On complex symmetric block Toeplitz operators

TL;DR

This paper characterizes complex symmetric block Toeplitz operators on vector-valued Hardy spaces, providing theoretical insights and concrete examples of such operators.

Contribution

It offers a new characterization of complex symmetric block Toeplitz operators with a specific conjugation on vector-valued Hardy spaces.

Findings

Characterization of complex symmetric block Toeplitz operators

Examples of such operators on vector-valued Hardy spaces

Insights into the structure of these operators

Abstract

In this paper, we study complex symmetry of Toeplitz operators and block Toeplitz operators. In particular, we give a characterization of complex symmetric block Toeplitz operators with the special conjugation on the vector-valued Hardy space $H_{{\mathbb C}^2}^2$. As some applications, we provide examples of such operators.

Equations

$$f(z)=\sum _{n=0}^{\infty }{a}_n{z}^n\text{mbox}with\sum _{n=0}^{\infty }|a_n{|}^2<\infty .$$

$$T_{\phi }f=P(\phi f)\text{mbox}forf\in H^2$$

$$T_{\Phi }h=P_2(\Phi h),h\in H_{{\mathbb{C}}^2}^2$$

$$T_{\Phi }=\left(\begin{array}{cc}{T}_{{\phi }_1}& T_{{\phi }_2}\\ T_{{\phi }_3}& T_{{\phi }_4}\end{array}\right).$$

$$T_{\Phi }$$

$$C_{\mu ,\lambda }f(z)=\mu \overline{f(\lambda \overline{z})}$$

$$C_{\mu ,\lambda }P=P{C}_{\mu ,\lambda },$$

$$P{C}_{\mu ,\lambda }{z}^n=P\mu {\overline{\lambda }}^n{z}^n=\mu {\overline{\lambda }}^n{z}^n=C_{\mu ,\lambda }{z}^n=C_{\mu ,\lambda }P{z}^n$$

$$C_{\mu ,\lambda }T{C}_{\mu ,\lambda }$$

$$=\mu I\cdot C_{1,\lambda }\cdot \mu I\cdot T\cdot C_{1,\lambda }$$

$$=C_{1,\lambda }\cdot \overline{\mu }I\cdot \mu I\cdot T\cdot C_{1,\lambda }$$

$$=C_{1,\lambda }\cdot T\cdot C_{1,\lambda }.$$

$$C_{1,\lambda }{T}_{\phi (z)}{C}_{1,\lambda }=T_{C_{1,\lambda }\phi (z)}=T_{\overline{\phi (\lambda \overline{z})}}.$$

$$C_{1,\lambda }{T}_{\phi (z)}{C}_{1,\lambda }h(z)$$

$$=C_{1,\lambda }P\left[\phi (z)\overline{h(\lambda \overline{z})}\right]$$

$$=P{C}_{1,\lambda }\left[\phi (z)\overline{h(\lambda \overline{z})}\right]$$

$$=P\left[\overline{\phi (\lambda \overline{z})}h(z)\right]=T_{\overline{\phi (\lambda \overline{z})}}h(z),$$

$$\phi (z)=\phi (\lambda \overline{z})\text{mbox}on\left|z\right|=1.$$

$$T_{\phi (z)}=C_{1,\lambda }{T}_{\phi (z)}^{\ast }{C}_{1,\lambda }=C_{1,\lambda }{T}_{\overline{\phi (z)}}{C}_{1,\lambda }.$$

$$C_{1,\lambda }{T}_{\overline{\phi (z)}}{C}_{1,\lambda }=T_{\phi (\lambda \overline{z})}.$$

$${\phi }_{+}=S^{\ast }{T}_{\phi }{e}_0=\sum _{n=0}^{\infty }\hat{\phi }(n+1)e_n\text{mbox}and{\phi }_{-}=S^{\ast }{T}_{\phi }^{\ast }{e}_0=\sum _{n=0}^{\infty }\overline{\hat{\phi }(-n-1)}{e}_n$$

$$\{\begin{array}{l}\phi (z)={\phi }_{+}(z)+{\phi }_0+\overline{{\phi }_{-}(z)}\text{mbox}and\\ \phi (\lambda \overline{z})={\phi }_{+}(\lambda \overline{z})+{\phi }_0+\overline{{\phi }_{-}(\lambda \overline{z}}),\end{array}$$

$$\phi (z)={\phi }_0+2\sum _{n=1}^{\infty }\hat{\phi }(n)Re\{z^n\}.$$

$$\phi (z)={\phi }_0+2\sum _{k=1}^{\infty }\hat{\phi }(2k)Re\{z^{2k}\}+2i\sum _{k=1}^{\infty }\hat{\phi }(2k-1)Im\{z^{2k-1}\}.$$

$$T_{\phi }S-S{T}_{\phi }=e_0\otimes {\phi }_{-}\text{mbox}and{T}_{\phi }^{\ast }S-S{T}_{\phi }^{\ast }=e_0\otimes {\phi }_{+}.$$

$$\{\begin{array}{l}{T}_{\phi }S-S{T}_{\phi }=e_0\otimes {\phi }_{-}\\ T_{\phi }CSC-CSC{T}_{\phi }=C{e}_0\otimes C{\phi }_{+}.\end{array}$$

$$e_0=\lambda C{e}_0\text{mbox}and{\phi }_{-}=\frac{1}{\overline{\lambda }}C{\phi }_{+}$$

