}) r^{2j}\omega (\sqrt{r}) dr, \end{aligned}$$

which are exactly the eigenvalues of the diagonal Toeplitz operator $T_{rad(a)}$, acting on the analytic space ${\mathcal {A}}$, that correspond to its even basis elements.

4.6 Radial Toeplitz operators in ${\mathcal {V}}_n$

By Example 4.8, in this case any $L_{\infty }$-function a generates the radial Toeplitz operators, and $T_a = T_{rad(a)}$. The basis elements of ${\mathcal {V}}_n$ have the form

$$\begin{aligned} e^{(n)}_j \ = \ {\left\{ \begin{array}{ll} e_{j-1+n,j-1}, &{} \textrm{if} \ \ n \ge 0 \\ e_{j-1, j-1-s}, &{} \textrm{if} \ \ n < 0 \end{array}\right. }, \qquad j \in {\mathbb {N}}, \end{aligned}$$

Thus, as in Lemma 4.10, the operators $T_a$ can be identified with the infinite symmetric matrix

$$\begin{aligned} T_a = \left( \gamma _a^{(n)}(j,k)\right) _{j,k \in {\mathbb {N}}}, \end{aligned}$$

where

$$\begin{aligned} \gamma _a^{(n)}(j,k) = \gamma _a^{(n)}(k,j) = \int _J rad(a)(\sqrt{r})\,r^{|n|}P^{(|n|)}_{j-1}(r)P^{(|n|)}_{k-1}(r)\omega (\sqrt{r})dr. \end{aligned}$$

4.7 Radial operators in ${\mathcal {A}}$ via the spectral theorem

We characterize here the radial operators in ${\mathcal {A}}$ using the spectral theorem and functional calculus for an unbounded self-adjoint operator, see, e.g., [17, Chapter VIII] or [24, Chapter 7].

We start with the unbounded Hermitian operator $L = z \frac{\partial }{\partial z} - {{\overline{z}}}\frac{\partial }{\partial {{\overline{z}}}}$ in ${\mathcal {H}}$, introduced in [25, Formula (4.9)] for the space $L_2({\mathbb {D}}, d\nu _{\lambda })$, called therein the operator of the (orbital) angular momentum. Formula (3.3) implies that the operator L is well defined on the linear span of the basis elements of ${\mathcal {H}}$ and acts on them as follows:

$$\begin{aligned} L e_{p,q} = (p-q) e_{p,q},\quad \mathrm {for \ all} \quad p,q \in {\mathbb {Z}}_+. \end{aligned}$$

Then, the operator L can be extended to the domain

$$\begin{aligned} D(L) = \left\{ f = \sum _{p,q \in {\mathbb {Z}}_+}f_{p,q}e_{p,q} \in {\mathcal {H}} : \, \sum _{p,q \in {\mathbb {Z}}_+}|(p-q) f_{p,q}|^2 < \infty \right\} \end{aligned}$$

by the rule

$$\begin{aligned} Lf = \sum _{p,q \in {\mathbb {Z}}_+}(p-q)f_{p,q}e_{p,q}. \end{aligned}$$

The images of the operators $L \pm i$ defined on D(L) coincide with ${\mathcal {H}}$, implying [17, Theorem VIII.3] that the operator L is self-adjoint. The operator L obviously commutes with all $\varpi (t)$, $t \in {\mathbb {T}}$, providing thus an example of the unbounded radial operator in ${\mathcal {H}}$.

Consider now its restriction to the space ${\mathcal {A}}$, the operator $N := L|_{{\mathcal {A}}} = z \frac{\partial }{\partial z}$, which is defined on $D(N) = D(L) \cap {\mathcal {A}}$ by

$$\begin{aligned} N\left( \sum _{p \in {\mathbb {Z}}_+} f_{p}e_{p,0}\right) = \sum _{p \in {\mathbb {Z}}_+} p\,f_{p}e_{p,0}. \end{aligned}$$

The operator N is also self-adjoint in ${\mathcal {A}}$, and on its domain admits the representation

$$\begin{aligned} N = \sum _{k \in {\mathcal {Z}}_+} k\, \textrm{P}_k. \end{aligned}$$

It is not difficult to figure out that its spectral measure $E=E(\eta )$, where $\eta \in {\mathbb {R}}$, is given by

$$\begin{aligned} E(\eta ) = \sum _{k \le \eta } \textrm{P}_k, \end{aligned}$$

and [24, Page 182]

$$\begin{aligned} E((a,b]) = E(b) - E(a),&\quad&E((a,b)) = E(b-0) - E(a), \\ E([a,b]) = E(b) - E(a-0),&\quad&E([a,b)) = E(b-0) - E(a-0). \end{aligned}$$

Observe that the set $(-\infty ,0) \bigcup _{k \in {\mathcal {Z}}_+} (k, k+1)$ has zero E-measure, while $E([k,k]) = \textrm{P}_k$.

