Conjugations on $\varvec{L^2(\mathbb {T}^N)}$ and Invariant Subspaces

Abstract

We study conjugations on $L^2(\mathbb {T}^N)$ and their behaviour with respect to multiplication operators. Full characterizations of conjugations commuting or intertwining with multiplication operators are obtained. We also characterize conjugations leaving invariant subspaces being invariant for multiplication by independent variables. The subspaces not being the multiplication of the Hardy space $H^2(\mathbb {D}^N)$ by inner function are also considered.

1 Introduction

Let $\mathbb {T}$ denote the unit circle and $\mathbb {D}$ denote the unit disc. For a positive integer N, $L^2(\mathbb {T}^N)$, $L^{\infty }(\mathbb {T}^N)$ denote the standard $L^2$ and $L^{\infty }$ spaces with respect to normalized Lebesque measure $m_N$ on N-torus $\mathbb {T}^N$.

Let $H^2(\mathbb {D}^N)$ denote the Hardy space. Notice that these functions have a dual nature. Once we can see them as a holomorphic function on $\mathbb {D}^N$. On the other hand, they can be treated as an element of $L^2(\mathbb {T}^N)$. For $\varphi \in L^\infty (\mathbb {T}^N)$, a (essentially) bounded element of $L^2(\mathbb {T}^N)$, $M_\varphi$ denotes the multiplication operator on $L^2(\mathbb {T}^N)$, i.e., $M_\varphi f= \varphi f$ for $f\in L^2(\mathbb {T}^N)$.

In the paper we study conjugation operators on N–torus. By definition it is a bounded, antilinear operator $C :L^2(\mathbb {T}^N)\rightarrow L^2(\mathbb {T}^N)$ such that $C^2 = I$ and $\langle C f, C g \rangle = \langle g, f \rangle$ for every f, $g \in L^2(\mathbb {T}^N)$. Recall that such operator is isometric and onto.

In the space $L^2(\mathbb {T}^N)$ we can consider two natural conjugations: J and $J^\#$. Operation J conjugates the value of the functions (see (2.1)), while $J^\#$ conjugates the Fourier coefficients of the functions (see (2.2)). The behaviour of these two conjugations with respect to $M_{z_i}$, $i=1,\dots ,N$ is different. The conjugation $J^\#$ commutes with all $M_{z_i}$. On the other hand, the conjugation J intertwines $M_z$ and $M_{\bar{z}}$ for all $i=1,\dots ,N$ (see (2.3)).

Theory of conjugations on the unit circle $\mathbb {T}$ (i.e., $N=1$) has been intensively studied in the last few years (see e.g. [4, 5, 8, 9]). In [5, Theorem 2.2 and 2.4] and [4, Theorem 4.1], all conjugations commuting with (or intertwining) multiplication operator by the independent variable were described. One of the consequences of those results are characterizations (see [5, Theorems 5.2, Proposition 5.6]) of all conjugations leaving invariant subspaces being invariant for multiplication by independent variable.

The importance of conjugations is underlined by its connections with physics, where $\mathcal{PT}\mathcal{}$ symmetries in $L^2(\mathbb {R})$ are considered, see [1,2,3]. For a connection between conjugations on $L^2(\mathbb {T})$ and $L^2(\mathbb {R})$ see [7]. $\mathcal{PT}\mathcal{}$ symmetries can also be considered in $L^2(\mathbb {R}^N).$

The investigation in the following paper goes to multivariable case ($N>1$). The latter case becomes more interesting because of two reasons – the holomorphic function theory is much more complicated for $N>1$ and the Beurling theorem characterizing invariant subspaces is no longer true.

The structure of the paper is as follows. In Sect. 2, we present some natural conjugations in $L^2(\mathbb {T}^N)$ and give their basic properties. In Sect. 3, the characterization of conjugations intertwining with multiplication operators and its adjoints is settled. Sect. 4 presents a characterization of conjugations commuting with multiplication operators. Sects. 5, 6, 7 are devoted to conjugations and invariant subspaces. In Sect. 5, it is shown that there are no conjugations intertwining multiplication operators and its adjoint and leaving any subspace of $H^2(\mathbb {D}^N)$ invariant. In Sects. 6 and 7, we characterize all conjugations commuting with multiplication operators and leaving invariant given invariant subspace of $H^2(\mathbb {D}^N)$. Sect. 6 deals with invariant subspaces of type $\theta H^2(\mathbb {D}^N)$, where $\theta$ is an inner function. In Sect. 7, we consider subspaces fulfilling property (*), introduced in [6], which are different from those considered in Sect. 6.

2 Preliminaries

For a vector $z=(z_1, \ldots , z_N) \in \mathbb {C}^N$, we define its conjugation by $\bar{z}=(\overline{z_1}, \ldots , \overline{z_N})$.