$$\frac{1}{\sqrt{2}}\left(\begin{array}[c]{cc}C_{\mu,\lambda}&C_{\mu,\lambda}\\ C_{\mu,\lambda}&-C_{\mu,\lambda}\end{array}\right)$$

$${\mathcal{C}}:=\frac{1}{\sqrt{2}}\left(\begin{array}[c]{cc}C_{1,\lambda}&C_{1,\lambda}\\ C_{1,\lambda}&-C_{1,\lambda}\end{array}\right)$$

$$\{\begin{array}{l}{\phi }_1(z)=\frac{1}{2}{\sum }_{j=1}^4{\phi }_j(\lambda \overline{z})\\ {\phi }_2(z)=\frac{1}{2}{\sum }_{j=1}^2[{\phi }_j(\lambda \overline{z})-{\phi }_{j+2}(\lambda \overline{z})]\\ {\phi }_3(z)=\frac{1}{2}{\sum }_{j=1}^4(-1{)}^{j+1}{\phi }_j(\lambda \overline{z})\\ {\phi }_4(z)=\frac{1}{2}{\sum }_{j=1}^2(-1{)}^{j+1}[{\phi }_j(\lambda \overline{z})-{\phi }_{j+1}(\lambda \overline{z})]\end{array}$$

$$\mathcal{C}{T}_{\Phi }^{\ast }=\frac{1}{\sqrt{2}}\left(\begin{array}{cc}{C}_{1,\lambda }(T_{{\phi }_1}^{\ast }+T_{{\phi }_2}^{\ast })& C_{1,\lambda }(T_{{\phi }_3}^{\ast }+T_{{\phi }_4}^{\ast })\\ C_{1,\lambda }(T_{{\phi }_1}^{\ast }-T_{{\phi }_2}^{\ast })& C_{1,\lambda }(T_{{\phi }_3}^{\ast }-T_{{\phi }_4}^{\ast })\end{array}\right)$$

$$T_{\Phi }\mathcal{C}=\frac{1}{\sqrt{2}}\left(\begin{array}{cc}(T_{{\phi }_1}+T_{{\phi }_2})C_{1,\lambda }& (T_{{\phi }_1}-T_{{\phi }_2})C_{1,\lambda }\\ (T_{{\phi }_3}+T_{{\phi }_4})C_{1,\lambda }& (T_{{\phi }_3}-T_{{\phi }_4})C_{1,\lambda }\end{array}\right).$$

$$\{\begin{array}{l}{T}_{{\phi }_1+{\phi }_2}=C_{1,\lambda }(T_{{\phi }_1+{\phi }_2}^{\ast })C_{1,\lambda }\\ T_{{\phi }_1-{\phi }_2}=C_{1,\lambda }(T_{{\phi }_3+{\phi }_4}^{\ast })C_{1,\lambda }\\ T_{{\phi }_3-{\phi }_4}=C_{1,\lambda }(T_{{\phi }_3-{\phi }_4}^{\ast })C_{1,\lambda }\\ T_{{\phi }_3+{\phi }_4}=C_{1,\lambda }(T_{{\phi }_1-{\phi }_2}^{\ast })C_{1,\lambda }.\end{array}$$

Full text

00footnotetext: 2010 Mathematics Subject Classification; Primary 47B35, 47A05. Keywords; Conjugation; Complex symmetric operator; Block Toeplitz operator. This third author was supported by Basic Science Research Program through the National Research Foundation of Korea(NRF) funded by the Ministry of Education(2016R1D1A1B03931937). The last author was supported by Basic Science Research Program through the National Research Foundation of Korea(NRF) funded by the Ministry of Education, Science and Technology(2019R1A2C1002653).

1 Introduction and Preliminaries

The study of complex symmetric operators is the interaction of the fields of operator theory and complex analysis. Recently, many authors have been interested in non-hermitian quantum mechanics and in the spectral analysis of certain complex symmetric operators (see [4] and [8]). In particular, several authors have been studied an antilinear operator which is the only type of nonlinear operators that are important in quantum mechanics (see [1], [18], and [19]). A conjugation on $\cal H$ is an antilinear operator ${C}:{\cal H}\rightarrow{\cal H}$ with ${C}^{2}=I$ which satisfies $\langle{C}x,{C}y\rangle=\langle y,x\rangle$ for all $x,y\in{\cal H}$. For the conjugation $C$, there exists an orthonormal basis $\{e_{n}\}_{n=0}^{\infty}$ for ${\cal H}$ such that ${C}e_{n}=e_{n}$ for all $n$ (see [6]). We call an operator $T\in{\cal L(H)}$ complex symmetric if there exists a conjugation $C$ on ${\cal H}$ such that $T={C}T^{\ast}{C}$. The class of complex symmetric operators includes all normal operators, Hankel operators, truncated Toeplitz operators, Volterra integration operators, and etc. We refer the reader to [6], [9], [13], and [14] for more details, including historical comments and references.

Let $L^{2}$ be the Lebesgue (Hilbert) space on the unit circle $\partial{\mathbb{D}}$ and let $L^{\infty}$ be the Banach space of all essentially bounded functions on $\partial{\mathbb{D}}$. If $f\in L^{2},$ then the function $f$ is expressed as $f(z)=\sum_{n=-\infty}^{\infty}\hat{f}(n)z^{n}$ where $\hat{f}(n)$ denotes the $n$th Fourier coefficient of $f.$ The Hilbert Hardy space, denoted by $H^{2}$, consists of all functions $f$ analytic on the open unit disk $\mathbb{D}$ with the power series representation

[TABLE]

For any $\varphi\in L^{\infty}$, the Toeplitz operator $T_{\varphi}:{H^{2}}\rightarrow{H^{2}}$ is defined by the formula

[TABLE]

where $P$ denotes the orthogonal projection of $L^{2}$ onto $H^{2}$.

