Commutants and complex symmetry of finite Blaschke product multiplication operator in $L^2(\mathbb{T})$

Abstract

Consider the multiplication operator M_B in $L^2(\mathbb{T})$, where the symbol B is a finite Blaschke product. In this article, we characterize the commutant of M_B in $L^2(\mathbb{T})$. As an application of this characterization result, we explicitly determine the class of conjugations commuting with $M_{z^2}$ or making $M_{z^2}$ complex symmetric by introducing a new class of conjugations in $L^2(\mathbb{T})$. Moreover, we analyse their properties while keeping the whole Hardy space, model space and Beurling-type subspaces invariant. Furthermore, we extended our study concerning conjugations in the case of finite Blaschke products.

1. Introduction

Suppose T is a bounded linear operator on a separable Hilbert space $\mathcal{H}$. Then, the commutant of T is usually denoted by $\{T\}^\prime$ and defined by $\{T\}^\prime :=\{S\in \mathcal{B}(\mathcal{H}): ST=TS\}$, where $\mathcal{B}(\mathcal{H})$ is the space of all bounded linear operators on $\mathcal{H}$. Given a bounded linear operator T, it is always interesting to find the commutant $\{T\}^\prime$ since it plays an important role in determining the structure of the operator. Another strong reason for identifying the commutant of an operator T is to recognize all reducing subspaces of T. In this direction, there are various nice works available in the literature, specially for analytic multiplication operators on some well-known function spaces like the Hardy space, the Bergman space and also for reproducing kernel Hilbert spaces. The details can be found in [13, 23, 24, 35, 39].

Let $\mathbb{D}=\{z\in \mathbb{C}:~|z| \lt 1\}$ denote the unit disc in the complex plane and $\mathbb{T}=\{z\in \mathbb{C}:~|z|=1\}$ be the unit circle. Let us denote by $L^2 (\mathbb{T})$ (or simply L ²) the space of all square-integrable functions over the unit circle $\mathbb{T}$. The Banach space of all essentially bounded functions on the unit circle $\mathbb{T}$ is denoted by $L^\infty(\mathbb{T})$ (or $L^\infty$ in short). Therefore, if $f\in L^2(\mathbb{T})$, then the function f can be expressed as:

\begin{equation*}f(z)=\sum_{n=-\infty}^{\infty}\hat{f}(n)z^n,\end{equation*}

where $\hat{f}(n)$ denotes the nth Fourier coefficient of f and $\sum\limits_{n=-\infty}^{\infty}|\hat{f}(n)|^2 \lt \infty$. For $\phi\in L^{\infty}$, let M_ϕ denote the multiplication operator on L ² defined as: $M_{\phi}(f)=\phi f,\quad f\in L^2.$

Recall that the classical Hardy space over the unit disc $\mathbb{D}$ is denoted by $H^2(\mathbb{D})$ and defined by:

\begin{equation*} H^2(\mathbb{D}):=\Big\{f(z)=\sum_{n=0}^{\infty}a_nz^n:~\|f\|_{H^2(\mathbb{D})}^2:=\sum_{n=0}^{\infty}|a_n|^2,~z\in \mathbb{D},~a_n\in \mathbb{C}\Big\}. \end{equation*}

For $f\in H^2(\mathbb{D})$, the non-tangential boundary limit (radial limit) $\tilde{f}(e^{i\theta})=\lim_{r\to 1-}f(re^{i\theta})$ exists almost everywhere on $\mathbb{T}$ [14, 15, 28]. Therefore, $H^2(\mathbb{D})$ can be embedded isometrically as a closed subspace of $L^2(\mathbb{T})$ by identifying $H^2(\mathbb{D})$ through the non-tangential boundary limits of the $H^2(\mathbb{D})$ functions. In other words, we can recognize the Hardy space $H^2(\mathbb{D})$ as a closed subspace of $L^2(\mathbb{T})$, say H ², consisting of L ² functions having negative Fourier coefficients zero. We denote by $H^{\infty}$ the space of all bounded analytic functions on $\mathbb{D}$ that can be identified with a closed subspace of $L^{\infty}$, that is $H^\infty:= L^\infty \cap H^2$. An inner function θ is an element in $H^\infty$ such that $|\theta | =1$ almost everywhere on $\mathbb{T}$. For $\phi\in L^{\infty}$, define the Toeplitz operator $T_{\phi}:H^2\to H^2$ by $T_{\phi}f=P(\phi f),$ where P is the orthogonal projection from L ² onto H ². Recall that the unilateral shift S on H ² is defined by $Sf(z)=zf(z), f\in H^2$, a restriction of the bilateral shift M_z on L ². A well-known theorem due to Beurling characterizes all the S-invariant subspaces of H ². In other words, any S-invariant subspace of H ² is of the form $\theta H^2$, where θ is an inner function. For any inner function θ, the model space K_θ is defined by $K_\theta := H^2\ominus \theta H^2$, which is $S^*$ invariant. For more on model spaces and related stuff, we refer the reader to [14, 15, 17, 28].

The commutant of S in H ² is $\{T_{\phi}:~\phi\in H^{\infty}\}$ and the commutant of M_z in L ² is $\{M_{\phi}:~\phi\in L^{\infty}\}$ [28, 32]. In general, it is highly non-trivial to describe the commutant of a given bounded linear operator. In this direction, the commutant of more general shift operators and Toeplitz operators in various function spaces has been studied by many authors [1, 6–8, 12, 25, 26, 30, 33].

For a finite Blaschke product B, let M_B be the operator of multiplication by B in $L^2(\mathbb{T})$. The commutants of finite Blaschke product Toeplitz operators T_B are ‘well known’ for the Hardy space H ² [6]. So in some sense, we can say that these are the ‘same’ operators that commute with M_B on other Hilbert spaces. In 2015, Cowen and Wahl [5] gave a detailed independent description of the operators on the Hilbert space $\mathcal{H}$ of analytic functions defined on $\mathbb{D}$ that commutes with M_B. Abkar et al. [1] gives a complete description of the commutant of $M_{z^n}$ on a family of analytic function spaces on $\mathbb{D}$, which include the Bergman space, the Hardy space and the Dirichlet space. Note that these are shift operators of higher multiplicity, in other words, special multiplication operator by particular finite Blaschke. More explicitly, the authors in [1] provide the description regarding the commutant of $M_{z^n}$ by identifying with n copies of the multiplier algebra of the underlying function spaces. Recently, Gallardo and Partington [16] provide a characterization of the commutant of T_B extending the work on the weighted Bergman spaces.

Our primary goal in this paper is to characterize the commutant of M_B independently in the $L^2(\mathbb{T})$-space and obtain a detailed description of the commutant in terms of $L^\infty$-functions. As a consequence, by using the precise structure of the commutator, we will obtain some applications towards characterizing conjugations in $L^2(\mathbb{T})$.

Now, the study of conjugations and complex symmetric operators has become a widely interested research area because of its significance in the fields of operator theory, complex analysis and physics. Recently, several authors have been interested in non-Hermitian quantum mechanics and the spectral analysis of certain complex symmetric operators (for more details, see [2, 18]). One of the motivations for studying conjugations comes from the study of complex symmetric operators. The study of complex symmetric (in short, C-symmetric) operators was initiated by Garcia et al. [19–22]. Technically speaking, a conjugation C is an antilinear and isometric involution on $\mathcal{H}$. In other words, a conjugate-linear operator $C:\mathcal{H} \to \mathcal{H}$ is said to be a conjugation if it satisfies the following two conditions:

1. (a) $C^2 =I_{\mathcal{H}}$, (the identity operator on $\mathcal{H}$);

2. (b) $\langle Cf,Cg \rangle = \langle g,f \rangle , \forall f,g \in \mathcal{H}$.

It is interesting to observe that the conjugation is a straightforward generalization of the conjugate-linear map $z\mapsto \bar{z}$ on the one-dimensional Hilbert space $\mathbb{C}$. Moreover, we can also interpret conjugation as a special kind of antilinear operator and note that the antilinear operator is the only type of nonlinear operator that is important in the field of quantum mechanics (for more details, see [29, 34, 38]). In this connection, it is remarkable to mention that Garcia and Putinar have shown in [19] that for any given conjugation C on $\mathcal{H}$, there exists an orthonormal basis $\{e_n: n\in \mathbb{N}_0\}$ such that $Ce_n=e_n$ for all $n\in \mathbb{N}_0$, where $\mathbb{N}_0$ denotes the set of all non-negative integers.

Definition 1.1. An operator $T\in \mathcal{B}(\mathcal{H})$ is called C-symmetric if there exists a conjugation C on $\mathcal{H}$ such that $CTC=T^*$, where $T^*$ is the adjoint of T. If T is C-symmetric for some conjugation C, then T is called complex symmetric.

The class of complex symmetric operators includes all normal operators, Hankel operators, truncated Toeplitz operators and Volterra integration operators. For more on complex symmetric operators and related topics, including historical comments, we refer the reader to [19–22, 27, 37] and the references cited therein. We denote by $\mathcal{B}A(\mathcal{H})$ the space of all bounded antilinear operators on $\mathcal{H}$. Recall that for $A \in \mathcal{B}A(\mathcal{H})$ there exists a unique antilinear operator $A^\sharp$, known as the antilinear adjoint of A, defined by the equality:

\begin{equation*} \langle Af,g\rangle =\overline{\langle f,A^\sharp g\rangle}.\end{equation*}

It is important to observe that $C^\sharp=C$ for any conjugation C, $(AB)^{\sharp}=B^*A^{\sharp}$ and $(BA)^{\sharp}=A^{\sharp}B^*$ for $A \in \mathcal{B}A(\mathcal{H})$, $B\in\mathcal{B}(\mathcal{H})$ (see e.g., [9]).

Two natural conjugations J and $J^\star$ on $L^2(\mathbb{T})$ are defined as follows:

(1.1)\begin{equation} Jf=\bar{f}, \quad J^\star f= f^\#,\quad \text{where}\quad \bar{f}(z)=\overline{f(z)}\quad \text{and} \quad f^\#(z)=\overline{f(\bar{z})}. \end{equation}

It is important to observe that the conjugation J intertwines the operators M_z and $M_{\bar{z}}$. In other words M_z is J-symmetric, that is $JM_zJ =M_{\bar{z}}$. Moreover, $JH^2 = \overline{H^2}$, that is J maps an analytic function to a co-analytic function. On the other hand, the conjugation $J^\star$ has different nature in comparison to J. For example, $J^\star M_z =M_zJ^\star$ and it preserves the Hardy space H ². Furthermore, it is worth mentioning that the map $J^\star$ has a connection with model spaces [4] and Hankel operators [28]. In this connection, recently, Câmara-Garlicka-Łanucha-Ptak raised a natural question in [10] regarding the characterization of conjugations that possess similar kinds of properties as J and $J^\star$. More precisely, they characterize the class of conjugations that commute with M_z and also classify the class of conjugations intertwining the operators M_z and $M_{\bar{z}}$ in scalar-valued $L^2(\mathbb{T})$ space [9, 10] as well as in vector-valued $L^2(\mathcal{H})$ space [11] over the unit circle $\mathbb{T}$. Therefore it is natural to study the complex symmetry of M_B in $L^2(\mathbb{T})$, where B is a finite Blaschke product. To this end, we begin with the study of the complex symmetry of $M_{z^2}$ in $L^2(\mathbb{T})$ since it will help to complement the study of M_B. It is important to observe that the commutant of M_z is properly contained in the commutant of $M_{z^2}$ and also the class of conjugations intertwining the operators $M_{z^2}$, $M_{\bar{z}^2}$ in L ² is bigger than the class of conjugations intertwining the operators M_z, $M_{\bar{z}}$ in L ². In this direction, we begin with the following two questions.

Question 1. Which class of conjugations commute with $M_{z^2}$ in $L^2(\mathbb{T})$ but not with M_z

Question 2. Which class of conjugations makes the operator $M_{z^2}$ complex symmetric in $L^2(\mathbb{T}) $ but does not possess the same characteristic with M_z?