In the space $L^2(\mathbb {T}^N)$ there are two conjugations, which seems very natural. First of them is J, which is simply a conjugation of the function, i.e.,

$$\begin{aligned} J f = \bar{f}, \quad f \in L^2(\mathbb {T}^N).\end{aligned}$$

(2.1)

The second one is $J^\#$, which derives from conjugating Fourier coefficients of functions in $L^2(\mathbb {T}^N)$. Precisely,

$$\begin{aligned} J^\# f = f^\#, \quad f \in L^2(\mathbb {T}^N), \end{aligned}$$

(2.2)

where

$$\begin{aligned} f^\# (z) = \overline{f(\bar{z})}, \quad z \in \mathbb {T}^N. \end{aligned}$$

The conjugation $J^\#$ keeps the Hardy space invariant ($J^\#H^2(\mathbb {D}^N)=H^2(\mathbb {D}^N)$). On the other hand, $JH^2(\mathbb {D}^N)=\overline{H^2(\mathbb {D}^N)}$.

Let $\mathcal {M}$ be a subspace of $H^2(\mathbb {D}^N)$. We denote the set $J^\# \mathcal {M}$ by $\mathcal {M}^\#$. Moreover, if $\mathcal {M}$ is closed, then $\mathcal {M}$ is called invariant if $z_i\mathcal {M}\subset \mathcal {M}$ for $i=1,\dots ,N$.

Let us summarize some basic properties below.

Proposition 2.1

Let $\mathcal {M}$ be a subspace of $H^2(\mathbb {D}^N)$. Then:

• (a) $\mathcal {M}^\# \subset H^2(\mathbb {D}^N)$,

• (b) If $\mathcal {M}$ is invariant, then $\mathcal {M}^\#$ is also invariant,

• (c) $J\mathcal {M}\subset \overline{H^2(\mathbb {D}^N)}$.

The behaviour of J and $J^\#$ with respect to the multiplication operators $M_{z_i}$, $i=1,\dots , N$, is also completely different. Precisely,

$$\begin{aligned} JM_{z_i}=M_{\bar{z}_i}J\qquad \text {and}\qquad J^\#M_{z_i}=M_{ z_i}J^\#\quad \text {for}\ i=1,\dots ,N. \end{aligned}$$

(2.3)

Let $\psi \in L^\infty (\mathbb {T}^N)$. Then

$$\begin{aligned} (M_\psi J)M_{z_i}=M_{\bar{z}_i}(M_\psi J)\ \text {and}\ (M_\psi J^\#)M_{z_i}=M_{ z_i}(M_\psi J^\#)\ \text {for}\ i=1,\dots ,N. \end{aligned}$$

(2.4)

A unimodular function $\psi \in L^\infty (\mathbb {T}^N)$ is called symmetric if $\psi \psi ^\#=1$ or equivalently $\psi (z)=\psi (\bar{z})$ for $z\in \mathbb {T}^N$.

Now, we set apart functions $\psi \in L^\infty (\mathbb {T}^N)$ such that operators $M_\psi J$ and $M_\psi J^\#$ are conjugations.

Proposition 2.2

Let $\psi \in L^\infty (\mathbb {T}^N)$. Then:

• (a) $C=M_\psi J$ is a conjugation if and only if $\psi$ is a unimodular function,

• (b) $C=M_\psi J^\#$ is a conjugation if and only if $\psi$ is a unimodular symmetric function.

Proof

We give a proof only for (b). Assume that $C= M_\psi J^\#$ is a conjugation. Then, for any $f,g\in L^2(\mathbb {T}^N)$, we have

$$\begin{aligned} \int g\bar{f}\,dm&=\langle g, f\rangle = \langle Cf,Cg\rangle =\langle \psi f^{\#},\psi g^{\#}\rangle \\ {}&=\int |\psi (z)|^2\,\overline{f(\bar{z})}g(\bar{z})\,dm(z)\\&=\int |\psi (\bar{z}) |^2\,\overline{f(z)}g(z)\,dm(z), \quad z \in \mathbb {T}^N. \end{aligned}$$

Hence $|\psi |=1$ a.e. on $\mathbb {T}^N$. For all $f\in L^2(\mathbb {T}^N)$ we have

$$\begin{aligned} f=C^2 f= M_\psi J^{\#}M_\psi J^{\#}f= M_\psi J^{\#}({\psi f^{\#}})=\psi \psi ^\# f, \end{aligned}$$

which implies that $\psi \psi ^\#=1$ a.e. on $\mathbb {T}^N$. Therefore, $\psi$ is symmetric, i.e., $\psi (z)=\psi (\bar{z})$ a.e. on $\mathbb {T}^N$. The reverse implication is clear. $\square$

Let $\alpha \in L^\infty (\mathbb {T}^N)$ be a unimodular function. If $\alpha \in H^2(\mathbb {D}^N)$, then we say that $\alpha$ is inner. We say that an inner function $\alpha$ is irreducible if it is not a product of two non-constant inner functions. For any inner functions $\alpha ,\beta$, we write $\alpha \leqslant \beta$ if $\frac{\beta }{\alpha }$ is inner. In such case $\beta H^2(\mathbb {D}^N)\subset \alpha H^2(\mathbb {D}^N)$. Finally, we say that $\alpha$ is self–reflected if $\alpha \leqslant \alpha ^\#$ and $\alpha ^\# \leqslant \alpha$.