Then, by the spectral theorem

$$\begin{aligned} N = \int _{{\mathbb {R}}}\eta \, dE(\eta ) = \sum _{k \in {\mathcal {Z}}_+} k\, \textrm{P}_k. \end{aligned}$$

Further, for each $f = \sum _{k \in {\mathcal {Z}}_+}c_ke_{k,0} \in {\mathcal {A}}$, we introduce the bounded, right continuous step function

$$\begin{aligned} \rho _f(\eta ) = \langle f, E(\eta )f\rangle = \Vert E(\eta )f\Vert ^2 = \sum _{k \le \eta }|c_k|^2, \end{aligned}$$

the function $\rho _f(\eta )$ in a standard way defines the measure on ${\mathbb {R}}$, which we will also denote by $\rho _f$. Recall the that a function $\varphi : {\mathbb {R}} \rightarrow {\mathbb {R}}$ is called E-measurable if it is $\rho _f$-measurable for all $f \in {\mathcal {A}}$.

Note that for each $f = \sum _{k \in {\mathcal {Z}}_+}c_ke_{k,0} \in {\mathcal {A}}$, the set $(-\infty ,0) \bigcup _{k \in {\mathcal {Z}}_+} (k, k+1)$ has zero $\rho _f$-measure, while $\rho _f([k,k]) = |c_k|^2$.

That is, identifying E-measurable functions $\varphi$, that differ on the zero measure set, we have that each such class of equivalent functions is uniquely defined by a sequence $\varphi = \{\varphi (k)\}_ {k \in {\mathcal {Z}}_+}$. Then, the spectral theorem (see, e.g., [17, Theorem VIII.6] or [24, Theorem 7.14]) implies

Proposition 4.11

Given a E-measurable function $\varphi$, the operator

$$\begin{aligned} \varphi (N) = \int _{{\mathbb {R}}} \varphi (\eta )\,dE(\eta ) = \sum _{k \in {\mathcal {Z}}_+} \varphi (k)\textrm{P}_k \end{aligned}$$

is well defined on its domain

$$\begin{aligned} D_{\varphi } = \left\{ f = \sum _{k \in {\mathcal {Z}}_+}c_ke_{k,0} \in {\mathcal {A}} : \int _{{\mathbb {R}}} |\varphi (\eta )|^2\, d\rho _f(\eta ) = \sum _{k \in {\mathcal {Z}}_+} |\varphi (k)c_k|^2 < \infty \right\} . \end{aligned}$$

The operator $\varphi (N)$ is bounded, and thus defined on the whole ${\mathcal {A}}$, if and only if the sequence $\varphi = \{\varphi (k)\}_ {k \in {\mathcal {Z}}_+}$ is bounded.

Corollary 4.12

The set of all operators $\varphi (N)$, defined by (the class of equivalency) of E-measurable functions $\varphi$ constitutes the class of all, unbounded in general, radial operators in ${\mathcal {A}}$, which are mutually commute to each other being considered on the set of all finite linear combinations of the basis elements $e_{k,0}$, $k \in {\mathcal {Z}}_+$.

The set of all bounded operators $\varphi (N)$, defined by (the class of equivalency) of E-measurable functions $\varphi$, with $\varphi = \{\varphi (k)\}_ {k \in {\mathcal {Z}}_+} \in \ell _{\infty }$, coincides with von Neumann algebra of radial operators in ${\mathcal {A}}$.

Given a radial function $a = a(|z|) = a(r) \in L_{\infty }(J)$, let

$$\begin{aligned} \varphi _a(\eta ) = {\left\{ \begin{array}{ll} 0, &{} \textrm{for} \quad \eta <0 \\ \left( \int _J r^{\eta }\omega (\sqrt{r})dr\right) ^{-1}\int _J a(\sqrt{r})r^{\eta }\omega (\sqrt{r})dr, &{} \textrm{for} \quad \eta \ge 0 \end{array}\right. }, \end{aligned}$$

(4.9)

then the radial Toeplitz operator $T_a$, with symbol a, admits the representation

$$\begin{aligned} T_a = \varphi _a(N) = \int _{{\mathbb {R}}} \varphi _a(\eta )\,dE(\eta ) = \sum _{k \in {\mathcal {Z}}_+} \varphi _a(k)\textrm{P}_k = \sum _{k \in {\mathbb {Z}}_+}\gamma _a(k)\textrm{P}_k, \end{aligned}$$

where $\gamma _a(k)$ is given by (4.8).

It is worth mentioning that there are unbounded radial functions $a=a(|z|)$, for which the corresponding function (4.9) is bounded, and thus, the Toeplitz operator $\varphi _a(N)=T_a$, with symbol a, is bounded. Moreover, an unbounded radial symbol can be chosen in such a way that the corresponding bounded Toeplitz operators $\varphi _a(N)=T_a$ does not belong to the $C^*$-algebra generated by all Toeplitz operators with bounded radial symbols. The corresponding examples for the Bergman space case are given, e.g., in [11, 12].