Later in $\S$ 4, we provide appropriate answers to the above questions. Note that our work mainly focus on the case for $only~M_{z^2}$commuting conjugations and $M_{z^2},M_{\bar{z}^2}$-intertwining (but not $M_{z},M_{\bar{z}}$-intertwining ) conjugations throughout this article. Note that the operator M_z in $\mathbb{C}^n$-valued L ² space over $\mathbb{T}$ is unitary equivalent to M_B in the classical L ² space over $\mathbb{T}$, where B is a finite Blaschke product of degree n. Since the characterization of the class of conjugations that commute with M_z and also the class of conjugations intertwining the operators M_z and $M_{\bar{z}}$ have been studied in the vector-valued $L^2(\mathcal{H})$ space in [11], then in some sense such characterization results regarding $M_{z^2}$ can also be given in $L^2(\mathbb{T})$ space. But the novelty of our characterization results lies in the fact that we have the precise structure of such conjugations, which cannot be achieved easily using the above-mentioned unitary equivalence. Moreover, a further benefit of our description is that one can independently classify the conjugations in $L^2(\mathbb{T})$, which do not possess similar properties with M_z.

In other words, the precise (or major) contributions in this article are the following:

• • We identify the commutant of M_B in L ², where B is a finite Blaschke product.

• • Using the above characterization, we classify the set of conjugations that commute with $M_{z^2}$ but fail to commute with M_z.

• • Class of conjugations intertwining the operators $M_{z^2}$ and $M_{\bar{z}^2}$ but not necessarily intertwining the operators M_z and $M_{\bar{z}}$ has been identified.

• • As a consequence of our main results, we investigate the properties of the above classes of conjugations keeping the whole Hardy space, model space and Beurling-type subspaces invariant.

• • At the end, we extend our discussions in connection to M_B, where B is a finite Blaschke product.

The paper is organized as follows: In $\S$ 2, we characterize the commutant of the multiplication operator M_B in $L^2(\mathbb{T})$ for any finite Blaschke product B. Some new conjugations in $L^2(\mathbb{T})$ have been introduced as a generalization of some well-known conjugations, in $\S$ 3. Next, we provide the answer to Question 1 and Question 2 in $\S$ 4. In $\S$ 5, we discuss the properties of such conjugations that keep invariant the entire Hardy space, model space and Beurling-type subspaces. Section 6 is devoted to studying the related results in the case of finite Blaschke product B and, in particular, for $B(z)=z^n$.

2. Commutant of M_B in $L^2(\mathbb{T})$

In this section, we provide a characterization of commutants of the multiplication operator M_B having symbol B, where B is a finite Blaschke product. It is worth mentioning that, in [1, 5, 16], the authors obtained the commutant of M_B in various reproducing kernel Hilbert spaces of analytic functions. In contrast with the existing results, we obtain the commutant of M_B { in $L^2(\mathbb{T})$, and our characterization result is different from the existing ones. We denote $\{M_B\}^\prime$ to be the set of all bounded linear operators on $L^2(\mathbb{T})$ that commutes with M_B. In other words, if $T\in \{M_B\}^\prime$, then $T:L^2(\mathbb{T})\to L^2(\mathbb{T})$ is a bounded linear operator such that $TM_B=M_BT$. Our main aim is to characterize the set $\{M_B\}^\prime$, and to achieve our goal; we need the following useful lemma.

Lemma 2.1. Let $\phi \in L^2(\mathbb{T})$ be such that $\phi f \in L^2(\mathbb{T})$ for all $f\in L^2(\mathbb{T})$, then $\phi \in L^\infty (\mathbb{T})$.

Proof. By the Closed-Graph theorem, we can show that the multiplication operator M_ϕ on $L^2(\mathbb{T})$, defined by:

\begin{align*} M_\phi (f) =\phi f, \end{align*}

is a well-defined bounded linear operator. Then using similar argument as in [31, Theorem 1.6.2], one can show that $\phi \in L^\infty (\mathbb{T})$.

Before going to the main theorem in this section, we need to recall some basic facts. For any inner function θ, by Wold-type Decomposition [36], we can view $L^2(\mathbb{T})$ as follows:

(2.1)\begin{equation} L^2(\mathbb{T}) =\bigoplus _{k=-\infty}^{\infty}\theta ^k K_\theta. \end{equation}

The proof of the above decomposition (2.1) is easily obtained from the fact that the corresponding Toeplitz operator T_θ is a pure isometry and using the decomposition $L^2(\mathbb{T})=H^2 \oplus \overline{H^2_0}$. For reader convenience, we briefly discuss the proof of the above decomposition. Applying the classical Von Neumann-Wold Decomposition theorem on H ² associated with the shift operator T_θ we obtain the following: $H^2 (\mathbb{T}) =\bigoplus\limits _{k=0}^{\infty}\theta ^k K_\theta .$ Therefore, for any $h\in H^2(\mathbb{T})$, there exists a sequence $\{h_k\}_{k=0}^\infty$ in K_θ such that $h=\sum\limits_{k=0}^{\infty}h_k\theta^k$ and $\|h\lVert^2=\sum\limits_{k=0}^{\infty}\|h_k\lVert^2$. Note that, $\theta^k K_\theta \perp \theta^m K_\theta$ for any $k\neq m\in \mathbb{Z}$. As a consequence, we have $\bigoplus\limits _{k=-\infty}^{\infty}\theta ^k K_\theta \subseteq L^2(\mathbb{T}).$ Let $f\in L^2(\mathbb{T})=H^2\oplus \overline{H^2_0}$, then there exist $f_1\in H^2$ and $f_2 \in \overline{H^2_0}$ such that $f=f_1\oplus f_2$. Again, $f_1=\sum\limits_{k=0}^{\infty}\tilde{f}_k\theta^k$, where $\{\tilde{f}_k\}_{k=0}^\infty \in K_\theta$ and $f_2=\overline{zg}$ for some $g\in H^2$. Moreover, we have $g=\sum\limits_{k=0}^{\infty}g_k\theta^k$, where $\{g_k\}_{k=0}^\infty \in K_\theta$ and hence,

\begin{align*} f_2 = \sum_{k=0}^{\infty}\overline{zg_k}{\theta}^{-k} =\sum_{k=0}^{\infty}\overline{zg_k}\theta \theta^{-(k+1)}=\sum_{k=0}^{\infty}(c_\theta g_k)\theta^{-(k+1)} &= \sum_{k=-\infty}^{-1}\tilde{g}_k\theta^k, \end{align*}

where $\tilde{g}_k=c_\theta g_k \in K_\theta $. Finally, combining all the above facts, we conclude, $f=\sum\limits_{k=-\infty}^{\infty}f_k\theta^k,$ where $f_k= \tilde{f}_k$ for $k\geq 0$ and $f_k= \tilde{g}_k$ for k < 0.

We know that for a finite Blaschke product B, the dimension of the model space $K_B:= H^2\ominus BH^2$ is finite. In fact, if the degree of B is a finite natural number n, then $dim(K_B)=n$. Let $\{h_1,h_2,\ldots,h_n \}$ be an orthonormal basis of K_B. Now, due to Wold-type decomposition as discussed above, the space $L^2(\mathbb{T})$ can be viewed as follows:

\begin{equation*} L^2(\mathbb{T})= \bigoplus_{k=-\infty}^{\infty}B^k K_B. \end{equation*}

Using the above decomposition we identify $L^2(\mathbb{T})$ as orthogonal direct sum of n-closed subspaces, that is:

(2.2)\begin{equation} L^2(\mathbb{T})= \mathcal{H}_1(h_1,B) \oplus \mathcal{H}_2(h_2,B) \oplus \cdots \oplus \mathcal{H}_n(h_n,B), \end{equation}

where

\begin{equation*} \mathcal{H}_1(h_1,B):=\bigvee_{k=-\infty}^\infty \{h_1 B^k\}, ~\mathcal{H}_2(h_2,B):=\bigvee_{k=-\infty}^\infty \{h_2 B^k\},\ldots ,\mathcal{H}_n(h_n,B):=\bigvee_{k=-\infty}^\infty \{h_nB^k\}. \end{equation*}

Therefore, for any $f\in L^2(\mathbb{T}) $ can be represented as $f=f_1+f_2+ \cdots +f_n$, where $f_i \in \mathcal{H}_i(h_i,B)$ for $i\in \{1,2,\ldots ,n\}$. For a finite sequence $\{\lambda_k \}_{1\leq k \leq n}$ (maybe repeating) in $\mathbb{D}$, we consider the following finite Blaschke product:

(2.3)\begin{equation} B(z)=\prod_{k=1}^n\dfrac{\lambda_k -z}{1-\bar{\lambda}_k z},quad z\in \overline{\mathbb{D}}. \end{equation}

For $1\leq j\leq n$, consider the following functions:

(2.4)\begin{align} e_j(z) = &\Big(\prod_{k=1}^{j-1}\dfrac{\lambda_k -z}{1-\bar{\lambda}_k z}\Big)\dfrac{\sqrt{1-|\lambda_j|^2}}{1-\bar{\lambda}_j z}, \end{align}

then it is easy to observe that $\{e_1,e_2,\ldots ,e_n \}$ forms an orthonormal basis of K_B. In fact, this is a standard orthonormal basis of K_B, and for more details, we refer to [[14], Theorem: 14.7]. It is worth mentioning some nice properties of e_j as follows:

1. (i) $e_j \in H^\infty (\mathbb{T})$ for $1\leq j\leq n$.

2. (ii) For $1\leq j\leq n$, each e_j is invertible in $L^\infty (\mathbb{T})$, and

   \begin{equation*} e_j ^{-1}(z)=\Big(\prod_{k=1}^{j-1}\dfrac{1-\bar{\lambda}_k z}{\lambda_k -z}\Big)\dfrac{1-\bar{\lambda}_j z}{\sqrt{1-|\lambda_j|^2}}. \end{equation*}

Now, we proceed to find the commutant of M_B. To this end, we use the decomposition of $L^2(\mathbb{T})$ as an orthogonal direct sum of n-subspaces given in (2.2), along with the fact that B is of the form (2.3) and $\{e_1,e_2,\ldots ,e_n \}$ is the required orthonormal basis as mentioned in (2.4). Therefore, for any $f\in L^2(\mathbb{T}) \big(= \mathcal{H}_1 (e_1,B) \oplus \mathcal{H}_2(e_2,B) \oplus \cdots \oplus \mathcal{H}_n(e_n,B)\big)$ can be written as $f=f_1\oplus f_2\oplus\cdots \oplus f_n$, where $f_i \in \mathcal{H}_i(e_i,B) $ for each $i\in \{1,2,\ldots ,n \}$. Let $S\in \{M_B \}^\prime$, then S is a bounded linear operator on $L^2(\mathbb{T})$ such that $M_BS=SM_B$. Since $f_j\in \mathcal{H}_j(e_j,B) $, then there exists $g_j=\sum\limits_{k=-\infty}^{\infty}a^j_kB^k \in L^2(\mathbb{T})$ such that $f_j=e_jg_j$ for $j\in \{1,2, \ldots ,n \}$. Since $e_j^{-1}\in L^\infty(\mathbb{T})$, then by using the fact that S commutes with M_B we have:

\begin{align*} Sf_j &= (Se_j)g_j =(Se_j)e_j^{-1}e_j g_j =(Se_j)e_j^{-1}f_j=\phi_jf_j, \end{align*}

where $\phi_j = (Se_j)e_j^{-1} \in L^2(\mathbb{T})$. Next we claim that $\phi_j \in L^\infty (\mathbb{T})$ for $j\in \{1,2,\ldots ,n\}$. We already proved that $\phi_jf_j =Sf_j\in L^2(\mathbb{T})$. Now for i ≠ j,

\begin{align*} \phi_jf_i & = \phi_j(e_ig_i)=e_j^{-1}e_j \phi_j(e_ig_i)=e_j^{-1}e_i(\phi_je_jg_i)=e_j^{-1}e_i S(h_j) \in L^2(\mathbb{T}), \end{align*}

where $h_j= e_jg_i \in \mathcal{H}_j(e_j,B)$. Thus, we have $\phi_jf=\phi_j f_1+\phi_jf_2+\cdots +\phi_jf_n\in L^2(\mathbb{T})$ for any $f\in L^2(\mathbb{T})$. Therefore by using Lemma 2.1 we conclude $\phi_j \in L^\infty(\mathbb{T})$ for $j\in \{1,2,\ldots ,n\}$. Since each $\phi_j\in L^\infty(\mathbb{T})$, then for $f\in L^2(\mathbb{T})$ the action of S is given as follows:

\begin{align*} Sf = & \phi_1f_1 + \phi_2f_2 + \cdots +\phi_nf_n \\ = & M_{\phi_1}f_1 + M_{\phi_2}f_2 + \cdots + M_{\phi_n}f_n. \end{align*}

From now onward, we use the notation $M_{[\phi_1,\phi_2,\ldots ,\phi_n]}$ to denote the bounded linear operator on $L^2(\mathbb{T})$ such that:

(2.5)\begin{align} M_{[\phi_1,\phi_2,\ldots ,\phi_n]}(f)=M_{\phi_1}f_1 + M_{\phi_2}f_2 + \cdots + M_{\phi_n}f_n. \end{align}

Summing up, we have the following main theorem in this section, and it is extremely useful in later sections to prove other characterization results. Moreover, we expect that the following theorem will have an independent interest in the future.