3 Characterization of $M_z$–Conjugations

Consider a conjugation C on $L^2(\mathbb {T}^N)$. It is called $M_z$–conjugation¹ if $M_{z_i}C=CM_{\bar{z}_i}$, for $i=1,\dots ,N$. Observe that the natural conjugation J on $L^2(\mathbb {T}^N)$ defined by (2.1) is an $M_z$-conjugation.

Let us define a space $\mathcal {L}^{\infty }(\mathbb {T}^N)=\{M_\varphi : \varphi \in L^{\infty }(\mathbb {T}^N)\}$. It will be used in the proof of the next theorem, which completely characterizes all $M_z$–conjugations on $L^2(\mathbb {T}^N)$.

Theorem 3.1

Let C be a conjugation on $L^2(\mathbb {T}^N)$. The following conditions are equivalent:

• (a) C is $M_z$–conjugation,

• (b) $M_\varphi C=CM_{\bar{\varphi }}$ for all $\varphi \in L^{\infty }(\mathbb {T}^N)$,

• (c) there is a unimodular function $\psi \in L^{\infty }(\mathbb {T}^N)$ such that $C=JM_{\psi }$,

• (d) there is a unimodular function $\psi ^\prime \in L^{\infty }(\mathbb {T}^N)$ such that $C=M_{\psi ^\prime } J$.

Note that function $\psi ^\prime$ from condition (d) corresponds to function $\bar{\psi }$ from condition (c).

Proof

Equivalence of (a) and (b) is a consequence of standard approximation procedure. Condition (c) is equivalent to (d). In fact, it is enough to show that (b) implies (c).

If $\varphi \in L^{\infty }(\mathbb {T}^N)$, then

$$\begin{aligned} JCM_\varphi = JM_{\bar{\varphi }}C=M_\varphi JC. \end{aligned}$$

Thus JC is a bounded linear operator and JC commutes with all $M_\varphi \in \mathcal {L}^{\infty }(\mathbb {T}^N)$. By [10, Theorem 1.20], $\mathcal {L}^{\infty }(\mathbb {T}^N)$ is a m.a.s.a. (maximal abelian selfadjoint algebra) and as a consequence $JC \in \mathcal {L}^{\infty }(\mathbb {T}^N)$. Therefore, there is a function $\psi \in L^{\infty }(\mathbb {T}^N)$ such that $JC=M_\psi$. Finally, $C=JM_\psi$. $\square$

4 Characterization of $M_z$–Commuting conjugations

Let C be a conjugation on $L^2(\mathbb {T}^N)$. C is called $M_z$–commuting conjugation if $M_{z_i} C=C M_{z_i}$, $i=1,\dots ,N$. Notice that the natural conjugation $J^\#$ on $L^2(\mathbb {T}^N)$ defined by (2.2) is $M_z$–commuting conjugation.

The theorem below completely characterizes all $M_z$–commuting conjugations on $L^2(\mathbb {T}^N)$.

Theorem 4.1

Let C be a conjugation on $L^2(\mathbb {T}^N)$. The following conditions are equivalent:

• (a) C is an $M_z$–commuting conjugation,

• (b) $M_\varphi C=C M_{\varphi ^{\#}}$ for all $\varphi \in L^\infty (\mathbb {T}^N)$,

• (c) there is a symmetric unimodular function $\psi \in L^\infty (\mathbb {T}^N)$ such that $C=J^{\#}\,M_{\psi }$,

• (d) there is a symmetric unimodular function $\psi ^\prime \in L^\infty (\mathbb {T}^N)$ such that $C=M_{\psi ^\prime } J^{\#}$.

Note that function $\psi ^\prime$ from condition (d) corresponds to function $\psi ^\#$ from condition (c).