Let $L_{{\mathbb{C}}^{2}}^{2}=L^{2}\otimes{\mathbb{C}}^{2}$, $H_{{\mathbb{C}}^{2}}^{2}=H^{2}\otimes{\mathbb{C}}^{2}$, and let $M_{2}$ be the set of $2\times 2$ complex matrices. For a matrix-valued function $\Phi\in L^{\infty}_{M_{2}}=L^{\infty}\otimes M_{2}$, the block Toeplitz operator with symbol $\Phi$ is the operator $T_{\Phi}$ on the vector-valued Hardy space $H_{{\mathbb{C}}^{2}}^{2}$ of the unit disk defined by

[TABLE]

where $P_{2}$ denotes the orthogonal projection of $L_{{\mathbb{C}}^{2}}^{2}$ onto $H_{{\mathbb{C}}^{2}}^{2}$ (see [11] and [12]). In particular, if $\Phi=\begin{pmatrix}\varphi_{1}&\varphi_{2}\cr\varphi_{3}&\varphi_{4}\end{pmatrix}$ where $\varphi_{j}\in L^{\infty}$ for $j=1,2,3,4$, then the block Toeplitz operator has the following representation;

[TABLE]

The study of Toeplitz operators and block Toeplitz operators provides deep and important connections with various problems in the field of physics (see [4] and [8]). In particular, the spectral theory of Toeplitz operators is applied to the research in quantum mechanics (see [17]). In addition, the study of the theory of block Toeplitz determinants plays a significant role in the study of high-temperature superconductivity (see [2]). In 1960’s, A. Brown and P. Halmos [3] proved that $T_{\varphi}$ is normal if and only if $\varphi=\alpha+\beta\rho$ where $\rho$ is a real-valued function in $L^{\infty}$ and $\alpha,\beta\in{\mathbb{C}}$. In general, $T_{\varphi}$ may not be a complex symmetric operator. In 2014, K. Guo and S. Zhu [10] have raised the following interesting question; characterize a complex symmetric Toeplitz operator on $H^{2}$. Recently, the authors in [14] gave a characterization of a complex symmetric Toeplitz operator $T_{\varphi}$ on $H^{2}$ with some conjugations. In view of this, we are interested in the following question;

[TABLE]

These results give us some motivations to study complex symmetric block Toeplitz operators.

In this paper, we study complex symmetry of Toeplitz operators and block Toeplitz operators. In particular, we provide a characterization of complex symmetric block Toeplitz operators with the special conjugation. From this, we give examples of normal complex symmetric block Toeplitz operators.

2 Remark on complex symmetry of a Toeplitz operator

In this section, we give a characterization of complex symmetric Toeplitz operator with the special conjugation on the Hardy space $H^{2}$. Recall that if $C_{\mu,\lambda}:H^{2}\rightarrow H^{2}$ is defined by

[TABLE]

for all $\lambda,\mu\in\partial{\mathbb{D}}$, then $C_{\mu,\lambda}$ is the conjugation operator on $H^{2}$ from [14]. Let $P$ denote the orthogonal projection of $L^{2}$ onto $H^{2}$. Then the operator $C_{\mu,\lambda}$ can also be defined on $L^{2}.$ Note that

[TABLE]

since for $n\geq 0$

[TABLE]

and for $n<0,PC_{\mu,\lambda}\overline{z}^{-n}=0=C_{\mu,\lambda}P\overline{z}^{-n}.$ We start with the following lemmas.

Lemma 2.1**.**

On $H^{2},$ for any operator $T,$ $C_{\mu,\lambda}TC_{\mu,\lambda}=C_{1,\lambda}TC_{1,\lambda}$. Hence, the operator $T$ is complex symmetric with the conjugation $C_{\mu,\lambda}$ if and only if $T$ is complex symmetric with the conjugation $C_{1,\lambda}.$

Proof.

Since $C_{\mu,\lambda}=\mu I\cdot C_{1,\lambda}=C_{1,\lambda}\cdot\overline{\mu}I,$ it follows that

[TABLE]

Therefore the operator $T$ is complex symmetric with the conjugation $C_{\mu,\lambda}$ if and only if $T$ is complex symmetric with the conjugation $C_{1,\lambda}.$ ∎

Lemma 2.2**.**

On $H^{2},$ for any Toeplitz operator $T_{\varphi},$

[TABLE]

Proof.

For any $h\in H^{2},$ we have

[TABLE]

where the third equality follows from (1). ∎

We prove one of our main results.

Theorem 2.3**.**

$T_{\varphi}$* is complex symmetric with the conjugation $C_{1,\lambda}$ if and only if*

[TABLE]

Proof.

By definition, $T_{\varphi}$ is complex symmetric with the conjugation $C_{1,\lambda}$ if and only if

[TABLE]

By Lemma 2.2, we have

[TABLE]

So we obtain $T_{\varphi(\lambda\overline{z})}=T_{\varphi(z)}$ and $\varphi(z)=\varphi(\lambda\overline{z}).$ ∎

Let us recall that $\varphi(z)=\sum_{n=-\infty}^{\infty}\hat{\varphi}(n)e_{n}$ belong to $L^{\infty}$. Set

[TABLE]

where $S$ denotes the unilateral shift on $H^{2}$.