Theorem 2.2. Let B be a finite Blaschke product of degree n and $S\in \{M_B\}^\prime$ in $L^2(\mathbb{T})$. Then there exist n-number of $L^\infty(\mathbb{T})$ functions $\phi_1,\phi_2,\ldots ,\phi_n$ such that $S=M_{[\phi_1,\phi_2,\ldots ,\phi_n]}$.

One can find the classical theorem for the commutant of the shift operator on $L^2(\mathbb{T})$-space in [28](see Theorem: 2.2.5), which says that the commutant of M_z in $L^2(\mathbb{T})$ is $\{M_\phi:\phi \in L^\infty(\mathbb{T})\}$. It is worth noticing that, if we consider $\phi_1=\phi_2=\cdots =\phi_n=\phi$ in (2.5), then $M_{[\phi_1,\phi_2,\ldots ,\phi_n]}$ turns out to be the classical multiplication operator M_ϕ. Therefore we have the following remark:

Remark 2.3. Suppose $B(z)=z^n$ and $S\in \{M_{z^n}\}^\prime$. Since $\{M_z\}^\prime \subset \{M_{z^n}\}^\prime$, then

1. (i) for $S\in \{M_z\}^\prime$, there exists $\phi \in L^\infty$ such that $S=M_{[\phi,\phi,\dots,\phi]}=M_{\phi}$,

2. (ii) for $S\notin \{M_z\}^\prime$, there exist $\phi_1,\phi_2,\ldots ,\phi_n \in L^\infty$ with $\phi_i\neq\phi_j$ for some $i,j\in \{1,2, \ldots ,n\}$ such that $S=M_{[\phi_1,\phi_2 , \ldots ,\phi_n]}$.

In a similar fashion, one can distinguish between the commutant of M_z and the commutant of M_B, where B is a finite Blaschke product.

3. Conjugations in $L^2(\mathbb{T})$

We begin the section by recalling two natural conjugations in $L^2(\mathbb{T})$ which have been mentioned earlier, namely J and $J^\star$, where J is defined by $Jf=\bar{f}$ and $J^\star$ is given by $J^\star f= f^\sharp$, where $f^\sharp(z)=\overline{f(\bar{z})}$. These two conjugations have some interesting properties. The bilateral shift M_z becomes complex symmetric in $L^2(\mathbb{T})$ with respect to the conjugation J, and it maps the analytic functions space $H^2(\mathbb{T})$ into the co-analytic functions space $\overline{H^2}(\mathbb{T})$, that is, $J(H^2(\mathbb{T}))=\overline{H^2}(\mathbb{T})$. On the other hand, the conjugation $J^\star$ commutes with M_z and keeps the analytic function space $H^2(\mathbb{T})$ invariant, that is, $J^\star (H^2(\mathbb{T}))\subset H^2(\mathbb{T})$. As discussed earlier, we will introduce some conjugations in $L^2(\mathbb{T})$ as a generalization of these two conjugations J and $J^\star$ from a different point of view. Going further, for any inner function θ, the natural conjugation on $K_\theta := H^2\ominus \theta H^2$ is given by:

\begin{align*} c_\theta:& K_{\theta} \to K_{\theta}\\ & h\mapsto \overline{zh}\theta, \end{align*}

and it has a connection with the truncated Toeplitz operator in model spaces. For example, the truncated Toeplitz operator is c_θ-symmetric (for more details, see [17, 19]).

For any inner function θ, by Wold-type Decomposition (2.1) in $\S$ 2, we define conjugations extending the natural conjugation c_θ from the model space K_θ to the entire $L^2(\mathbb{T})$ space in two different ways. First, we define $\mathscr{C}_\theta$ on $L^2(\mathbb{T})$ as follows:

\begin{align*} \mathscr{C}_\theta: & L^2(\mathbb{T}) \to L^2(\mathbb{T})\\ & f \mapsto \sum_{k=-\infty}^{\infty}(c_\theta f_k) \theta ^{-k}, \quad \text{where}~ f=\sum_{k=-\infty}^{\infty}f_k\theta ^{k} ,f_k \in K_\theta, ~\text{that is}: \end{align*}

(3.1)\begin{equation} \mathscr{C}_\theta(\sum_{k=-\infty}^{\infty}f_k\theta ^k )= \sum_{k=-\infty}^{\infty}\overline{zf_k}\theta ^{-k+1}. \end{equation}

It is easy to check that $\mathscr{C}_\theta$ is a conjugation on $L^2(\mathbb{T})$. In particular, if $\theta (z)=z$, then the action of $\mathscr{C}_z$ is given by:

(3.2)\begin{equation} \mathscr{C}_z(\sum_{k=-\infty}^{\infty}a_kz^k)=\sum_{k=-\infty}^{\infty}\bar{a}_k\bar{z}^k. \end{equation}

Note that the action of $\mathscr{C}_\theta$ given in (3.1) can also be rewritten as:

(3.3)\begin{equation} \mathscr{C}_\theta (f)=\theta \overline{zf}, \end{equation}

and hence $\mathscr{C}_z(f)=\bar{f}$. In other words, $\mathscr{C}_z=J$, and we can think $\mathscr{C}_\theta $ is a generalization of the conjugation J. These types of conjugations (3.3) have already been introduced in [10], where they use it to characterize all M_z-commuting and M_z, $M_{\bar{z}}$-intertwining conjugations preserving model spaces.

Now we define another conjugation (different from $\mathscr{C}_\theta$ above) corresponding to an inner function θ, namely $\mathscr{C}_\theta ^\star :L^2(\mathbb{T})\to L^2(\mathbb{T})$ such that if $f=\sum\limits_{k=-\infty}^{\infty}f_k\theta ^{k}$ with $f_k \in K_\theta$, then $\mathscr{C}_\theta^\star(f)=\sum\limits_{k=-\infty}^{\infty}(c_\theta f_k) \theta ^{k},$ which implies that:

(3.4)\begin{equation} \mathscr{C}_\theta^\star(\sum_{k=-\infty}^{\infty}f_k\theta ^k )= \sum_{k=-\infty}^{\infty}\overline{zf_k}\theta ^{k-1}. \end{equation}

In particular, for $\theta(z)=z$, we have $\mathscr{C}_z ^\star=J^\star$. These types of conjugations can also be found in [3].

In the beginning, we have already discussed certain properties of two natural conjugations J and $J^\star$. It is easy to observe that those properties also hold for $\mathscr{C}_\theta$ and $\mathscr{C}_\theta ^\star$. To be more specific, $\mathscr{C}_\theta M_\theta =M_{\bar{\theta}}\mathscr{C}_\theta $, $\mathscr{C}_\theta (H^2)=\overline{H^2}$, $\mathscr{C}_\theta^\star M_\theta =M_\theta \mathscr{C}_\theta^\star$, and $\mathscr{C}_\theta^\star (H^2)=H^2$. Moreover, the conjugation $\mathscr{C}_{z^2}^\star$ has one more extra property, namely $\mathscr{C}_{z^2}^\star M_z \neq M_z \mathscr{C}_{z^2}^\star$ and, in general, $\mathscr{C}_\theta^\star M_z \neq M_z \mathscr{C}_\theta^\star$ for an inner function θ.

4. Conjugations intertwining $M_{z^2}$, $M_{\bar{z}^2}$ and commuting with $M_{z^2}$ In L ²

The definition of the conjugations J, $J^\star$ in L ² (see (1.1)) and some of their properties have already been introduced in $\S$ 1. In [9, 10], the authors obtained the characterization of all conjugations C intertwining the operators M_z, $M_{\bar{z}}$ in terms J in L ² (see [10, Theorem 2.2]). Moreover, they also characterized all conjugations C that commute with the operator M_z in terms of $J^\star$ in L ² (see [10, Theorem 2.4]). Motivated by the characterization results mentioned above, we raise Question 1 and Question 2 in $\S$ 1. Our main aim in this section is to provide appropriate answers of these questions. First, we characterize all conjugations C that satisfies the identity $CM_{z^2}=M_{\bar{z}^2}C$. One of the importance of our Theorem 4.1 below is that the class of conjugations C intertwining the operators $M_{z^2}$, $M_{\bar{z}^2}$ in L ² is bigger than the class of conjugations C intertwining the operators M_z, $M_{\bar{z}}$ in L ². Before going to the main theorems, we define:

\begin{equation*}\phi^e(z)=\dfrac{\phi(z)+\phi(-z)}{2} \quad \text{and}\quad \phi^o(z)=\dfrac{\phi(z)-\phi(-z)}{2}, \quad \text{for}\quad\phi \in L^2(\mathbb{T}).\end{equation*}

Note that, $\phi^e \in \bigvee_{k=-\infty}^{\infty}\{z^{2k}\}=\mathcal{H}_1(1,z^2)$ and $\phi^o \in \bigvee_{k=-\infty}^{\infty}\{z^{2k+1}\}=\mathcal{H}_2(z,z^2)$. Next, we state and prove one of our main results in this section which involve the conjugation $\mathscr{C}_{z^2}(f)=z\overline{f}$ for $f\in L^2$ (see (3.3)).

Theorem 4.1. Let $C:L^2(\mathbb{T})\to L^2(\mathbb{T})$ be an arbitrary conjugation such that $CM_{z^2}=M_{\bar{z}^2}C$, then there exist $\phi_1,\phi_2\in L^\infty$ such that $C= M_{[\phi_1,\phi_2]}\mathscr{C}_{z^2}$, where $\phi_1,\phi_2$ satisfies $|\phi_1|^2+|\phi_2|^2=2$ and $\phi_1^e= \phi_2^e$. Conversely, any conjugation C of the form $M_{[\phi_1,\phi_2]}\mathscr{C}_{z^2}$ for $\phi_1, \phi_2 \in L^\infty$ satisfies $CM_{z^2}=M_{\bar{z}^2}C$.

Proof. Suppose C is a conjugation on $L^2(\mathbb{T})$ such that $CM_{z^2}=M_{\bar{z}^2}C$. Then, it is easy to observe that the unitary operator $C\circ\mathscr{C}_{z^2}$ commutes with $M_{z^2}$. Therefore by applying Theorem 2.2, we have:

(4.1)\begin{equation} C\circ\mathscr{C}_{z^2}= M_{[\phi_1,\phi_2]}, \quad \text{for}\quad \phi_1,\phi_2 \in L^\infty(\mathbb{T}), \end{equation}

and hence $C = M_{[\phi_1,\phi_2]} \mathscr{C}_{z^2} $. By considering the decomposition $L^2(\mathbb{T})=\mathcal{H}_1(1,z^2) \oplus \mathcal{H}_2(z,z^2)$, we have the following block-matrix representation of $M_{[\phi_1,\phi_2]}$:

\begin{equation*} M_{[\phi_1,\phi_2]}= \begin{bmatrix} M_{\phi_1^e} && M_{\phi_2^o}\\ \\ M_{\phi_1^o} && M_{\phi_2^e}\\ \end{bmatrix} \text{,~and~hence~} M_{[\phi_1,\phi_2]}^*= \begin{bmatrix} M_{\bar{\phi_1^e}} && M_{\bar{\phi_1^o}}\\ \\ M_{\bar{\phi_2^0}} && M_{\bar{\phi_2^e}}\\ \end{bmatrix}. \end{equation*}

From (4.1) it follows that $M_{[\phi_1,\phi_2]}$ is a unitary operator, and therefore we obtain the following conditions on the symbols:

(4.2)\begin{equation} \begin{cases} |\phi_1^e|^2+|\phi_1^o|^2=1 ~,~ |\phi_1^e|^2+|\phi_1^o|^2=1,\\ |\phi_1^e|^2+|\phi_2^o|^2=1 ~,~ |\phi_1^o|^2 + |\phi_2^e|^2=1,\\ \bar{\phi_1^e}\phi_2^o+\bar{\phi_1^o}\phi_2^e=0 ~,~ \bar{\phi_1^o}\phi_1^e+\bar{\phi_2^e}\phi_2^o=0, \end{cases} \end{equation}

which by simplifying, reduces to:

(4.3)\begin{equation} \begin{cases} |\phi_1|^2+|\phi_2|^2=2\\ |\phi_1^o+\phi_2^e|^2+|\phi_1^e+\phi_2^o|^2=2. \end{cases} \end{equation}

Moreover, due to the involutive property of the conjugation C, we have:

\begin{equation*} M_{[\phi_1,\phi_2]} \mathscr{C}_{z^2}M_{[\phi_1,\phi_2]} \mathscr{C}_{z^2} = I\implies \mathscr{C}_{z^2}M_{[\phi_1,\phi_2]} \mathscr{C}_{z^2} = M_{[\phi_1,\phi_2]}^*, \end{equation*}

which by applying on the basis elements 1 or z yields $\phi_1^e= \phi_2^e$. Therefore, the conditions in (4.3) merged to $|\phi_1|^2+|\phi_2|^2=2$.