Proof

Assume (a) and note that $M_\varphi C=C M_{\varphi ^{\#}}$ if $\varphi$ is a polynomial. Now, equivalence of (a) and (b) is a consequence of standard approximation procedure. Clearly, condition (c) is equivalent to (d). It remains to show the implication from (b) to (c). If $\varphi \in L^{\infty }(\mathbb {T}^N)$, then

$$\begin{aligned} J^\#CM_\varphi = J^\#M_{\varphi ^\#}C=M_\varphi J^\#C. \end{aligned}$$

Using again the fact that $\mathcal {L}^{\infty }(\mathbb {T}^N)$ is a m.a.s.a., we get $J^\#C \in \mathcal {L}^{\infty }(\mathbb {T}^N)$. Therefore, there is a function $\psi \in L^\infty (\mathbb {T}^N)$ such that $J^{\#}C=M_{\psi }$. Hence $C=J^\#M_{\psi }=M_{\psi ^{\#}}J^\#$. By Proposition 2.2, the function $\psi$ is unimodular and symmetric. $\square$

5 Invariant Subspaces of $H^2(\mathbb {D}^N)$ and $M_z$–Conjugations

In this section, we study the image of a given invariant subspace of the Hardy space under $M_z$–conjugations.

Theorem 5.1

Let $\mathcal {M}\subset H^2(\mathbb {D}^N)$ be a non–zero invariant subspace. Then there does not exist $M_z$–conjugation C such that $C\mathcal {M}\subset H^2(\mathbb {D}^N)$.

Proof

Let C be an $M_z$–conjugation such that $C\mathcal {M}\subset H^2(\mathbb {D}^N)$. Let $n_1,\dots ,n_N$ be non–negative integers. Since $z_i\mathcal {M}\subset \mathcal {M}$, we get

$$\begin{aligned} z_1^{n_1}\cdots z_N^{n_N} \mathcal {M}\subset \mathcal {M}. \end{aligned}$$

From the fact that C is an $M_z$–conjugation, we deduce

$$\begin{aligned} \bar{z}_1^{n_1}\cdots \bar{z}_N^{n_N}\,C\mathcal {M}=C\, \big (z_1^{n_1}\cdots z_N^{n_N}\mathcal {M}\big ) \subset C\mathcal {M}\subset H^2(\mathbb {D}^N). \end{aligned}$$

Finally,

$$\begin{aligned} C\mathcal {M}\subset z_1^{n_1}\cdots z_N^{n_N} H^2(\mathbb {D}^N). \end{aligned}$$

Therefore,

$$\begin{aligned} C\mathcal {M}\subset \bigcap _{(n_1,\dots , n_N)\in \mathbb {N}^N} z_1^{n_1}\cdots z_N^{n_N} H^2(\mathbb {D}^N)=\{0\} \end{aligned}$$

and as a consequence $\mathcal {M}=\{0\}$. $\square$

Corollary 5.2

Let $\mathcal {M}\subset H^2(\mathbb {D}^N)$ be a non–zero invariant subspace. Then there does not exist $M_z$–conjugation C such that $C\mathcal {M}\subset \mathcal {M}$.

Corollary 5.3

There are no $M_z$–conjugations on $L^2(\mathbb {T}^N)$ which preserve $H^2(\mathbb {D}^N)$.

6 Invariant Subspaces of Type $\alpha H^2(\mathbb {D}^N)$ and $M_z$–Commuting Conjugations

Let $\alpha \in H^2(\mathbb {D}^N)$ be an inner function. Then $\alpha H^2(\mathbb {D}^N)$ is invariant. We will now investigate conjugations in $L^2(\mathbb {T}^N)$ which preserve subspaces of this form.

Let $\alpha$, $\beta$ be two inner functions. Since $H^2(\mathbb {D}^N)$ is invariant for $J^\#$, the operator

$$M_{\beta }J^{\#}M_{\bar{\alpha }}=M_{\beta \overline{\alpha ^\#}}J^\#\ :\ L^2(\mathbb {T}^N)\rightarrow L^2(\mathbb {T}^N)$$

is an antilinear isometry which maps $\alpha H^2(\mathbb {D}^N)$ onto $\beta H^2(\mathbb {D}^N)$ and commutes with $M_{z_i},\ 1,\dots ,N$. However, this operator does not have to be an involution. By Proposition 2.2, the necessary and sufficient condition is that $\beta \overline{\alpha ^\#}$ is symmetric (or equivalently $\alpha \alpha ^{\#}=\beta \beta ^{\#}$).

The theorem below characterizes all $M_z$–commuting conjugations mapping one invariant subspace of the considered type into another.

Theorem 6.1

Let $\theta$, $\alpha \in H^2(\mathbb {D}^N)$ be two inner functions. The following conditions are equivalent:

• (a) there exists an $M_z-$commuting conjugation C such that

  $$\begin{aligned} C(\alpha H^2(\mathbb {D}^N))\subset \theta H^2(\mathbb {D}^N),\end{aligned}$$

  (6.1)

• (b) there exists an inner function $\beta \in H^2(\mathbb {D}^N)$ such that $\theta \leqslant \beta$, $\alpha \alpha ^\# = \beta \beta ^\#$.

