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 [Reference Abkar, Cao and Zhu1, Reference Cowen and Wahl5, Reference Gallardo-Gutiérrez and Partington16], 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.

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 [Reference Partington31, 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 [Reference Sz.-Nagy, Foias, Bercovici and Kérchy36], 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 [[Reference Fricain and Mashreghi14], 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.

One can find the classical theorem for the commutant of the shift operator on $L^2(\mathbb{T})$-space in [Reference Martínez-Avendaño and Rosenthal28](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:

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 [Reference Garcia, Mashreghi and Ross17, Reference Garcia and Putinar19]).

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 [Reference Câmara, Kliś-Garlicka, Łanucha and Ptak10], 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 [Reference Chattopadhyay, Das, Pradhan and Sarkar3].

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 [Reference Câmara, Kliś-Garlicka and Ptak9, Reference Câmara, Kliś-Garlicka, Łanucha and Ptak10], the authors obtained the characterization of all conjugations C intertwining the operators M_z, $M_{\bar{z}}$ in terms J in L ² (see [Reference Câmara, Kliś-Garlicka, Łanucha and Ptak10, 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 [Reference Câmara, Kliś-Garlicka, Łanucha and Ptak10, 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)).

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:

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.

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.

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.

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 [Reference Câmara, Kliś-Garlicka, Łanucha and Ptak10].

• • 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 [Reference Câmara, Kliś-Garlicka, Łanucha and Ptak10]. The significant difference between our study and the study in [Reference Câmara, Kliś-Garlicka, Łanucha and Ptak10] 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 [Reference Câmara, Kliś-Garlicka, Łanucha and Ptak10, 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 [Reference Cima, Garcia, Ross and Wogen4] 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 [Reference Câmara, Kliś-Garlicka, Łanucha and Ptak10, Theorem 4.2] and [Reference Câmara, Kliś-Garlicka, Łanucha and Ptak10, 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 [Reference Câmara, Kliś-Garlicka, Łanucha and Ptak10], 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.

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 $B(z)=z^n$, $n\geq 3$.

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.

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?