For the converse, suppose $C=M_{[\phi_1,\phi_2]}\mathscr{C}_{z^2}$ for $\phi_1, \phi_2 \in L^\infty$. Then for $f\in L^2(\mathbb{T})$, there exist $f^e \in \mathcal{H}_1(1,z^2)$ and $f^o \in \mathcal{H}_2(z,z^2)$ such that $f=f^e\oplus f^o$. Since $\mathscr{C}_{z^2}(f)= z\overline{f}$, then we have:

\begin{align*} CM_{z^2}(f)=C(z^2f^e\oplus z^2f^o)= & M_{[\phi_1,\phi_2]}\mathscr{C}_{z^2}(z^2f^e\oplus z^2f^o)\\ = & M_{[\phi_1,\phi_2]}(\overline{zf^o}\oplus\overline{zf^e})=\phi_1\overline{zf^o}+\phi_2\overline{zf^e}. \end{align*}

Similarly, we obtain $M_{\bar{z}^2}C(f)=M_{\bar{z}^2}M_{[\phi_1,\phi_2]}(z\overline{f^o}\oplus z\overline{f^e}) = \phi_1\overline{zf^o}+\phi_2\overline{zf^e}$. This completes the proof.

As a consequence of the above theorem, we characterize all conjugations C intertwining the operators $M_{z^2}$, $M_{\bar{z}^2}$ in L ² in terms of J as follows:

Corollary 4.2. Let C be a conjugation on L ² such that $CM_{z^2}=M_{\bar{z}^2}C$, then $C=M_{[\psi_1,\psi_2]}J$ for $\psi_1,\psi_2\in L^\infty$ with $\psi_1^o= \psi_2^o$ and $|\psi_1|^2+|\psi_2|^2=2$. Conversely, any conjugation C of the form $M_{[\psi_1,\psi_2]}J$ for $\psi_1,\psi_2 \in L^\infty$ satisfies $CM_{z^2}=M_{\bar{z}^2}C$.

Proof. By definition of $\mathscr{C}_{z^2}$ it follows that $\mathscr{C}_{z^2} = M_z J$. Therefore, by using Theorem 4.1, we conclude $C=M_{[\phi_1,\phi_2]} \mathscr{C}_{z^2}$, where $\phi_1^e= \phi_2^e$ and $|\phi_1|^2+|\phi_2|^2=2$. Note that $M_{[\phi_1,\phi_2]}M_z= M_{[z\phi_2,z\phi_1]}$ and hence $C=M_{[\psi_1,\psi_2]}J$, where $\psi_1 (=z\phi_2) , \psi_2 (= z\phi_1)\in L^\infty$. Moreover, the involutive property of the conjugation C implies $\psi_1^o= \psi_2^o$, and since $M_{[\psi_1,\psi_2]}$ is unitary, then we have $|\psi_1|^2+|\psi_2|^2=2$. The converse is straightforward.

Remark 4.3. Note that, the conditions (4.2) in Theorem 4.1 after substituting $\mathscr{C}_{z^2} = M_z J$ and $\psi_1 =z\phi_2 , \psi_2= z\phi_1$, change to:

(4.4)\begin{equation} \begin{cases} |\psi_2^o|^2+|\psi_2^e|^2=1 ~,~ |\psi_2^o|^2+|\psi_2^e|^2=1,\\ |\psi_2^o|^2+|\psi_1^e|^2=1 ~,~ |\psi_2^e|^2 + |\psi_1^o|^2=1,\\ \bar{\psi_2^o}\psi_1^e+\bar{\psi_2^e}\psi_1^o=0,~ \bar{\psi_2^e}\psi_2^o+\bar{\psi_1^o}\psi_1^e=0. \end{cases} \end{equation}

It should be noted that if we combine the above set of conditions, it reduces to the single condition $|\psi_1|^2+|\psi_2|^2=2$ obtained in Corollary 4.2. The usefulness of the conditions in (4.4) will be realized in a later section.

Remark 4.4. It is important to note that if we assume $\psi_1 =\psi_2$ in Corollary (4.2), then $C=M_{\psi_1}J$ intertwining the operators M_z, $M_{\bar{z}}$ and hence intertwining the operators $M_{z^2}$, $M_{\bar{z}^2}$ as well in L ². On the other hand, if $\psi_1 \neq \psi_2$, then $C=M_{[\psi_1,\psi_2]}J$ intertwining the operators $M_{z^2}$, $M_{\bar{z}^2}$ but not intertwining the operators M_z and $M_{\bar{z}}$ in L ².

According to the previous discussion, our next aim is to characterize all the conjugations commuting with $M_{z^2}$. For instance, the conjugation $\mathscr{C}_{z^2}^\star$ (see (3.4)) commutes with $M_{z^2}$. To be more specific, for $f(z)=\sum\limits_{k=-\infty}^{\infty}a_nz^n\in L^2$, the action of $\mathscr{C}_{z^2}^\star$ is given by:

\begin{align*} \mathscr{C}_{z^2}^\star(\sum_{k=-\infty}^{\infty}a_nz^n)=& \sum_{k=-\infty}^{\infty}\bar{a}_{2n}z^{2n+1} + \sum_{k=-\infty}^{\infty}\bar{a}_{2n+1}z^{2n}~\implies~ \mathscr{C}_{z^2}^\star(f(z))\\ =& z\overline{f^e(\bar{z})} +\bar{z}\overline{f^o(\bar{z})}. \end{align*}

Now, we have the following theorem concerning $M_{z^2}$-commuting conjugations.

Theorem 4.5. If C is a conjugation on $L^2(\mathbb{T})$ such that $CM_{z^2}=M_{z^2}C$, then there exist $\zeta_1,\zeta_2 \in L^\infty$ satisfying $|\zeta_1|^2+|\zeta_2|^2=2, |\bar{z}\zeta_1^o+z\zeta_2^e|^2+|\bar{z}\zeta_1^e+z\zeta_2^o|^2=2, \zeta_2^e(\bar{z})=\zeta_1^e(z), \zeta_2^o(\bar{z})=z^2\zeta_2^0(z), z^2\zeta_1^o(\bar{z})=\zeta_1^o(z)$ such that:

\begin{equation*}C=M_{[\zeta_1,\zeta_2]}\mathscr{C}_{z^2}^\star.\end{equation*}

Conversely, any conjugation C of the form $M_{[\zeta_1,\zeta_2]}\mathscr{C}_{z^2}^\star$ for $\zeta_1,\zeta_2 \in L^\infty$ commutes with $M_{z^2}$.

Proof. Let C be a conjugation on $L^2(\mathbb{T})$ such that $CM_{z^2}=M_{z^2}C$. Since $\mathscr{C}_{z^2}^\star M_{z^2}=M_{z^2}\mathscr{C}_{z^2}^\star$, then the unitary operator $C\circ \mathscr{C}_{z^2}^\star$ also commutes with $M_{z^2}$ in $L^2(\mathbb{T})$ . Therefore by applying Theorem 2.2, there exist $\zeta_1,\zeta_2\in L^\infty(\mathbb{T})$ such that $C\circ \mathscr{C}_{z^2}^\star = M_{[\zeta_1,\zeta_2]}$. Now by considering the decomposition $L^2(\mathbb{T})=\mathcal{H}_1(1,z^2) \oplus \mathcal{H}_2(z,z^2)$, we have the following block-matrix representation of $M_{[\zeta_1,\zeta_2]}$:

\begin{equation*} M_{[\zeta_1,\zeta_2]}= \begin{bmatrix} M_{\zeta_1^e} && M_{\zeta_2^o}\\ \\ M_{\zeta_1^o} && M_{\zeta_2^e}\\ \end{bmatrix} \text{,~and~hence~} M_{[\zeta_1,\zeta_2]}^*= \begin{bmatrix} M_{\bar{\zeta_1^e}} && M_{\bar{\zeta_1^o}}\\ \\ M_{\bar{\zeta_2^0}} && M_{\bar{\zeta_2^e}}\\ \end{bmatrix}. \end{equation*}

Using the fact that $M_{[\zeta_1,\zeta_2]}$ is a unitary operator we obtain the following conditions:

(4.5)\begin{equation} \begin{cases} |\zeta_1^e|^2+|\zeta_1^o|^2=1,~ |\zeta_1^e|^2+|\zeta_1^o|^2=1,\\ |\zeta_1^e|^2+|\zeta_2^o|^2=1,~ |\zeta_1^o|^2 + |\zeta_2^e|^2=1,\\ \bar{\zeta_1^e}\zeta_2^o+\bar{\zeta_1^o}\zeta_2^e=0,~ \bar{\zeta_1^o}\zeta_1^e+\bar{\zeta_2^e}\zeta_2^o=0, \end{cases} \end{equation}

which by simplifying, becomes:

\begin{equation*} \begin{cases} |\zeta_1|^2+|\zeta_2|^2=2,\\ |\bar{z}\zeta_1^o+z\zeta_2^e|^2+|\bar{z}\zeta_1^e+z\zeta_2^o|^2=2. \end{cases} \end{equation*}

Moreover, the involutive property of the conjugation C implies:

\begin{equation*} \mathscr{C}_{z^2}^\star M_{[\zeta_1,\zeta_2]} \mathscr{C}_{z^2}^\star = M_{[\zeta_1,\zeta_2]}^*, \end{equation*}

which by applying on the basis vectors 1 and z, we get $\zeta_2^e(\bar{z})=\zeta_1^e(z)$, $\zeta_2^o(\bar{z})=z^2\zeta_2^0(z)$ and $z^2\zeta_1^o(\bar{z})=\zeta_1^o(z)$, $\zeta_1^e(\bar{z}) =\zeta_2^e(z)$, respectively.

Conversely, if $C=M_{[\zeta_1,\zeta_2]}\mathscr{C}_{z^2}^\star$ for $\zeta_1,\zeta_2 \in L^\infty$ , then it is easy to verify that $CM_{z^2}=M_{z^2}C$. This completes the proof.

Remark 4.6. It is interesting to note that the conjugation $\mathscr{C}_{z^2}^\star$ commutes with $M_{z^2}$ but not with M_z. But in the above Theorem 4.5, we characterize $M_{z^2}$-commuting conjugations in terms of $M_{[\zeta_1,\zeta_2]}$ and $\mathscr{C}_{z^2}^\star$, where the composition operator $M_{[\zeta_1,\zeta_2]}\mathscr{C}_{z^2}^\star$ may commutes with M_z.

In general, it is quite difficult to distinguish between the class of conjugations commuting with M_z and the class of conjugations not commuting with M_z inside $M_{z^2}$-commuting conjugations. To get a better understanding of the same, we revisit the study of $M_{z^2}$-commuting conjugations in the following corollary.