Corollary 2.4**.**

Let $S$ denote the unilateral shift on $H^{2}$. If $T_{\varphi}$ is a complex symmetric operator with the conjugation $C_{1,\lambda}$, then $\varphi_{-}(z)=\overline{\varphi_{+}(\lambda\overline{z})}$ with $|\lambda|=1$. Moreover, $\hat{\varphi}(-n-1)={\lambda}^{n+1}\hat{\varphi}(n+1)$ for all $n=0,1,2,\cdots$.

Proof. By Theorem 2.3, if we write

[TABLE]

then $\varphi(z)=\varphi(\lambda\overline{z})$ if and only if $\varphi_{-}(z)=\overline{\varphi_{+}(\lambda\overline{z})}$, which in terms of Fourier coefficients is $\hat{\varphi}(-n-1)={\lambda}^{n+1}\hat{\varphi}(n+1)$ for all $n\geq 0$ and $\left|\lambda\right|=1.$

As a consequence of Theorem 2.3, we recapture the following corollary.

Corollary 2.5**.**

([14, Corollary 2.6])(i)* $T_{\varphi}$ is complex symmetric with the conjugation $C_{1,1}$ if and only if*

[TABLE]

(ii)* $T_{\varphi}$ is complex symmetric with the conjugation $C_{1,-1}$ if and only if*

[TABLE]

Remark that if $\varphi(z)=\phi(z)+\alpha+\phi(\overline{z})$ and $\psi(z)=\phi(z)+\beta+\phi(-\overline{z})$ for $\phi\in zH^{2}$ and $\alpha,\beta\in{\mathbb{C}}$, then $T_{\varphi}$ and $T_{\psi}$ are complex symmetric operators on $H^{2}$ ([14]).

Proposition 2.6**.**

Let $\{e_{n}\}$ be an orthonormal basis of $H^{2}$ and let $C$ be the conjugation on $H^{2}$ with $Ce_{n}=e_{n}$. If $T_{\varphi}$ is a complex symmetric operator with the conjugation $C$ and $CSC=S$ where $S$ denotes the unilateral shift on $H^{2}$, then $\varphi_{-}={\lambda}C\varphi_{+}$ and $Ce_{0}=\overline{\lambda}e_{0}$ for some $\lambda\in{\partial{\mathbb{D}}}.$

Proof. Suppose that $T_{\varphi}$ is a Toeplitz operator with a symbol $\varphi\in L^{\infty}$. If $\varphi_{+}=S^{\ast}T_{\varphi}e_{0}\ \mbox{and}\ \varphi_{-}=S^{\ast}T_{\varphi}^{\ast}e_{0}$, then we have

[TABLE]

Since $T_{\varphi}^{\ast}=CT_{\varphi}C$, it follows from (2) that

[TABLE]

If $CSC=S$, then $e_{0}\otimes\varphi_{-}=Ce_{0}\otimes C\varphi_{+}$ implies that

[TABLE]

for some $\lambda\in{\mathbb{C}}.$ Hence $\varphi_{-}={\lambda}C\varphi_{+}$ and $Ce_{0}=\overline{\lambda}e_{0}$ for some $\lambda\in{\partial{\mathbb{D}}}.$

3 Complex symmetry of a block Toeplitz operator

In this section, we study complex symmetric block Toeplitz operators. We recently realized the operator conjugations were rarely studied. So we consider these block Toeplitz operators with special conjugation

[TABLE]

on $H_{{\mathbb{C}}^{2}}^{2}$ from [15]. From Lemma 2.1, we may assume

[TABLE]

with $\mu=1.$

Theorem 3.1**.**

T_{\Phi}=\left(\begin{array}[c]{cc}T_{\varphi_{1}}&T_{\varphi_{2}}\\ T_{\varphi_{3}}&T_{\varphi_{4}}\end{array}\right)* is complex symmetric with the conjugation {\mathcal{C}}=\frac{1}{\sqrt{2}}\left(\begin{array}[c]{cc}C_{1,\lambda}&C_{1,\lambda}\\ C_{1,\lambda}&-C_{1,\lambda}\end{array}\right) if and only if*

[TABLE]

on $\left|z\right|=1$ and $|\lambda|=1$.

Proof. Since $T_{\Phi}$ is complex symmetric with the conjugation ${\cal C}$, it follows that

[TABLE]

and

[TABLE]

Then (3) and (4) imply that

[TABLE]

By Theorem 2.3, we obtain that

[TABLE]

Applying the proof of Lemma 2.2, we have

[TABLE]

From the second and fourth equations of (5), we get that

[TABLE]

Therefore we have

[TABLE]

Hence from (6), we get the followings;

[TABLE]

where $|\lambda|=1$.

Conversely, if (7) holds, then $(\ref{E_03-1})$ clearly holds. Thus we get (5). By Theorem 2.3 and Lemma 2.2, $T_{\Phi}$ is complex symmetric with the conjugation ${\cal C}$.    $\Box$

The following result is the special case for Theorem 3.1 with the assumption that $\varphi_{j}(z)$ are trigonometric polynomials.