Moreover, every conjugation C fullfiling condition (a) has a form $C = M_{\overline{\alpha ^\#} \beta } J^\#$ for some $\beta \in H^2(\mathbb {D}^N)$ such that $\theta \leqslant \beta$, $\alpha \alpha ^\# = \beta \beta ^\#$, and $C(\alpha H^2(\mathbb {D}^N))= \beta H^2(\mathbb {D}^N)$.

Proof

Assume firstly that C is an $M_z$–commuting conjugation satisfying (6.1). By Theorem 4.1, $C=M_{\psi }J^{\#}$ for some unimodular symmetric function $\psi \in L^{\infty }(\mathbb {T}^N)$. In particular,

$$\begin{aligned} \theta H^2(\mathbb {D}^N)\ni C(\alpha ) =M_{\psi }J^{\#}\,\alpha =\psi \alpha ^{\#}, \end{aligned}$$

and there exists $u\in H^2(\mathbb {D}^N)$ such that $\psi \alpha ^{\#}=\theta u$. Note that u must be inner and so $\psi =\beta \overline{\alpha ^{\#}}$ with $\beta =\theta u$, $\theta \leqslant \beta$. Hence $C = M_{\overline{\alpha ^\#} \beta } J^\#$. Moreover, $\beta \overline{\alpha ^{\#}}$ is symmetric, i.e., $\beta \beta ^{\#}=\alpha \alpha ^{\#}$.

Now, let us assume that (b) holds. We define

$$\begin{aligned} C=M_{\overline{\alpha ^\#} \beta } J^\#. \end{aligned}$$

The condition $\beta \beta ^{\#}=\alpha \alpha ^{\#}$ guarantees that C is a conjugation. Then, for any $f\in H^2(\mathbb {D}^N)$, since $\theta \leqslant \beta$, we have

$$\begin{aligned} C(\alpha f)=M_{\beta \overline{\alpha ^{\#}}}J^{\#}(\alpha f)=M_{\beta \overline{\alpha ^{\#}}}(\alpha ^{\#}f^\#)=\beta f^\#\in \beta H^2(\mathbb {D}^N)\subset \theta H^2(\mathbb {D}^N). \end{aligned}$$

$\square$

The corollary below characterizes all possible subspaces of $H^2(\mathbb {D}^N)$ which are images of $\alpha H^2(\mathbb {D}^N)$ under some $M_z$–commuting conjugation. We can take $\theta =1$ in Theorem 6.1 (b) to get the following.

Corollary 6.2

Let $\alpha \in H^2(\mathbb {D}^N)$ be an inner function and let $\mathcal {M}\subset H^2(\mathbb {D}^N)$ be a closed subspace. Then the following conditions are equivalent:

• (a) there exists an $M_z$–commuting conjugation C such that $C(\alpha H^2(\mathbb {D}^N))=\mathcal {M}$,

• (b) there exists an inner function $\beta \in H^2(\mathbb {D}^N)$ such that $\beta \beta ^{\#}=\alpha \alpha ^{\#}$ and $\mathcal {M}= \beta H^2(\mathbb {D}^N)$

The another consequence of Theorem 6.1 is the following corollary.

Corollary 6.3

Let $\alpha$ be an inner function and let C be an $M_z$–commuting conjugation in $L^2(\mathbb {T}^N)$. Then:

• (a) $C(\alpha H^2(\mathbb {D}^N))\subset \alpha H^2(\mathbb {D}^N)$ if and only if $C= M_{\alpha \overline{\alpha ^\#}}J^{\#}$,

• (b) $C(\alpha H^2(\mathbb {D}^N))\subset \alpha ^{\#} H^2(\mathbb {D}^N)$ if and only if $C=J^{\#}$.

Moreover, if $\alpha$ is self-reflected, then $C(\alpha H^2(\mathbb {D}^N))\subset \alpha H^2(\mathbb {D}^N)$ if and only if $C= J^{\#}$.

Proof

By Theorem 6.1, $C(\alpha H^2(\mathbb {D}^N))\subset \alpha H^2(\mathbb {D}^N)$ if and only if there exists an inner function $\beta$ such that $\alpha \leqslant \beta$ and $\beta \beta ^{\#}=\alpha {\alpha ^{\#}}$. Thus $\beta =u\alpha$ for some inner function u and $u\alpha u^{\#}\alpha ^{\#}=\alpha {\alpha ^{\#}}$. Hence $uu^{\#}=1$. This is only possible if $u=\lambda \in \mathbb {T}$ since $u\in H^2(\mathbb {D}^N)$. This gives (a). The proof of (b) is similar. $\square$

In particular, we have the following fact.