Corollary 4.7. Any $ M_{z^2}$-commuting conjugation C on L ², there exist two $L^\infty$-functions $\xi_1,\xi_2$ on $\mathbb{T}$ such that $C=M_{[\xi_1,\xi_2]}J^\star$, where ξ ₁ and ξ ₂ satisfy $|\xi_1|^2+|\xi_2|^2=2,|\xi_2^e+\xi_1^o|^2+| \xi_2^o+\xi_1^e|^2=2,\xi_1^o(\bar{z})=\xi_2^o(z), \xi_1^e(\bar{z})=\xi_1^e(z), \text{and}~ \xi_2^e(\bar{z})=\xi_2^e(z)$. Conversely, any conjugation C of the form $M_{[\xi_1,\xi_2]}J^\star$ for $\xi_1,\xi_2 \in L^\infty $ commutes with $M_{z^2}$.

Proof. Let $C:L^2(\mathbb{T})\to L^2(\mathbb{T})$ be a conjugation such that $CM_{z^2}=M_{z^2}C$. Then by using the above Theorem 4.5, we get two $L^\infty$-functions $\zeta_1,\zeta_2$ such that $C= M_{[\zeta_1,\zeta_2]}\mathscr{C}_{z^2}^\star$, and

(4.6)\begin{equation} \begin{cases} |\zeta_1|^2+|\zeta_2|^2=2, |\bar{z}\zeta_1^o+z\zeta_2^e|^2+|\bar{z}\zeta_1^e+z\zeta_2^o|^2=2,\\ \zeta_2^e(\bar{z})=\zeta_1^e(z), \zeta_2^o(\bar{z})=z^2\zeta_2^0(z), z^2\zeta_1^o(\bar{z})=\zeta_1^o(z). \end{cases} \end{equation}

Moreover, for any $f(=f^e\oplus f^o)\in L^2$ and $z\in \mathbb{T}$, we have

\begin{equation*}(\mathscr{C}_{z^2}^\star f) (z)= z\overline{f^e(\bar{z})}+\bar{z}\overline{f^o(\bar{z})}=\bar{z}\overline{f^o(\bar{z})} \oplus z\overline{f^e(\bar{z})} =(M_{[z,\bar{z}]}J^\star f)(z) ,\end{equation*}

and hence $\mathscr{C}_{z^2}^\star = M_{[z,\bar{z}]}J^\star $ which immediately implies:

(4.7)\begin{equation} C=M_{[\zeta_1,\zeta_2]}M_{[z,\bar{z}]}J^\star=M_{[\xi_1,\xi_2]}J^\star, \end{equation}

where $\xi_1=z\zeta_2$ and $\xi_2=\bar{z}\zeta_1$. Next by identifying the relation between $\{\zeta_1,\zeta_2\}$ and $\{\xi_1,\xi_2\}$ we obtain:

(4.8)\begin{equation} \begin{cases} \zeta_1^e = z\xi_2^o ,~ \zeta_1^o = z\xi_2^e,\\ \zeta_2^e = \bar{z}\xi_1^o ,~ \zeta_2^o = \bar{z}\xi_1^e. \end{cases} \end{equation}

Thus, the conditions in (4.6) becomes:

(4.9)\begin{equation} \begin{cases} |\xi_1|^2+|\xi_2|^2=2, |\xi_2^e+\xi_1^o|^2+| \xi_2^o+\xi_1^e|^2=2, \\ z\xi_1^o(\bar{z})=z\xi_2^o(z), z\xi_1^e(\bar{z})=z^2\bar{z}\xi_1^e(z), z^2\bar{z}\xi_2^e(\bar{z})=z\xi_2^e(z), \end{cases} \end{equation}

which by simple modification reduces to:

(4.10)\begin{equation} \begin{cases} |\xi_1|^2+|\xi_2|^2=2,|\xi_2^e+\xi_1^o|^2+| \xi_2^o+\xi_1^e|^2=2, \\ \xi_1^o(\bar{z})=\xi_2^o(z), \xi_1^e(\bar{z})=\xi_1^e(z), \xi_2^e(\bar{z})=\xi_2^e(z). \end{cases} \end{equation}

For the converse, it is easy to verify that for any conjugation C on L ² having the form $C=M_{[\xi_1,\xi_2]}J^\star$ commutes with $M_{z^2}$. This completes the proof.

Remark 4.8. It is important to observe that the conditions $|\xi_1|^2+|\xi_2|^2=2$ and $|\xi_2^e+\xi_1^o|^2+| \xi_2^o+\xi_1^e|^2=2 $ in (4.10) of the above Corollary 4.7 can be reduced from the original conditions (4.5) of Theorem 4.5. Going further, one should always remember that the original explicit conditions behind the above two reduced conditions are:

(4.11)\begin{equation} \begin{cases} |\xi_1^e|^2+|\xi_1^o|^2=1,~ |\xi_1^e|^2+|\xi_1^o|^2=1,\\ |\xi_1^e|^2+|\xi_2^o|^2=1,~ |\xi_1^o|^2 + |\xi_2^e|^2=1,\\ \bar{\xi_2^o}\xi_1^e+\bar{\xi_2^e}\xi_1^o=0,~ \bar{\xi_2^e}\xi_2^o+\bar{\xi_1^o}\xi_1^e=0. \end{cases} \end{equation}

Now, we are in a position to distinguish between the class of conjugations commuting with M_z and the class of conjugations not commuting with M_z inside $M_{z^2}$-commuting conjugations with the help of the above Corollary 4.7.

• • If the two $L^\infty$-functions ξ ₁ and ξ ₂ appearing in the representation of $M_{z^2}$-commuting conjugation C as described in 4.7 becomes identical, that is, if $\xi_1=\xi_2$, then the conclusion can be drawn that C actually commutes with M_z. Conversely, if C commutes with M_z, then by using Corollary 4.7 and Remark 2.3, we conclude $C=M_{[\xi,\xi]}J^\star$, where $\xi \in L^\infty $ satisfying $|\xi|^2+|\xi|^2=2,\xi^o(\bar{z})=\xi^o(z), \xi^e(\bar{z})=\xi^e(z), $ and $\xi^e(\bar{z})=\xi^e(z)$ which implies $C=M_{\xi}J^\star$, where $\xi \in L^\infty $ such that $|\xi|=1$ and $\xi(z)=\xi(\bar{z})$ and it is consistent with the condition obtained in [10].

• • As a matter of fact, if $\xi_1 \neq \xi_2$, then $C=M_{[\xi_1,\xi_2]}J^\star$ commutes with $M_{z^2}$ but not with M_z. Conversely, if C commutes only with $M_{z^2}$, then there exist $\xi_1 \neq \xi_2$ satisfying:

  \begin{align*} \hspace{0.5in} |\xi_1|^2+|\xi_2|^2=2,|\xi_2^e+\xi_1^o|^2+| \xi_2^o+\xi_1^e|^2=2, \xi_1^o(\bar{z})=&\xi_2^o(z), \xi_1^e(\bar{z})=\xi_1^e(z),\\& ~\text{and}~ \xi_2^e(\bar{z})=\xi_2^e(z), \end{align*}

  such that $C=M_{[\xi_1,\xi_2]}J^\star$ and we call such type of conjugations as $only~M_{z^2}$-commuting conjugations.

Therefore, the above discussions completely describe the class of conjugations that commute with $M_{z^2}$ in L ²-space.

5. The class of conjugations that keep invariant various subspaces of $L^2(\mathbb{T})$

Our main aim in this section is to study the behaviour of two special classes of conjugations, namely $M_{z^2},M_{\bar{z}^2}$-intertwining, and $M_{z^2}$-commuting conjugations that keep invariant various subspaces of L ² like the Hardy space H ², model spaces and shift-invariant subspaces. The study of $M_{z},M_{\bar{z}}$-intertwining, and M_z-commuting conjugations preserving H ², model spaces and shift-invariant subspaces have been done recently in [10]. The significant difference between our study and the study in [10] is that we characterize $only~M_{z^2}$-commuting and $M_{z^2},M_{\bar{z}^2}$-intertwining (but not $M_{z},M_{\bar{z}}$-intertwining ) conjugations preserving the above mentioned subspaces.

5.1. Hardy space preserving conjugations

We have already identified the class of conjugations that intertwine the operators $M_{z^2}$, $M_{\bar{z}^2}$ and also the class of conjugations that commute with $M_{z^2}$ in terms of known conjugations in the previous section. Note that the involutive property of a conjugation guarantees that if H ² is invariant under any conjugation C in L ², then $C(H^2)=H^2$. Now, we will discuss the class of $only~M_{z^2}-commuting$ conjugations that keep the Hardy space invariant.

Theorem 5.1. Let C be a ‘$only~M_{z^2}-commuting$’ conjugation on L ² such that $C(H^2)\subset H^2$, then $C=M_{[\xi_1,\xi_2]}J^\star$, where

1. (i) $\xi_1=a_0+a_1z$, $\xi_2 =a_1\bar{z}+b_0$ for some $a_0,a_1,b_0 \in \mathbb{C}, $ and

2. (ii) $|a_0|^2+|a_1|^2=1,|b_0|^2+|a_1|^2=1,a_0\bar{a_1}+a_1\bar{b}_0=0.$

Proof. Let C be a $only~M_{z^2}-commuting$ conjugation on L ². Then by Corollary 4.7, there exist $\xi_1,\xi_2\in L^\infty$ such that $C=M_{[\xi_1,\xi_2]}J^\star $, where $|\xi_1|^2+|\xi_2|^2=2,|\xi_2^e+\xi_1^o|^2+| \xi_2^o+\xi_1^e|^2=2,\xi_1^o(\bar{z})=\xi_2^o(z), \xi_1^e(\bar{z})=\xi_1^e(z), \xi_2^e(\bar{z})=\xi_2^e(z)$. Moreover, $C(1)\in H^2$ and $C(z)\in H^2$ implies $\xi_1\in H^\infty$ and $z\xi_2 \in H^\infty$ respectively. Suppose $\xi_1=\sum\limits_{k=0}^{\infty}a_kz^k $ and $\xi_2 =\sum\limits_{k=-1}^{\infty}b_kz^k$. Now, combining the facts $\xi_1^e\in H^\infty$ and $\xi_1^e(\bar{z})=\xi_1^e(z)$, we conclude $ \xi_1^e= constant $. Moreover, the condition $\xi_2^e(\bar{z})=\xi_2^e(z)$ implies $\xi_2^e= constant $ and the condition $\xi_1^o(\bar{z})=\xi_2^o(z)$ implies $\xi_1^o(z)=a_1z$ and $\xi_2^o(z)=b_{-1}\bar{z}$ with $a_1=b_{-1}$, which immediately yield $\xi_1=a_0+a_1z$ and $\xi_2 =a_1\bar{z}+b_0$ for some $a_0,a_1,b_0 \in \mathbb{C} $. It is important to note that, we obtain the conditions $|\xi_1|^2+|\xi_2|^2=2$ and $|\xi_2^e+\xi_1^o|^2+| \xi_2^o+\xi_1^e|^2=2$ by reducing the conditions listed in (4.11) in Remark 4.8 and hence we get the following relations between a ₀, a ₁ and b ₀

\begin{equation*} |a_0|^2+|a_1|^2=1,|b_0|^2+|a_1|^2=1,a_0\bar{a_1}+a_1\bar{b}_0=0. \end{equation*}

This completes the proof.

5.2. Model space preserving conjugations

In this subsection, we characterize $M_{z^2}$-commuting and M_z, $M_{\bar{z}}$-intertwining conjugations preserving model spaces in terms of $\mathscr{C}_\theta$. The following two propositions will be useful to prove our main results.

Proposition 5.2. If C is a conjugation on L ² such that $CM_{z^2}=M_{\bar{z}^2}C$, then $M_{z}CM_{\bar{z}}$ is also a conjugation on L ² intertwining $M_{z^2}, M_{\bar{z}^2}$, and $M_{z}CM_{\bar{z}}=M_{[z^2\psi_2 , z^2\psi_1 ]}J$ for some $\psi_1,\psi_2 \in L^\infty$ satisfying $\psi_1^o= \psi_2^o$ and $|\psi_1|^2+|\psi_2|^2=2$.

