Complex symmetry of Toeplitz operators and composition operators
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Denote by $H_{\gamma}(\mathbb{D})$ the Hilbert space of holomorphic functions over the open unit disk $\mathbb{D}$ with the reproducing kernel $K_{\omega}^{(\gamma)}(z)=\frac{1}{(1-\overline{\omega}z)^{\gamma}}$, $\gamma>0$. Let $J_{\alpha}: J_{\alpha}f(z)=\overline{f(\overline{\alpha z})}$ for $z\in \mathbb{D}$ and $\alpha \in \partial \mathbb{D}$, and let $W_{u,v}$ be the weighted composition operator. In this paper, we prove that if $b\in \mathbb{R}$ and $b\neq \frac{k\pi}{2}$, for any integer $k$, then $(\cos b+i\sin bW_{u,v})J_{\alpha}$ is a conjugation if and only if $v(z)=\frac{c-z}{1-\overline{c}z}$, $u(z)=\pm k_{c}^{(\gamma)}(z)$, $c\in\mathbb{D}$ and $\overline{c}=c\alpha$, where $k_c^{(\gamma)}(z)$ is the normalized reproducing kernel, or $(\cos b+i\sin bW_{u,v})J_{\alpha}=\lambda J_{\alpha}$ for some $\lambda\in \mathbb{C}$ with $|\lambda|=1$. We then derive a necessary and sufficient condition for a Toeplitz operator to be complex symmetric with respect to the conjugation $(\cos b+i\sin bW_{u,v})J_{\alpha}$ on $H_{\gamma}(\mathbb{D})$ with $\gamma \geq 1$. Similarly, we also conduct a similar study for composition operators on $H_{\gamma}(\mathbb{D})$ with $\gamma >0$.
Abstract
Denote by $H_{\gamma}(\mathbb{D})$ the Hilbert space of holomorphic functions over the open unit disk $\mathbb{D}$ with the reproducing kernel $K_{\omega}^{(\gamma)}(z)=\frac{1}{(1-\overline{\omega}z)^{\gamma}}$, $\gamma>0$. Let $J_{\alpha}: J_{\alpha}f(z)=\overline{f(\overline{\alpha z})}$ for $z\in \mathbb{D}$ and $\alpha \in \partial \mathbb{D}$, and let $W_{u,v}$ be the weighted composition operator. In this paper, we prove that if $b\in \mathbb{R}$ and $b\neq \frac{k\pi}{2}$, for any integer $k$, then $(\cos b+i\sin bW_{u,v})J_{\alpha}$ is a conjugation if and only if $v(z)=\frac{c-z}{1-\overline{c}z}$, $u(z)=\pm k_{c}^{(\gamma)}(z)$, $c\in\mathbb{D}$ and $\overline{c}=c\alpha$, where $k_c^{(\gamma)}(z)$ is the normalized reproducing kernel, or $(\cos b+i\sin bW_{u,v})J_{\alpha}=\lambda J_{\alpha}$ for some $\lambda\in \mathbb{C}$ with $|\lambda|=1$. We then derive a necessary and sufficient condition for a Toeplitz operator to be complex symmetric with respect to the conjugation $(\cos b+i\sin bW_{u,v})J_{\alpha}$ on $H_{\gamma}(\mathbb{D})$ with $\gamma \geq 1$. Similarly, we also conduct a similar study for composition operators on $H_{\gamma}(\mathbb{D})$ with $\gamma >0$.
참고문헌 (14)
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