Corollary 3.2**.**

Let ${\Phi}=\begin{pmatrix}{\varphi_{1}}&{\varphi_{2}}\cr{\varphi_{3}}&{\varphi_{4}}\end{pmatrix}$ where $\varphi_{j}(z)=\sum_{n=-m}^{N}\hat{\varphi}_{j}(n)z^{n}$ for $N\geq m>0$ and $j=1,2,3,4$. Then $T_{\Phi}$ is complex symmetric with the conjugation {\mathcal{C}}=\frac{1}{\sqrt{2}}\left(\begin{array}[c]{cc}C_{1,\lambda}&C_{1,\lambda}\\ C_{1,\lambda}&-C_{1,\lambda}\end{array}\right) if and only if $N=m$ and

[TABLE]

*for all $n=1,2,3,\cdots,N$ and $|\lambda|=1$. *

Proof. Applying the proof of Theorem 3.1, we get that $T_{\Phi}$ is complex symmetric with the conjugation ${\mathcal{C}}$ if and only if $N=m$ and

[TABLE]

equivalently, $N=m$ and

[TABLE]

where $|\lambda|=1$.    $\Box$

We illustrate the following example as some applications of Corollary 3.2.

Example 3.3**.**

Let

[TABLE]

and

[TABLE]

Then (8) holds for $T_{\Phi}$. Hence $T_{\Phi}$ is complex symmetric with the conjugation ${\mathcal{C}}$ (with $\lambda=-1$) from Corollary 3.2. However, (8) does not hold for $T_{\Psi}$. Hence $T_{\Psi}$ is not complex symmetric with the conjugation ${\mathcal{C}}$ (with $\lambda=1$) from Corollary 3.2.

Let us recall that $\Phi(z)=\sum_{n=-\infty}^{\infty}\widehat{\Phi}(n)z^{n}$ where

[TABLE]

Put $\Phi_{+}(z)=\sum_{n=1}^{\infty}\widehat{\Phi}(n)z^{n}$, $\Phi_{-}(z)=\sum_{n=1}^{\infty}\overline{\widehat{\Phi}(-n)}z^{n},$ and $\Phi_{0}=\begin{pmatrix}\hat{\varphi_{1}}(0)&\hat{\varphi_{2}}(0)\cr\hat{\varphi_{3}}(0)&\hat{\varphi_{4}}(0)\end{pmatrix}$. Then $\Phi_{+}(z),\Phi_{-}(z)\in H^{\infty}_{M_{2}}$ and

[TABLE]

where $\ast$ denotes the complex conjugate. Hence $\Phi(z)=\Phi_{+}(z)+\Phi_{0}+\Phi_{-}^{\ast}(z)\in L^{\infty}_{M_{2}}$ where $\Phi_{0}$ is a constant matrix. Using Theorem 3.1, we give a neccessary and sufficient condition of complex symmetric block Toeplitz operators with the special conjugation.

Theorem 3.4**.**

*If $T_{\varphi_{j}}(j=1,2,3,4)$ are complex symmetric with the conjugation $C_{1,\lambda}$ where ${\varphi_{1}}={\varphi_{2}+\varphi_{3}+\varphi_{4}}$, then the following statements are equivalent:

(i) $T_{\Phi}$ is complex symmetric with the conjugation {\mathcal{C}}=\frac{1}{\sqrt{2}}\left(\begin{array}[c]{cc}C_{1,\lambda}&C_{1,\lambda}\\ C_{1,\lambda}&-C_{1,\lambda}\end{array}\right).

(ii) $\widehat{\Phi}(n)\lambda^{n}=\widehat{\Phi}(-n)$ for all $n\in{\mathbb{Z}}$ and $|\lambda|=1$.

(iii) $\Phi(z)=\Phi_{0}+\sum_{n=1}^{\infty}\widehat{\Phi}(n)(z^{n}+\lambda^{n}\overline{z}^{n})$ for $|\lambda|=1$.

(iv) $\Phi(z)=\Phi_{+}(z)+\Phi_{0}+{\Phi_{+}(\lambda\overline{z})}$ for $\Phi_{+}\in zH^{2}_{{\mathbb{C}}^{2}}$ and $|\lambda|=1$. *

Proof. By Theorem 3.1 and Lemma 2.1, $T_{\Phi}$ is complex symmetric with the conjugation ${\cal C}$ if and only if $T_{\varphi_{j}}(j=1,2,3,4)$ are complex symmetric with the conjugation $C_{1,\lambda}$.

(i)$\Leftrightarrow$(ii): Since $T_{\varphi_{j}}(j=1,2,3,4)$ are complex symmetric with the conjugation $C_{1,\lambda}$ if and only if $\hat{\varphi_{j}}(-n)=\lambda\hat{\varphi_{j}}(n)$ for all $n\in{\mathbb{Z}}$ and for $j=1,2,3,4$ by [14], we conclude that

[TABLE]

for all $n\in{\mathbb{Z}}$ and $|\lambda|=1$.

(i) $\Leftrightarrow$ (iii): Let $T_{\Phi}$ be complex symmetric with the conjugation ${\cal C}$. Then, from assertion (ii),

[TABLE]

and therefore we have the statement (iii).

Conversely, if $\Phi(z)=\Phi_{0}+\sum_{n=1}^{\infty}\widehat{\Phi}(n)(z^{n}+\lambda^{n}\overline{z}^{n})$ with $|\lambda|=1$, then (9) and [14] implies that

[TABLE]

for all $n\in{\mathbb{Z}}$. By (i) $\Leftrightarrow$ (ii), $T_{\Phi}$ is complex symmetric with the conjugation ${\cal C}$.