Proof. Suppose C is a conjugation on L ² such that $CM_{z^2}=M_{\bar{z}^2}C$, then by Corollary 4.2, there exist $\psi_1,\psi_2 \in L^\infty$ such that $C=M_{[\psi_1,\psi_2]}J$, where $\psi_1^o= \psi_2^o$ and $|\psi_1|^2+|\psi_2|^2=2$. It is easy to see that $M_{z}CM_{\bar{z}}$ is a conjugation on L ² and $M_{z}CM_{\bar{z}}M_{z}=M_{\bar{z}}M_{z}CM_{\bar{z}}$. Therefore, by applying Corollary 4.2 again, we have $M_{z}CM_{\bar{z}}=M_{[\vartheta_1,\vartheta_2]}J$ for some $\vartheta_1,\vartheta_2\in L^\infty $. On the other hand, the following computation for any $f(=f^e\oplus f^o)\in L^2$ yields:

\begin{align*} M_{z}CM_{\bar{z}}(f) = M_{z}M_{[\psi_1,\psi_2]}JM_{\bar{z}}(f^e\oplus f^o) =z^2\psi_2 \bar{f}^e + z^2\psi_1 \bar{f}^o = M_{[z^2\psi_2 , z^2\psi_1 ]}J(f), \end{align*}

and hence $ M_{z}CM_{\bar{z}} = M_{[z^2\psi_2 , z^2\psi_1 ]}J$. This completes the proof.

Proposition 5.3. Let C be a conjugation on L ² intertwining $M_{z^2}$ and $M_{\bar{z}^2}$. Then $CJ=M_{[\psi_1,\psi_2]} $ for some $\psi_1,\psi_2 \in L^\infty$ satisfying $\psi_1^o= \psi_2^o$ and $|\psi_1|^2+|\psi_2|^2=2$. Furthermore, we also have $M_{z}CJM_{\bar{z}}=M_{[\psi_2,\psi_1]}$.

Proof. Since $CM_{z^2}=M_{\bar{z}^2}C$, then by Corollary 4.2, $C=M_{[\psi_1,\psi_2]}J $ for some $\psi_1,\psi_2 \in L^\infty$ with $\psi_1^o= \psi_2^o$ and $|\psi_1|^2+|\psi_2|^2=2$. Note that J is also a conjugation on L ² and hence $CJ=M_{[\psi_1,\psi_2]}$. For the second part, we proceed similarly as Proposition 5.2 and obtain the following:

\begin{align*} M_{z}CJM_{\bar{z}}(f) = M_{z}M_{[\psi_1,\psi_2]}M_{\bar{z}}(f^e\oplus f^o) = \psi_2 f^e + \psi_1 f^o = M_{[\psi_2,\psi_1]}(f). \end{align*}

Consequently, $M_{z}CJM_{\bar{z}}=M_{[\psi_2,\psi_1]} $. This completes the proof.

Now, we are in a position to prove one of our main results related to [10, Theorem 4.2].

Theorem 5.4. Let $\alpha, \theta $ be two non-constant inner functions and consider a $M_{z^2},M_{\bar{z}^2}$-intertwining conjugation C on L ². If C satisfies the following conditions:

1. (i) $C(K_\alpha) \subset K_\theta $ and also, $M_zCM_z(K_\alpha) \subset K_\theta $,

2. (ii) $Re[(M_{z}CJM_{\bar{z}})^*(CJ)] =I$,

then there exists an inner function γ such that $C+M_zCM_z = \sqrt{2}\mathscr{C}_\gamma$, where γ is divisible by α and γ divides θ, in other words, $\alpha \leq \gamma \leq \theta$.

Proof. Since $CM_{z^2}=M_{\bar{z}^2}C$, then by Proposition 5.3 , $C=M_{[\psi_1,\psi_2]}J$ and $M_{z}CJM_{\bar{z}}=M_{[\psi_2,\psi_1]} $ for some $\psi_1,\psi_2 \in L^\infty$ satisfying $\psi_1^o= \psi_2^o$ and $|\psi_1|^2+|\psi_2|^2=2$. Next, we consider the reproducing kernel function at 0, that is $k^\alpha_0= 1-\overline{\alpha(0)}\alpha$ and its conjugate $\tilde{k}^\alpha_0 =\mathscr{C}_\alpha k^\alpha_0 =\bar{z}( \alpha -\alpha(0))$ in K_α. Therefore by simple calculations, we get

\begin{align*} C(\tilde{k}^\alpha_0) = M_{[\psi_1,\psi_2]}J(\bar{z}( \alpha -\alpha(0))) = M_{[\psi_1,\psi_2]}(zf^e+zf^o) = \psi_1 zf^o + \psi_2 zf^e, \end{align*}

where $f=\overline{(\alpha -\alpha(0))} $. Similarly, we also have,

\begin{align*} M_z C M_z(\tilde{k}^\alpha_0) = M_z M_{[\psi_1,\psi_2]}J( \alpha -\alpha(0)) = M_z M_{[\psi_1,\psi_2]}\overline{(\alpha -\alpha(0))} = \psi_1 zf^e + \psi_2 zf^o. \end{align*}

Since $C(\tilde{k}^\alpha_0),M_zCM_z(\tilde{k}^\alpha_0) \in K_\theta $, then $C(\tilde{k}^\alpha_0)+M_zCM_z(\tilde{k}^\alpha_0)\in K_\theta$. In other words, $(\psi_1 +\psi_2)zf \in K_\theta $ which implies that there exist $g\in K_\theta $ such that $g=(\psi_1 +\psi_2)z\overline{(\alpha -\alpha(0))}=(\psi_1 +\psi_2)z\bar{\alpha} (1-\bar{\alpha}(0)\alpha )$. Note that $(1-\overline{\alpha(0)}\alpha)^{-1}$ is a bounded analytic function and hence,

\begin{equation*} z\bar{\alpha}(\psi_1 +\psi_2) = g(1-\overline{\alpha(0)}\alpha)^{-1} \in H^2, \end{equation*}

which yields $z(\psi_1+\psi_2)\in H^\infty$. Our next aim is to show that $\dfrac{z(\psi_1+\psi_2)}{2}=\gamma(say)$ is an inner function. Since $CJ=M_{[\psi_1,\psi_2]}$ and $M_{z}CJM_{\bar{z}}=M_{[\psi_2,\psi_1]} $, then it follows that:

\begin{equation*}(M_{z}CJM_{\bar{z}})^*(CJ) = M_{[\psi_2,\psi_1]}^*M_{[\psi_1,\psi_2]}.\end{equation*}

By considering the decomposition $L^2(\mathbb{T})=\mathcal{H}_1(1,z^2) \oplus \mathcal{H}_2(z,z^2)$, we have the following block-matrix representation:

\begin{align*} M_{[\psi_2,\psi_1]}^*M_{[\psi_1,\psi_2]} &= \begin{bmatrix} M_{\bar{\psi_2^e}\psi_1^e+\bar{\psi_2^o}\psi_1^o}&& M_{\bar{\psi_2^e}\psi_2^o+\bar{\psi_2^o}\psi_2^e}\\ \\ M_{\bar{\psi_1^o}\psi_1^e+\bar{\psi_1^e}\psi_1^o} && M_{\bar{\psi_1^o}\psi_2^o+\bar{\psi_1^e}\psi_2^e}\\ \end{bmatrix}. \end{align*}

Therefore, the condition $Re[(M_{z}CJM_{\bar{z}})^*(CJ)] =I$ boils down to $ Re\left[M_{[\psi_2,\psi_1]}^*M_{[\psi_1,\psi_2]}\right]=I$ which produces the following two conditions:

(5.1)\begin{equation} \begin{cases} & Re[\bar{\psi_2^e}\psi_1^e+\bar{\psi_2^o}\psi_1^o] =1, \\ & \bar{\psi_2^e}\psi_2^o+\bar{\psi_2^o}\psi_2^e + \psi_1^o\bar{\psi_1^e}+\psi_1^e\bar{\psi_1^o}=0. \end{cases} \end{equation}

Recall that, to characterize the class of conjugations C intertwining $M_{z^2}$ and $M_{\bar{z}^2}$, we initially obtain the conditions (4.4) in Remark 4.3. Therefore, combining the conditions (5.1) and $\bar{\psi_2^o}\psi_1^e+\bar{\psi_2^e}\psi_1^o=0$ in (4.4), we get,

\begin{align*} 2Re[\psi_2\bar{\psi_1]} =\psi_2\bar{\psi_1}+\psi_1\bar{\psi_2} = 2Re[\bar{\psi_2^e}\psi_1^e+\bar{\psi_2^o}\psi_1^o] + 2Re[\bar{\psi_2^o}\psi_1^e+\bar{\psi_2^e}\psi_1^o]=2. \end{align*}

Thus, $|\psi_1+\psi_2|^2=|\psi_1|^2+|\psi_2|^2+2Re[\psi_2\bar{\psi_1]}=4$, and hence $|\gamma|=|\dfrac{z(\psi_1+\psi_2)}{2}|=1$ a.e. on $\mathbb{T}$. Therefore, $\bar{\alpha}\gamma\in H^2$ shows that γ is divisible by α. Moreover, we also obtain,

\begin{align*} \mathscr{C}_\theta (C+M_zCM_z)(k^\alpha_0) & = \mathscr{C}_\theta (M_{[\psi_1,\psi_2]}J+M_zM_{[\psi_1,\psi_2]}JM_z)(1-\overline{\alpha(0)}\alpha)\\ & = \theta \overline{z(\psi_1+\psi_2)}(1-\overline{\alpha(0)}\alpha)\in K_\theta. \end{align*}

Therefore, $\theta \bar{\gamma} \in H^2$ implies that θ is divisible by γ. Moreover, one can easily conclude that $C+M_zCM_z=2\mathscr{C}_\gamma$.

Next, we discuss the relationship between $only~M_{z^2}-commuting$ conjugations and model spaces. Before we start, recall some useful facts obtained in [4] as follows:

(5.2)\begin{equation} J^\star(K_\alpha)=K_{\alpha^\#},~ J^\star \mathscr{C}_\alpha = \mathscr{C}_{\alpha^\#}J^\star,~\quad \text{and} \quad J^\star(\alpha H^2) = \alpha^\# H^2. \end{equation}

Note that, for an antilinear operator T, $T^\sharp$ denotes the antilinear adjoint of T.

Theorem 5.5. Let $\alpha, \theta $ be two non-constant inner functions and let C be a $only~M_{z^2}-commuting$ conjugation C on L ². If C satisfies the following conditions:

1. (i) $J^\star C$ is $\mathscr{C}_\alpha $-symmetric,

2. (ii) $C(K_\alpha) \subset K_\theta $ and $M_zCM_{\bar{z}}(K_\alpha) \subset K_\theta $,

3. (iii) $Re\left[(M_{z}C\mathscr{C}_\alpha JM_{\bar{z}})^\sharp(C\mathscr{C}_\alpha J)\right] =I$,

then there exists an inner function ψ satisfying $\alpha \leq \psi \leq \theta^\#$ such that $C+M_zCM_{\bar{z}} = J^\star M_{\frac{2\psi}{\alpha}}$.

Proof. Since C is a $only~M_{z^2}-commuting$ conjugation, then by Corollary 4.7, there exist $\xi_1,\xi_2\in L^\infty$ such that $C=M_{[\xi_1,\xi_2]}J^\star$ and $\xi_1,\xi_2$ satisfy $|\xi_1|^2+|\xi_2|^2=2,|\xi_2^e+\xi_1^o|^2+| \xi_2^o+\xi_1^e|^2=2,\xi_1^o(\bar{z})=\xi_2^o(z), \xi_1^e(\bar{z})=\xi_1^e(z), \xi_2^e(\bar{z})=\xi_2^e(z)$. Note that $J^\star C=M_{[\xi_1,\xi_2]}^*$ and by hypothesis, it is $\mathscr{C}_\alpha$-symmetric. Next, we consider the antilinear operator $J^\star C \mathscr{C}_\alpha $ on L ², and we have the following observations:

1. (i) $(J^\star C \mathscr{C}_\alpha)^2 =J^\star C \mathscr{C}_\alpha J^\star C \mathscr{C}_\alpha =M_{[\xi_1,\xi_2]}^*\mathscr{C}_\alpha M_{[\xi_1,\xi_2]}^*\mathscr{C}_\alpha=M_{[\xi_1,\xi_2]}^*M_{[\xi_1,\xi_2]}=I$,

2. (ii) $\langle J^\star C \mathscr{C}_\alpha(f),J^\star C \mathscr{C}_\alpha(g)\rangle =\langle g,f\rangle $ for any $f,g \in L^2$.