Corollary 6.4

Let C be an $M_z$–commuting conjugation on $L^2(\mathbb {T}^N)$. Then

$$\begin{aligned} C(H^2(\mathbb {D}^N))\subset H^2(\mathbb {D}^N)\text { if and only if } C=J^{\#}. \end{aligned}$$

Remark 6.5

Let $\alpha$ and $\theta$ be two inner functions satisfying condition $\theta H^2(\mathbb {D}^N)\varsubsetneq \alpha H^2(\mathbb {D}^N).$ Then there does not exist $M_z$–commuting conjugation C such that

$$\begin{aligned} C(\alpha H^2(\mathbb {D}^N))\subset \theta H^2(\mathbb {D}^N). \end{aligned}$$

Indeed, if so,

$$\begin{aligned} C(\alpha H^2(\mathbb {D}^N))\subset \theta H^2(\mathbb {D}^N)\subset \alpha H^2(\mathbb {D}). \end{aligned}$$

Thus, by Corollary 6.3(a), $C= M_{\alpha \overline{\alpha ^\#}}J^{\#}$ and $\theta H^2(\mathbb {D}^N)= \alpha H^2(\mathbb {D})$. This is a contradiction. Compare this remark with Example 7.5.

Notice that, for given inner functions $\alpha$, $\theta$, the number of existing $M_z$–commuting conjugations fulfilling (6.1) can be different. We provide three examples. First of them is based on [11, Theorem 5.5.2.].

Example 6.6

Let $c\in (0,\frac{1}{8})$ and let $\alpha \in H^2(\mathbb {T}^2)$ be an inner function defined by

$$\begin{aligned} \alpha (z_1,z_2)=\frac{2z_1^2z_2-z_1+c}{2-z_1z_2+cz_1^2z_2}. \end{aligned}$$

Then $\alpha ^{\#}=\alpha$. Suppose that C is an $M_z$–commuting conjugation such that

$$\begin{aligned} C(\alpha H^2(\mathbb {T}^2))\subset \theta H^2(\mathbb {T}^2)\end{aligned}$$

(6.2)

for some inner function $\theta \in H^2(\mathbb {T}^2)$. By Theorem 6.1, there is an inner function $\beta$ such that $\theta \leqslant \beta$ and $\alpha ^2=\beta \beta ^{\#}$. Note that $\alpha$ is continuous on $\overline{\mathbb {D}^2}$. Then, by [12, Theorem 2.3.], $\alpha ^2$ has the unique Rudin-Ahern factorization. Since $\alpha$ is irreducible and $\alpha ^2=\beta \beta ^{\#}$, $\beta =\beta ^\#=\alpha$. Hence there is only one possible conjugation C fulfilling (6.2), namely $C=M_{\alpha \overline{\alpha ^\#}}J^\#=J^\#$. On the other hand, there are only two functions $\theta$ ($\theta =1$ or $\theta =\alpha$) which can satisfy (6.2), since $\theta \leqslant \beta =\alpha$.

Example 6.7

Let $\alpha _1$ and $\alpha _2$ be two different inner functions such that

• $\alpha _1$ and $\alpha _2$ are irreducible,

• $\alpha _j^\#\ne \alpha _k$ for $j,k\in \{1,2\},$

• both $\alpha _1(z_1,\cdot )$ and $\alpha _2(z_1, \cdot )$ are rational for $z_1$ in some subset of positive measure in $\mathbb {T}$.²

We can consider $\alpha _j(z_1,z_2)=\frac{z_1z_2-\lambda _j}{1-\overline{\lambda _j}z_1z_2}$ for $j=1,2$, where $\lambda _1,\lambda _2\in \mathbb {D}\,\setminus \,\mathbb {R}$ and $\lambda _1\ne \lambda _2$, $\lambda _1\ne \bar{\lambda }_2$. These functions satisfy the above three condition. The conditions will allow us to apply [12, Theorem 2.3].

Now, let us take $\alpha =\alpha _1\alpha _2$ and $\theta =\alpha ^\#_2$. Note that $\alpha H^2(\mathbb {D})\not \subset \theta H^2(\mathbb {D})$. We would like to find all $M_z$–commuting conjugations such that

$$\begin{aligned} C(\alpha H^2(\mathbb {D}))\subset \theta H^2(\mathbb {D}).\end{aligned}$$

(6.3)

Realize that, by the uniqueness of the Rudin-Ahern factorization, there are exactly two inner function satisfying condition (b) in Theorem 6.1: $\beta _1=\alpha _1\alpha ^\#_2$ and $\beta _2=\alpha ^\#_1\alpha ^\#_2$. Applying this theorem we obtain only two conjugations $C_1=M_{\alpha _1\overline{\alpha ^\#_1}} J^\#$ and $C_2=J^\#$.