(i) $\Leftrightarrow$ (iv): Since $T_{\Phi}$ is complex symmetric with the conjugation ${\cal C}$, it follows from (ii) that

[TABLE]

where $\ast$ denotes the complex conjugate. Therefore, $\Phi(z)=\Phi_{+}(z)+\Phi_{0}+{\Phi_{+}(\lambda\overline{z})}$ with $|\lambda|=1$.

Conversely, assume that $\Phi(z)=\Phi_{+}(z)+\Phi_{0}+{\Phi_{+}(\lambda\overline{z})}$ with $|\lambda|=1$. Then

[TABLE]

with $|\lambda|=1$. By (i) $\Leftrightarrow$ (ii), we know that $T_{\Phi}$ is complex symmetric with the conjugation ${\cal C}$.    $\Box$

As an application of Theorem 3.4, we recapture the following result.

Corollary 3.5**.**

([15]) *If $T_{\varphi_{j}}(j=1,2,3,4)$ are complex symmetric with the conjugation $C_{1,1}$ where ${\varphi_{1}}={\varphi_{2}+\varphi_{3}+\varphi_{4}}$, then the following statements are equivalent:

(i) $T_{\Phi}$ is complex symmetric with the conjugation ${\mathcal{C}}$ (with $\lambda=1$).

(ii) $\widehat{\Phi}(n)=\widehat{\Phi}(-n)$ for all $n\in{\mathbb{Z}}$.

(iii) $\Phi(z)=\Phi_{0}+\sum_{n=1}^{\infty}\widehat{\Phi}(n)(z^{n}+\overline{z}^{n})$.

(iv) $\Phi(z)=\Phi_{+}(z)+\Phi_{0}+{\Phi_{+}(\overline{z})}$ for $\Phi_{+}\in zH^{2}_{{\mathbb{C}}^{2}}$. *

We say that a block Toeplitz operator $T_{\Phi}$ is analytic if $\Phi\in H^{\infty}_{M_{2}}$ and coanlaytic if $\overline{\Phi}\in H^{\infty}_{M_{2}}$, respectively. If M=\left(\begin{array}[c]{cc}T_{1}&T_{2}\\ T_{3}&T_{4}\end{array}\right) where $T_{j}$ are bounded linear operators on a Hilbert space, then $det(M):=T_{1}T_{4}-T_{2}T_{3}$.

Theorem 3.6**.**

*For $\Phi\in L^{\infty}_{M_{2}},$ let $T_{\Phi}$ be a complex symmetric operator on $H^{2}_{{\mathbb{C}}^{2}}$ and let $CK_{\mu}$ be inner and $det(CK_{\mu})$ be nonzero where $C$ is the conjugation on $H^{2}_{{\mathbb{C}}^{2}}$. If $T_{\Phi}$ is analytic or coanalytic, then $\Phi$ is either identically zero on $\mathbb{D}$ or a nonzero constant function on ${\mathbb{D}}.$ *

Proof. Suppose $T_{\Phi}$ is analytic. For $\mu\in{\mathbb{D}}$, let $K_{\mu}=k_{\mu}I_{M_{2}}$, where $k_{\mu}=\frac{1}{1-\overline{\mu}z}$ is the reproducing kernel of $H^{2}$ at $\mu\in\mathbb{D}$. Observe for any $f\in H^{\infty}$ and $g\in H^{2}$,

[TABLE]

which implies that $T_{\overline{f}}k_{\mu}=\overline{f(\mu)}k_{\mu}$. Now it is easy to verify $T_{\Phi}^{\ast}K_{\mu}=\overline{\Phi(\mu)}K_{\mu}$. Observe

[TABLE]

Therefore, we find that ${\Phi(\mu)}-{\Phi}$ is a zero (matrix) function, that is, $\Phi$ actually is a constant matrix.

On the other hand, if $T_{\Phi}$ is coanalytic, then $\overline{\Phi}\in H^{\infty}_{M_{2}}$. Hence $\Phi$ must be a constant function or $\Phi=0$ by a similar argument.    $\Box$

We observe that if $T_{\Phi}$ is complex symmetric where $\Phi\in H^{\infty}_{M_{2}}$, then it must be normal from [20, Lemma 3.1]. We say that $\Phi$ is a nonconstant inner function on ${\mathbb{D}}$ which means that $\Phi^{\ast}\Phi=I$ and the entries of $\Phi$ are nonconstant functions.

Corollary 3.7**.**

Let $CK_{\mu}$ be inner and let $det(CK_{\mu})$ be nonzero where $C$ is the conjugation on $H^{2}_{{\mathbb{C}}^{2}}$ and $K_{\mu}=k_{\mu}I_{M_{2}}$. If $\Phi$ is a nonconstant inner function on ${\mathbb{D}},$ then $T_{\Phi}$ is not a complex symmetric operator.

Proof. The proof immediately follows from Theorem 3.6.    $\Box$

Next, we consider the relations among of the Fourier coefficients of $\varphi_{j}(j=1,2,3,4)$ for complex symmetric block Toeplitz operators $T_{\Phi}$.

Proposition 3.8**.**

If $T_{\Phi}$ is a complex symmetric operator with the conjugation {\mathcal{C}}=\frac{1}{\sqrt{2}}\left(\begin{array}[c]{cc}C_{1,\lambda}&C_{1,\lambda}\\ C_{1,\lambda}&-C_{1,\lambda}\end{array}\right), then the following identities hold.