Therefore, $J^\star C \mathscr{C}_\alpha $ is a conjugation satisfying $J^\star C \mathscr{C}_\alpha M_{z^2} = M_{\bar{z}^2} J^\star C \mathscr{C}_\alpha $, but observe that $J^\star C \mathscr{C}_\alpha M_z \neq M_{\bar{z}}J^\star C \mathscr{C}_\alpha$. Moreover, using the hypothesis, we get:

\begin{equation*} J^\star C \mathscr{C}_\alpha (K_\alpha)=J^\star C (K_\alpha) \subset J^\star(K_\theta)=K_{\theta^{\#}},\end{equation*}

and $M_zJ^\star C \mathscr{C}_\alpha M_z(K_\alpha)=J^\star M_zC M_{\bar{z}}\mathscr{C}_\alpha (K_\alpha)=J^\star M_zC M_{\bar{z}}(K_\alpha)\subset J^\star(K_\theta)=K_{\theta^{\#}} $. Furthermore,

\begin{align*} Re[(M_{z}J^\star C \mathscr{C}_\alpha JM_{\bar{z}})^*(J^\star C \mathscr{C}_\alpha J)] = Re[(M_{z}C\mathscr{C}_\alpha JM_{\bar{z}})^\sharp(C\mathscr{C}_\alpha J)] =I. \end{align*}

Finally, by applying Theorem 5.4 and using the fact $\mathscr{C}_\alpha M_z=M_{\bar{z}}\mathscr{C}_\alpha$ , there exists an inner function ψ satisfying $\alpha \leq \psi \leq \theta ^{\#}$ such that:

\begin{align*} & J^\star C \mathscr{C}_\alpha +M_z J^\star C \mathscr{C}_\alpha M_z =2\mathscr{C}_\psi \implies J^\star C \mathscr{C}_\alpha + J^\star M_z C M_{\bar{z}}\mathscr{C}_\alpha = 2\mathscr{C}_\psi \\ & \implies C + M_z C M_{\bar{z}} =J^\star 2\mathscr{C}_\psi \mathscr{C}_\alpha \implies C + M_z C M_{\bar{z}} = J^\star M_{\frac{2\psi}{\alpha}}. \end{align*}

This completes the proof.

It is important to note that assumptions made in Theorem 5.4 and Theorem 5.5 above are fully consistent with the assumptions made in [10, Theorem 4.2] and [10, Theorem 4.6], respectively.

5.3. S-invariant subspaces preserving conjugations

In this subsection, we characterize all conjugations C preserving shift invariant subspaces and intertwining $M_{z^2}$, $M_{\bar{z}^2}$. Moreover, we also characterize all $M_{z^2}$-commuting conjugations C that preserve shift-invariant subspaces.

Theorem 5.6. There exist no $M_{z^2}$ and $M_{\bar{z}^2}$ intertwining conjugations in L ² that map any Beurling type subspace $\alpha H^2$ into another $\theta H^2$ for any two inner functions α and θ.

Proof. Let C be a $M_{z^2},M_{\bar{z}^2}$-intertwining conjugation on L ² such that $C(\alpha H^2)\subset \theta H^2$. Since C satisfies $CM_{z^2}=M_{\bar{z}^2}C$, then by Corollary (4.2), there exist $\psi_1,\psi_2 \in L^\infty$ such that $C=M_{[\psi_1,\psi_2]}J$ and $\psi_1^o= \psi_2^o$, $|\psi_1|^2+|\psi_2|^2=2$. Since $C(\alpha H^2)\subset \theta H^2$, then there exists $f\in H^2$ such that:

\begin{align*} C(\alpha) = M_{[\psi_1,\psi_2]}J(\alpha) = M_{[\psi_1,\psi_2]}J(\alpha^e \oplus \alpha^o ) = \psi_1\bar{\alpha}^e + \psi_2 \bar{\alpha}^o = \theta f. \end{align*}

Similarly, for $n=2k,k\geq 0$, we get,

\begin{align*} C(\alpha z^n) = M_{[\psi_1,\psi_2]}J(\alpha z^n) = M_{[\psi_1,\psi_2]}J(\alpha^e z^n \oplus \alpha^o z^n) = \bar{z}^n(\psi_1\bar{\alpha}^e + \psi_2 \bar{\alpha}^o)=\theta h_n, \end{align*}

for some $h_n\in H^2$ and hence $\bar{z}^n f = h _n \in H^2$. Thus for any non-negative even integer n,

\begin{align*} \langle \bar{z}^nf , \bar{z} \rangle =0 \implies \langle f, z^{n-1} \rangle =0 \implies f^o =0. \end{align*}

Similarly, $\langle \bar{z}^nf, \bar{z}^2 \rangle =0$ implies $f^e =0$ and hence we conclude that f = 0. Thus $C(\alpha)=0$, a contradiction.

Now, we move to the characterization of $M_{z^2}$-commuting conjugations that map a shift-invariant subspace $\alpha H^2$ into another one, say $\theta H^2$.

Theorem 5.7. Let α and β be two inner functions, and let C be any $M_{z^2}$-commuting conjugation on L ² satisfying $Re[(M_{\bar z}CJ^\star M_z)^*(CJ^\star)]=I$. If $C(\alpha H^2)\subset \beta H^2$, then $\beta \beta^\# \leq \alpha \alpha^\#$ and $C+M_zCM_z =2\mathscr{C}_{\theta}J^*\mathscr{C}_{\alpha}$, where θ is an inner function such that $\beta \leq \theta$, $\theta \theta ^\# =\alpha \alpha^\#$ and $\theta\overline{\alpha ^\#}$ is symmetric.

Proof. Since $CM_{z^2}=M_{z^2}C$, then by Corollary (4.2), there exist $\xi_1,\xi_2\in L^\infty$ such that $C=M_{[\xi_1,\xi_2]}J^\star$ and $|\xi_1|^2+|\xi_2|^2=2,|\xi_2^e+\xi_1^o|^2+| \xi_2^o+\xi_1^e|^2=2,\xi_1^o(\bar{z})=\xi_2^o(z), \xi_1^e(\bar{z})=\xi_1^e(z), \xi_2^e(\bar{z})=\xi_2^e(z)$. Now, by using the condition $C(\alpha H^2)\subset \beta H^2$, we conclude that

\begin{equation*}C(\alpha)=M_{[\xi_1,\xi_2]}J^\star (\alpha) = \xi_1 {\alpha^e}^{\#} +\xi_2 {\alpha^o}^{\#} \in \beta H^2.\end{equation*}

Moreover, we also have $M_z C M_z (\alpha H^2)\subset M_zC(\alpha)\subset M_z(\beta H^2)\subset \beta H^2$. Similarly,

\begin{equation*}M_zCM_z(\alpha )= \xi_1 {\alpha^o}^{\#} +\xi_2 {\alpha^e}^{\#} \in \beta H^2.\end{equation*}

Therefore, $\xi_1 {\alpha^e}^{\#} +\xi_2 {\alpha^o}^{\#} + \xi_1 {\alpha^o}^{\#} +\xi_2 {\alpha^e}^{\#} \in \beta H^2$ implies that there exists $h\in H^2$ such that:

(5.3)\begin{equation} (\xi_1+\xi_2)\alpha^{\#} =\beta h. \end{equation}

On the other hand, $CJ^\star = M_{[\xi_1,\xi_2]}$ and $M_{\bar z} CJ^\star M_z = M_{[\xi_2,\xi_1]}$, and hence $Re\left[(M_{\bar z}CJ^\star M_z)^*(CJ^\star)\right]=I$, which yields the following conditions:

(5.4)\begin{equation} \begin{cases} & Re\left[\bar{\xi_2^e}\xi_1^e+\bar{\xi_2^o}\xi_1^o\right] =1 , \\ & \bar{\xi_2^e}\xi_2^o+\bar{\xi_2^o}\xi_2^e + \xi_1^o\bar{\xi_1^e}+\xi_1^e\bar{\xi_1^o}=0. \end{cases} \end{equation}

Next, by combining conditions obtained in (5.4) and (4.11), we conclude:

\begin{align*} 2Re[\xi_2\bar{\xi_1}] = 2Re[\bar{\xi_2^e}\xi_1^e+\bar{\xi_2^o}\xi_1^o] + 2Re[\bar{\xi_2^o}\xi_1^e+\bar{\xi_2^e}\xi_1^o] = 2, \end{align*}

and hence $|\xi_1+\xi_2|^2=|\xi_1|^2 +|\xi_2|^2+ 2Re[\xi_2\bar{\xi_1}]=4$. Suppose $\psi = \dfrac{\xi_1 + \xi_2}{2}$, then $|\psi|=1$. Therefore, from (5.3) we get $\psi\alpha^{\#} = \beta f$ for some $f\in H^2$. On the other hand, ψ is a symmetric function since $\psi(\bar{z})=\frac{1}{2}[\xi_1(\bar{z})+\xi_2(\bar{z})]= \frac{1}{2}[(\xi_1^e(\bar{z})+\xi_1^o(\bar{z})+(\xi_2^e(\bar{z})+\xi_2^o(\bar{z})]=\frac{1}{2}[(\xi_1^e(z)+\xi_2^o(z)) + (\xi_2^e(z)+\xi_1^o(z))]= \frac{1}{2}[(\xi_1+\xi_2)(z)]=\psi(z)$. Let $\theta = \beta f$, then $\theta \overline{\alpha^{\#}}=\psi$ is symmetric. Moreover, θ is an inner function such that $\beta \leq \theta$ and $\theta \theta^{\#}=\alpha \alpha^{\#}$ which immediately implies $\beta \beta ^{\#} \leq \alpha \alpha^\#$. Furthermore, by simple calculations, one can also obtain the following identity:

\begin{equation*}C+M_zCM_z =2\mathscr{C}_\theta J^* \mathscr{C}_\alpha.\end{equation*}

This completes the proof.

Remark 5.8. Note that the converse of Theorem 5.7 may not be true in general. In other words, for the converse, if we assume $\beta \beta ^{\#} \leq \alpha \alpha^\#$, then by mimicking the proof of Theorem 5.2 in [10], one can show that $C+M_zCM_z =2\mathscr{C}_\theta J^* \mathscr{C}_\alpha$, which maps $\alpha H^2$ onto $\theta H^2 \subset \beta H^2$ but not necessarily $C(\alpha H^2) \subset \beta H^2$.

6. Conjugations related to M_B

In this section, we want to investigate some of the earlier characterization results in the context of M_B, where B is a finite Blaschke product. Let B be a finite Blaschke product of degree (or order, say) n, then the dimension of the model space $K_B:=H^2\ominus BH^2$ is n. Let $\{e_1,e_2,\ldots e_n\}$ be an orthonormal basis of K_B. Recall that (see $\S$ 3, (2.2)),

\begin{equation*}\text{L}^2(\mathbb{T}) = \mathcal{H}_1 (e_1,B)\oplus \mathcal{H}_2 (e_2,B)\oplus \cdots \oplus \mathcal{H}_n (e_n,B).\end{equation*}

For example, if $B(z)=z^2$, then for any $f\in L^2(\mathbb{T})$, we have $f=f^e\oplus f^o$, where $f^e \in \bigvee\limits_{k=-\infty}^{\infty}\{z^{2k}\} =\mathcal{H}_1(1,z^2)$ and $f^o \in \bigvee\limits_{k=-\infty}^{\infty}\{z^{2k+1}\}=\mathcal{H}_1(z,z^2)$, and we are already familiar with such factorization in earlier sections. Now if $B(z)=z^n$, then for any $f(z)=\sum\limits_{k=-\infty}^{\infty}a_kz^k \in L^2(\mathbb{T})$ has the following decomposition:

(6.1)\begin{equation} f(z) = \sum_{k=-\infty}^{\infty}a_{nk}z^{nk} +\sum_{k=-\infty}^{\infty}a_{nk+1}z^{nk+1}+\cdots +\sum_{k=-\infty}^{\infty}a_{nk+n-1}z^{nk+n-1}, \end{equation}

where $f_i(z)=\sum\limits_{k=-\infty}^{\infty}a_{nk+i-1}z^{nk+i-1}\in \mathcal{H}_i(z^i,z^n)$ for $i\in\{0,1,\ldots ,n-1\}$. Recall the following two well-known facts:

(6.2)\begin{equation} \mathscr{C}_BM_B = M_{\bar{B}}\mathscr{C}_B \quad \text{and}\quad \mathscr{C}_B^\star M_B = M_B\mathscr{C}_B^\star. \end{equation}

The following theorem is one of the main results in this section, which generalize Theorem 4.1 and Theorem 4.5 of $\S$ 4.