Example 6.8

Consider $\alpha$ as above and $\theta =1$, i.e., condition $C(\alpha H^2(\mathbb {D}))\subset H^2(\mathbb {D})$. Then there are exactly four inner functions satisfying condition (b) in Theorem 6.1: $\beta _1=\alpha _1\alpha _2$, $\beta _2=\alpha _1^\#\alpha _2$, $\beta _3=\alpha _1\alpha _2^\#$, and $\beta _4=\alpha _1^\#\alpha _2^\#$. Hence, we obtain four possible conjugations: $C_1=M_{\alpha _1\alpha _2\overline{\alpha _1^\#}\overline{\alpha _2^\#}}J^\#$, $C_2=M_{\alpha _2\overline{\alpha _2^\#}}J^\#$, $C_3=M_{\alpha _1\overline{\alpha _1^\#}}J^\#$, and $C_4=J^\#$.

7 Invariant Subspaces With Condition (*) and $M_z$–Commuting Conjugations

The aim of this section is to study different invariant subspaces fulfilling condition (*) introduced in [6]. This class of subspaces is completely different from those one considered in Sect. 6 if $N>1$.

Following [11, Chapter 2], we introduce some function theory notation. We define multivariable Poisson kernel by the formula

$$\begin{aligned} P(z,w) = \prod _{i=1}^N P_{r_i}(\theta _i-\varphi _i), \quad z_j=r_je^{i\theta _j}, \quad w_j=e^{i\varphi _j}, \end{aligned}$$

where $P_r$ denotes one dimensional Poisson kernel

$$\begin{aligned} P_r(\theta )= \frac{1-r^2}{1-2r \cos \theta +r^2}. \end{aligned}$$

For a complex Borel measure $\mu$ on $\mathbb {T}^N$ we define its Poisson integral by the formula

$$\begin{aligned} P[\mu ](z) = \int _{\mathbb {T}^N} P(z,w) d \mu (w), \quad z \in \mathbb {D}^N. \end{aligned}$$

Let $f\in H^2(\mathbb {D}^N)$. Then there is a real singular measure $\sigma _f$ on $\mathbb {T}^N$ such that the least harmonic majorant of function $\log |f|$, denoted by $u(\log |f|)$, is given by

$$\begin{aligned} u(\log |f|)(z)= P[\log |f|dm_N+d\sigma _f](z), \quad z \in \mathbb {D}^N. \end{aligned}$$

(7.1)

It is known that $d\sigma _f\leqslant 0$ for every $f\in H^2(\mathbb {D}^N)$.

For the invariant subspace $\mathcal {M}$, following [6], we define the zero set

$$\begin{aligned} Z(\mathcal {M})=\{z\in \mathbb {D}^N:f(z)=0 \text { for all } f\in \mathcal {M}\} \end{aligned}$$

and the measure

$$\begin{aligned} Z_{\partial }(\mathcal {M})=\inf \{-d\sigma _f:f\in \mathcal {M}, f\not =0\}. \end{aligned}$$

The invariant subspace $\mathcal {M}$ is said to satisfy condition (*) if $Z_\partial (\mathcal {M})=0$ and the real $(2N-2)$–dimensional Hausdorff measure of $Z(\mathcal {M})$ is 0.

Proposition 7.1

Let $\mathcal {M}\subset H^2(\mathbb {D}^N)$ be an invariant subspace. If $\mathcal {M}$ satisfies condition (*), then $\mathcal {M}^\#$ satisfies condition (*).

Proof

Let $\mu$ be a measure on $\mathbb {T}^N$ and let $\mu ^c$ be defined as follows

$$\begin{aligned} \mu ^c(\omega )=\mu (\{\bar{w}:w\in \omega \}). \end{aligned}$$

Notice that $m_N^c=m_N$. Moreover,

$$\begin{aligned} h_\#^{(r)}(z)&= \int _{\mathbb {T}^N} P\Big (\frac{z}{r}, w\Big ) \log |f^\#(rw)| d m_N(w) \\&=\int _{\mathbb {T}^N} P\Big (\frac{z}{r}, w\Big ) \log |f(r\bar{w})| d m_N(w) \\&=\int _{\mathbb {T}^N} P\Big (\frac{z}{r}, \bar{w}\Big ) \log |f(rw)| d m_N(w) \\&=\int _{\mathbb {T}^N} P\Big (\frac{\bar{z}}{ r}, w \Big ) \log |f(rw)| d m_N(w)=h^{(r)}(\bar{z}). \end{aligned}$$

From [11, Theorem 3.2.4] we get the connection between least N–harmonic majorants

$$u(\log |f|)(z) = \lim \limits _{r\rightarrow 1^-} h^{(r)}(z)= \lim \limits _{r\rightarrow 1^-} h_\#^{(r)}(\bar{z}) = u(\log |f^\#|)(\bar{z}).$$