[TABLE]

[TABLE]

[TABLE]

[TABLE]

for all $k$ where $|\lambda|=1.$

Proof. Suppose that $C$ is the conjugation on $H^{2}$. Let $h(z)=\sum_{k=1}^{\infty}\alpha_{k}z^{k}$ and $g(z)=\sum_{k=1}^{\infty}\beta_{k}z^{k},$ where

[TABLE]

Then $Ch(z)=\sum_{k=1}^{\infty}\widetilde{\alpha_{k}}z^{k}$ and $Cg(z)=\sum_{k=1}^{\infty}\widetilde{\beta_{k}}z^{k}$. If $C=C_{\mu,\lambda}$, then $\widetilde{a_{j}}=\mu\lambda^{j}\overline{a_{j}}$ with $|\mu|=|\lambda|=1$ and hence for any $\varphi$, we have

[TABLE]

Then

[TABLE]

and, similarly,

[TABLE]

Now, by using (15) and (18), we obtain that

[TABLE]

where $|\lambda|=1.$ Similarly, we get that

[TABLE]

[TABLE]

[TABLE]

for all $k$ where $|\lambda|=1.$ Since $T_{\Phi}$ is complex symmetric with the conjugation ${\mathcal{C}}$, we know that (a),(b),(c), and (d) always hold.    $\Box$

Corollary 3.9**.**

Let ${\Phi}=\begin{pmatrix}{\varphi}&{\psi}\cr-{\varphi}&{\psi}\end{pmatrix}$. If $T_{\Phi}$ is a complex symmetric operator with the conjugation {\mathcal{C}}=\frac{1}{\sqrt{2}}\left(\begin{array}[c]{cc}C_{1,\lambda}&C_{1,\lambda}\\ C_{1,\lambda}&-C_{1,\lambda}\end{array}\right), then

[TABLE]

[TABLE]

Proof. The proof follows from Proposition 3.8.    $\Box$

Corollary 3.10**.**

Let ${\Phi}=\begin{pmatrix}{\varphi}&{\varphi}\cr-{\varphi}&{\varphi}\end{pmatrix}$. If $T_{\Phi}$ is a complex symmetric operator with the conjugation {\mathcal{C}}=\frac{1}{\sqrt{2}}\left(\begin{array}[c]{cc}C_{1,\lambda}&C_{1,\lambda}\\ C_{1,\lambda}&-C_{1,\lambda}\end{array}\right), then

[TABLE]

Proof. The proof follows from Corollary 3.9.    $\Box$

4 Complex symmetry and normality of a block Toeplitz operator

Let ${\cal{L}}({\cal{H}})$ denote the algebra of all bounded linear operators on a separable complex Hilbert space ${\cal{H}}$. In this section, we study normal block Toeplitz operators as some examples of complex symmetric block Toeplit operators with some conjugation ${\mathcal{C}}$. For this, we need the following lemma.

Lemma 4.1**.**

([16, Theorem 2.1]) For $T=\begin{pmatrix}A&B\cr C&D\end{pmatrix}\in{\cal L}({\cal H}\oplus{\cal H})$, let

[TABLE]

*Then $T$ is normal if and only if $t_{j}=s_{j}$ for all $1\leq j\leq 3$. *

In general, even if $A,B,C,D$ are normal, then $T$ may not be normal. So we investigate the necessary and sufficient conditions so that the $2\times 2$ operator matrix $T$ to be normal.

Lemma 4.2**.**

*For $R,S\in{\mathcal{L}}(\mathcal{H}),$ put $[R,S]=RS-SR.$

(i) For $T=\begin{pmatrix}A&B\cr B&D\end{pmatrix}\in{\cal L}({\cal H}\oplus{\cal H})$, if $A$, $B$, and $D$ are normal, then $T$ is normal if and only if $A^{*}B+B^{*}D=AB^{*}+BD^{*}$ holds.

(ii) For $T=\begin{pmatrix}A&B\cr C&A\end{pmatrix}\in{\cal L}({\cal H}\oplus{\cal H})$, if $A$ is normal and $[A,B]=[A,C]=0$, then $T$ is normal if and only if $B^{*}B=CC^{\ast}$ and $BB^{*}=C^{\ast}C$ holds.

(iii) For $T=\begin{pmatrix}A&B\cr B^{\ast}&A\end{pmatrix}\in{\cal L}({\cal H}\oplus{\cal H})$, $T$ is normal if and only if $A$ and $D$ are normal and $AB=BD$ holds.

(iv) For $T=\begin{pmatrix}A&B\cr B&A\end{pmatrix}\in{\cal L}({\cal H}\oplus{\cal H})$, $T$ is normal if and only if $A$ and $B$ are normal and $[A,B]=0$ holds. *

Proof. (ii) Assume that $A$ is normal, $AB=BA$, and $AC=CA$. By the Fuglede-Putnam theorem, $A^{\ast}B=BA^{\ast}$ and $A^{*}C=CA^{\ast}$ (see [5]). Since

[TABLE]

and

[TABLE]

it follows that $t_{2}={s}_{2}$. Hence $T$ is normal if and only if $B^{*}B=CC^{\ast}$ and $BB^{*}=C^{\ast}C$ holds.