Theorem 6.1. Let $C:L^2(\mathbb{T}) \to L^2(\mathbb{T})$ be a conjugation.

1. (i) If $CM_B = M_{\bar{B}}C$, then there exist $\phi_1,\phi_2,\ldots ,\phi_n \in L^{\infty}$ such that $C=M_{[\phi_1,\phi_2,\ldots ,\phi_n]} \mathscr{C}_B $, where $M_{[\phi_1,\phi_2,\ldots ,\phi_n]}$ is unitary and $\mathscr{C}_B$-symmetric.

2. (ii) If $CM_{B} =M_BC$, then there exist $\psi_1,\psi_2,\ldots ,\psi_n \in L^{\infty}$ such that $C=M_{[\psi_1,\psi_2,\ldots ,\psi_n]} \mathscr{C}_B^\star $, where $M_{[\psi_1,\psi_2,\ldots ,\psi_n]}$ is unitary and $\mathscr{C}_B^\star$-symmetric.

Proof.

1. (i) Let $C:L^2(\mathbb{T}) \to L^2(\mathbb{T})$ be a conjugation such that $CM_B = M_{\bar{B}}C$, then using (6.2), we conclude that $C\circ \mathscr{C}_B$ is a unitary operator on L ² satisfying $C\circ \mathscr{C}_B M_B =M_B C\circ \mathscr{C}_B$. Therefore, from Theorem 2.2, we get $\phi_1, \phi_2, \ldots, \phi_n\in L^\infty$ such that $C\circ \mathscr{C}_B = M_{[\phi_1,\phi_2,\cdots ,\phi_n]}$, and hence $C=M_{[\phi_1,\phi_2,\ldots ,\phi_n]}\mathscr{C}_B$. It is easy to observe that $M_{[\phi_1,\phi_2,\ldots ,\phi_n]}$ is a unitary operator and

   \begin{equation*} M_{[\phi_1,\phi_2,\ldots ,\phi_n]}\mathscr{C}_B M_{[\phi_1,\phi_2,\ldots ,\phi_n]}\mathscr{C}_B =I, \end{equation*}

   and hence

   \begin{equation*} \mathscr{C}_B M_{[\phi_1,\phi_2,\ldots ,\phi_n]}\mathscr{C}_B = M^*_{[\phi_1,\phi_2,\ldots ,\phi_n]}, \end{equation*}

   implying that $M_{[\phi_1,\phi_2,\ldots ,\phi_n]}$ is $\mathscr{C}_B$-symmetric.

2. (ii) Let C be a conjugation on L ² such that $CM_B=M_BC$. Then again, using (6.2), we conclude that the unitary operator $C\circ \mathscr{C}_B^\star$ also commutes with M_B, and hence by Theorem 2.2, there exist $\psi_1,\psi_2, \ldots , \psi_n \in L^\infty$ such that

   \begin{equation*}C \circ \mathscr{C}^\star_B = M_{[\psi_1,\psi_2,\ldots ,\psi_n]}\end{equation*}

   which implies $C = M_{[\psi_1,\psi_2,\ldots ,\psi_n]}\mathscr{C}^\star_B $. Similarly, as in (i), $M_{[\psi_1,\psi_2,\ldots ,\psi_n]}$ is unitary and $\mathscr{C}_B^\star$-symmetric.

Now, we turn our discussion to find the relations between M_B-commuting conjugations and model spaces for $B(z)=z^n$. Observe that the actions of $\mathscr{C}_{B}$ and $\mathscr{C}_B^\star$ are same on K_B. The action of $\mathscr{C}_{z^n}$ on the set $\{1,z,\ldots,z^{n-1}\}$ is given by $\mathscr{C}_{z^n}(z^k)= z^{(n-1)-k}$ for $0\leq k \leq n-1$. Depending on n, we have the following two cases:

1. (1) If n is odd, then the action of $\mathscr{C}_{z^n}$ is the following

   (6.3)\begin{align} \begin{array}{|c| c| c| c| c| c| c| c| c|c|c|} \hline &&&&&&&&&&\\[-9pt] 1 & z & z^2 & \cdots & z^{\frac{n-3}{2}} & z^{\frac{n-1}{2}} & z^{\frac{n+1}{2}} & \cdots & z^{n-3} & z^{n-2} & z^{n-1} \\ \hline &&&&&&&&&&\\[-9pt] z^{n-1} & z^{n-2} & z^{n-3} & \cdots & z^{\frac{n+1}{2}} & z^{\frac{n-1}{2}} & z^{\frac{n-3}{2}} & \cdots & z^2 & z & 1 \\ \hline \end{array}, \end{align}

   where the first row contains the basis elements and the second row contains their images, respectively.

2. (2) If n is even, then as above the action of $\mathscr{C}_{z^n}$ is as follows:

   (6.4)\begin{align} \begin{array}{|c| c| c| c| c| c| c| c|c|c|} \hline &&&&&&&&&\\[-9pt] 1 & z & z^2 & \cdots & z^{\frac{n}{2}-1} & z^{\frac{n}{2}} & \cdots & z^{n-3} & z^{n-2} & z^{n-1} \\ \hline &&&&&&&&&\\[-9pt] z^{n-1} & z^{n-2} & z^{n-3} & \cdots & z^{\frac{n}{2}} & z^{\frac{n}{2}-1} & \cdots & z^2 & z & 1 \\ \hline \end{array}. \end{align}

Also, note that $\mathscr{C}_{z^n}=M_{z^{n-1}}J$, and

(6.5)\begin{equation} \mathscr{C}^\star_{z^n}= \begin{cases} M_{[z^{n-1},z^{n-3},\ldots ,z^2,1,\bar{z}^2, \ldots ,\bar{z}^{n-3},\bar{z}^{n-1}]} J^\star , ~\text{if~n~is ~odd,}\\ M_{[z^{n-1},z^{n-3},\ldots ,z,\bar{z}, \ldots ,\bar{z}^{n-3},\bar{z}^{n-1}]} J^\star , ~\text{if ~n ~is ~even}. \end{cases} \end{equation}

The above two relations follow from the decomposition of $f\in L^2$ mentioned in (6.1) along with the action of $\mathscr{C}_{z^n}$ and $\mathscr{C}^\star_{z^n}$, respectively. Now we are in a position to state and prove our second main result in this section, which generalizes Theorem 5.1. Nevertheless, we have more explicit expressions of ξ ₁ and ξ ₂ in Theorem 5.1 for $B(z)=z^2$, in compare to the following theorem for the case $B(z)=z^n$, $n\geq 3$.

Theorem 6.2. Let C be a conjugation on L ² such that $CM_{z^n} =M_{z^n} C$ and let C preserve the Hardy space H ², then $C = M_{[\xi_1,\xi_2, \ldots ,\xi_n]}J^\star$, where $\xi_1 ,z\xi_2,z^2\xi_3, \ldots ,z^{n-1}\xi_n \in H^\infty $ and $M_{[\xi_1,\xi_2, \ldots ,\xi_n]}$ is a $J^\star$-symmetric unitary operator.

Proof. By using Theorem 6.1, there exist $\psi_1, \psi_2 , \ldots , \psi_n \in L^\infty$ such that $C=M_{[\psi_1,\psi_2, \ldots ,\psi_n]}C^\star_{z^n}$. Now from (6.5), we get,

\begin{equation*} C= \begin{cases} M_{[\psi_1,\psi_2, \ldots ,\psi_n]}M_{[z^{n-1},z^{n-3},\ldots ,z^2,1,\bar{z}^2, \ldots ,\bar{z}^{n-3},\bar{z}^{n-1}]} J^\star ,~\text{if~n~is ~odd},\\ M_{[\psi_1,\psi_2, \ldots ,\psi_n]}M_{[z^{n-1},z^{n-3},\ldots ,z,\bar{z},\bar{z}^3, \ldots ,\bar{z}^{n-3},\bar{z}^{n-1}]} J^\star , ~\text{if ~n ~is ~even}. \end{cases} \end{equation*}

Next, we divide our analysis into the following two cases.

1. (i) Let n be an odd natural number.

   Now, we rewrite C as follows:

   \begin{align*} C & = M_{[\xi_1,\xi_2, \ldots ,\xi_n]}J^\star, \end{align*}

   where $\xi_1 = z^{n-1}\psi_n, \xi_2 = z^{n-3}\psi_{n-1} ,\ldots ,\xi_{\frac{n-1}{2}}= z^2 \psi_{\frac{n+3}{2}} ,\xi_{\frac{n+1}{2}} = \psi_{\frac{n+1}{2}} ,\xi_{\frac{n+3}{2}} = \bar{z}^2\psi_{\frac{n-1}{2}}, \ldots$, $ \xi_n = \bar{z}^{n-1}\psi_1 $. Therefore, by proceeding similarly as in Theorem 6.1, we conclude that $M_{[\xi_1,\xi_2, \ldots ,\xi_n]}$ is a unitary operator and $J^\star$-symmetric. Since $C(H^2)\subset H^2$, then we conclude $\xi_1 ,z\xi_2,$ $z^2\xi_3, \ldots ,z^{n-1}\xi_n \in H^\infty$.

2. (ii) Let n be an even natural number.

   Like case (i), we again rewrite C as follows

   \begin{align*} C & = M_{[\xi_1,\xi_2, \ldots ,\xi_n]}J^\star, \end{align*}

   where $\xi_1 = z^{n-1}\psi_n, \xi_2 = z^{n-3}\psi_{n-1} ,\ldots ,\xi_{\frac{n}{2}}= z \psi_{\frac{n}{2}+1} ,\xi_{\frac{n}{2}+1} = \bar{z}\psi_{\frac{n}{2}} ,\ldots$, $ \xi_n = \bar{z}^{n-1}\psi_1 $. Similarly, using the invariance property $C(H^2)\subset H^2$, we conclude $\xi_1 ,z\xi_2,z^2\xi_3, \ldots ,z^{n-1}\xi_n \in H^\infty $.

Therefore in both cases $C = M_{[\xi_1,\xi_2, \ldots ,\xi_n]}J^\star$, where $\xi_1 ,z\xi_2,z^2\xi_3, \ldots ,z^{n-1}\xi_n \in H^\infty $. Moreover, it is easy to check that $M_{[\xi_1,\xi_2, \ldots ,\xi_n]}$ is a unitary operator and $J^\star$-symmetric.

Finally, we have the following interesting result.

Theorem 6.3. There is no conjugation C on L ² for which $M_{z^n}$ is $C-$symmetric and $C(H^2)\subset H^2$.

Proof. Let C be a conjugation on L ² such that $CM_{z^n}C =M_{\bar{z}^n} $ and $C(H^2)\subset H^2$. Therefore by applying Theorem 6.1 we get,

(6.6)\begin{align} C = M_{[\phi_1,\phi_2,\ldots ,\phi_n]} \mathscr{C}_{z^n} = M_{[\phi_1,\phi_2,\ldots ,\phi_n]}M_{z^{n-1}}J = M_{[\zeta_1,\zeta_2,\ldots ,\zeta_n]}J, \end{align}

where $\zeta_1=z^{n-1} \phi_n ,\zeta_2 = z^{n-1} \phi_1 ,\zeta_3 = z^{n-1} \phi_2, \ldots ,\zeta_n =z^{n-1}\phi_{n-1} .$ Moreover, the fact $C(H^2)\subset H^2$ implies $\zeta_1,\zeta_2, \ldots , \zeta_n \in H^\infty$. Now by using (6.6), we get for any $k\geq 0$.

\begin{align*} \langle C(z^{nk}),\bar{z}^j \rangle =0\implies \langle \zeta_1, z^{nk-j} \rangle =0,\quad \text{for}\quad 1\leq j\leq n, \end{align*}

and hence $\zeta_1 =0$. By mimicking a similar kind of argument, we conclude $\zeta_{i} = 0$ for $2\leq i \leq n$. On the other hand, $ M_{[\zeta_1,\zeta_2,\ldots ,\zeta_n]}=CJ$ is an unitary operator, a contradiction. This completes the proof.

We end the article with the following question:

Question 3. What is the complete characterization for the class of conjugations that either commute with M_B or intertwine the operators M_B and $M_{\overline{B}}$ together with the invariant property of various subspaces of L ² corresponding to any arbitrary finite Blaschke product B?