Hence by (7.1)

$$\begin{aligned} \int _{\mathbb {T}^N} P(z,w)d\sigma _{f^\#}(w)= \int _{\mathbb {T}^N} P(z,w)d\sigma ^c_{f}(w), \quad z \in \mathbb {D}^N. \end{aligned}$$

Consequently, by uniqueness of the measure, $\sigma _{f^\#}=\sigma _f^c$. Now, one can easily see that $Z_{\partial }(\mathcal {M})=Z_{\partial }(\mathcal {M}^\#)$. The easy observation $Z(\mathcal {M}^\#)=\overline{Z(\mathcal {M})}$ finishes the proof. $\square$

Theorem 7.2

Let C be an $M_z$–commuting conjugation on $L^2(\mathbb {T}^N)$. Let $\mathcal {M}\subset H^2(\mathbb {D}^N)$ be an invariant subspace fulfilling condition (*). If $C\mathcal {M}\subset H^2(\mathbb {D}^N)$ then there is a symmetric inner function $\psi$ such that $C=M_\psi J^\#$ and $C\mathcal {M}=\psi \mathcal {M}^\#$.

Proof

Since C is $M_z$–commuting conjugation thus by Theorem 4.1 there is a unimodular symmetric function $\psi \in L^\infty (\mathbb {T}^N)$ such that $C=M_\psi J^\#=J^\# \psi ^\#$. Thus

$$\begin{aligned} \psi ^\#\mathcal {M}=J^\# J^\# {\psi }^\#\mathcal {M}=J^\#C\mathcal {M}\subset J^\#H^2(\mathbb {D}^N)=H^2(\mathbb {D}^N). \end{aligned}$$

The subspace $\mathcal {M}$ is invariant subspace fulfilling condition (*). Moreover $\psi ^\#\mathcal {M}\subset H^2(\mathbb {D}^N)$. Hence by [6, Theorem 1] function $\psi ^\#$ is holomorphic and in particular inner. Therefore $\psi$ is a symmetric inner function. $\square$

Notice that the above theorem can also be proved using Proposition 7.1.

Now, for any $\alpha \in \mathbb {D}^N$, let $H^2_\alpha$ be defined as

$$\begin{aligned} H^2_\alpha =\{f\in H^2(\mathbb {D}^N):f(\alpha )=0\}. \end{aligned}$$

In [13, Example 2.9] it was shown that $H^2_\alpha$ is invariant and fulfills property (*).

As a consequence of Theorem 7.2 and the fact that $J^\# H^2_\alpha =H^2_{\bar{\alpha }}$ for any $\alpha \in \mathbb {D}^N$, we get the following.

Theorem 7.3

Let C be an $M_z$–commuting conjugation in $L^2(\mathbb {T}^N)$. Let $\alpha \in \mathbb {D}^N$. If $C H^2_\alpha \subset H^2(\mathbb {D}^N)$ then there is an inner symmetric function $\psi$ such that $C=M_\psi J^\#$ and $C H^2_\alpha =\psi H^2_{\bar{\alpha }}$.

To illustrate the above theorem we give two examples, showing that there exists a lot of conjugations leaving invariant space $H^2_\alpha$.

Example 7.4

Let $\alpha \in \mathbb {D}^N\cap \mathbb {R}^N$. Then there are infinitely many conjugations C on $L^2(\mathbb {T}^N)$ such that $C H^2_\alpha \subset H^2_\alpha$. In fact, there is a one-to-one correspondence between such conjugations and the set of all inner symmetric functions.

Example 7.5

Let $\alpha =(\alpha _1,\ldots ,\alpha _N)\in \mathbb {D}^N\, \setminus \,\mathbb {R}^N$. Then there are infinitely many conjugations C on $L^2(\mathbb {T}^N)$ such that $C H^2_\alpha \subset H^2_\alpha$. If $\psi$ is any symmetric inner function such that $\psi (\alpha )=0$, then $C=M_\psi J^\#$ is a conjugation such that $C H^2_\alpha \subset H^2_\alpha$. It is clear that, in this case, the inclusion is strict. Compare this example with Remark 6.5.

To give a concrete example of $\psi$, denote by $B_\alpha$ the Blaschke product, i.e.,

$$\begin{aligned} B_\alpha (z)=\prod _{j=1}^N\frac{z_j-\alpha _j}{1-\bar{\alpha _j}z_j} \end{aligned}$$

for $z=(z_1,\ldots ,z_N)\in \mathbb {D}^N$. Now, take a function $\psi$ defined by $\psi (z)=B_\alpha (z)B_{\overline{\alpha }}(z)\varphi$, $z\in \mathbb {D}^N$, where $\varphi$ is a symmetric inner function (e.g. $\varphi \equiv 1$). It is easily seen that $\psi$ is symmetric and $C=M_\psi J^\#$ satisfies $CH^2_\alpha \subset H^2_\alpha$.