The statements (i), (iii), and (iv) hold by a similar way.    $\Box$

Proposition 4.3**.**

*Let $\varphi_{3}=\varphi_{2}$ and $\varphi_{j}=\alpha_{j}+\beta_{j}\rho_{j}$ where $\rho_{j}$ is a real-valued function in $L^{\infty}$ and $\alpha_{j},\beta_{j}\in{\mathbb{C}}$ for $j=1,2,4$. Assume that $\widehat{\Phi}(n)=\widehat{\Phi}(-n)$ for all $n\in{\mathbb{Z}}$ and ${\varphi_{1}}={2\varphi_{2}+\varphi_{4}}$. Then the following properties hold.

(i) If $T_{\varphi_{1}}$ and $T_{\varphi_{2}}$ commute, then $T_{\Phi}$ is normal complex symmetric with the conjugation ${\mathcal{C}}$(with $\lambda=1$).

(ii) If $D^{\ast}\not=-D$ where $D:=T_{\varphi_{1}}T_{\varphi_{2}}^{\ast}-T_{\varphi_{2}}^{\ast}T_{\varphi_{1}}$, then $T_{\Phi}$ is nonnormal complex symmetric with the conjugation ${\mathcal{C}}$(with $\lambda=1$).*

Proof. By Corollary 3.5, we know that $T_{\Phi}$ is complex symmetric with the conjugation ${\cal C}$ (with $\lambda=1$). By hypothesis, $T_{\varphi_{j}}$ are normal for $j=1,2,4$ from [3].

(i) Since $T_{\varphi_{1}}T_{\varphi_{2}}=T_{\varphi_{2}}T_{\varphi_{1}}$ and ${\varphi_{4}}=\varphi_{1}-{2\varphi_{2}}$, it follows from Fuglede-Putnam theorem that

[TABLE]

Hence $T_{\Phi}$ is normal from Lemma 4.2(i).

(ii) Let $D^{\ast}\not=-D$ where $D=T_{\varphi_{1}}T_{\varphi_{2}}^{\ast}-T_{\varphi_{2}}^{\ast}T_{\varphi_{1}}$. Since ${\varphi_{4}}=\varphi_{1}-{2\varphi_{2}}$ and $T_{\varphi_{2}}$ is normal, it follows that

[TABLE]

Hence $T_{\Phi}$ is not normal from Lemma 4.2(i).    $\Box$

Proposition 4.4**.**

*Let ${\varphi_{3}}=\overline{\varphi_{2}}$ and $\varphi_{4}=\varphi_{1}$ and $\varphi_{j}=\alpha_{j}+\beta_{j}\rho_{j}$ where $\rho_{j}$ is a real-valued function in $L^{\infty}$ and $\alpha_{j},\beta_{j}\in{\mathbb{C}}$ for $j=1,2$. Assume that $\widehat{\Phi}(n)=\widehat{\Phi}(-n)$ for all $n\in{\mathbb{Z}}$ and $Re(\varphi_{2})=0$. Then the following properties hold.

(i) If $T_{\varphi_{1}}$ and $T_{\varphi_{2}}$ commute, then $T_{\Phi}$ is normal complex symmetric with the conjugation ${\mathcal{C}}$ (with $\lambda=1$).

(ii) If $D^{\ast}\not=-D$ where $D:=T_{\varphi_{1}}T_{\varphi_{2}}^{\ast}-T_{\varphi_{2}}^{\ast}T_{\varphi_{1}}$, then $T_{\Phi}$ is nonnormal complex symmetric with the conjugation ${\mathcal{C}}$ (with $\lambda=1$).*

Proof. By Corollary 3.5, we know that $T_{\Phi}$ is complex symmetric with the conjugation ${\cal C}$ (with $\lambda=1$). By hypothesis, $T_{\varphi_{j}}$ are normal for $j=1,2$ from [3].

(i) Since $T_{\varphi_{1}}T_{\varphi_{2}}=T_{\varphi_{2}}T_{\varphi_{1}}$, it follows from Fuglede-Putnam theorem that

[TABLE]

Therefore $T_{\Phi}$ is normal from Lemma 4.2(i).

(ii) If $D^{\ast}\not=-D$ where $D=T_{\varphi_{1}}T_{\varphi_{2}}^{\ast}-T_{\varphi_{2}}^{\ast}T_{\varphi_{1}}$, then

[TABLE]

Hence $T_{\Phi}$ is not normal from Lemma 4.2(i).    $\Box$

Lemma 4.5**.**

([7], 2011, Gu-Hendricks-Rutherford) Let $\Phi=\Phi_{+}+\Phi_{0}+\Phi_{-}^{\ast}\in L^{\infty}_{M_{2}}$. Assume that $det(\Phi_{+})$ is not identically zero. Then $T_{\Phi}$ is normal if and only if $\Phi^{\ast}\Phi=\Phi\Phi^{\ast}$ and $\Phi_{+}=\Phi_{-}U$ for some constant unitary matrix $U$.

Proposition 4.6**.**

Assume that $T_{\varphi_{j}}(j=1,2,3,4)$ are complex symmetric with the conjugation $C_{1,1}$ where ${\varphi_{1}}={\varphi_{2}+\varphi_{3}+\varphi_{4}}$. If

[TABLE]

*and ${\Phi_{-}}^{\ast}(z)=\Phi_{+}(\overline{z})=U\Phi_{+}^{\ast}(z)$ for some constant unitary matrix $U$, then $T_{\Phi}$ is normal complex symmetric with the conjugation ${\mathcal{C}}$ (with $\lambda=1$). *

Proof. Since note that $\Phi^{\ast}\Phi=\Phi\Phi^{\ast}$ if and only if

[TABLE]

the proof follows from Corollary 3.5 and Lemma 4.5.   $\Box$