In this paper we provide necessary and sufficient conditions for the boundedness and compactness of weighted composition operators $W_{\psi,\varphi}$ between vector-valued Bloch-type spaces $\mathcal{B}_{\alpha}\left( X\right)$ and $\mathcal{B}_{\beta}\left( Y\right)$ for $\alpha, \beta >0$ in terms of $\psi,\varphi$, their derivatives, and the $n$th power $\varphi^n$ of $\varphi$.
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Objective & Hypothesis:Öz Let $X$ and $Y$ be complex Banach spaces and $\mathbb{D}$ be the open unit disc in the complex plane $\mathbb{C}$.
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Key Findings & Data:Let $\varphi$ be an analytic self-map of $\mathbb{D}$ and $\psi$ be an analytic operator-valued function from $\mathbb{D}$ into the space of all bounded linear operators from $X$ to $Y.$ The weighted composition operator $W_{\psi,\varphi}:\mathcal{H} \rightarrow\mathcal{H}(\mathbb{D}, Y)$ is defined by $$ W_{\psi,\varphi}( f)(z)=\psi(z)(f(\varphi(z))), \quad \quad (z\in\mathbb{D}, f\in \mathcal{H}),$$ where $\mathcal{H}$ is the space of all analytic $X$-valued functions on $\mathbb{D}$.
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Conclusion & Impact:In this paper we provide necessary and sufficient conditions for the boundedness and compactness of weighted composition operators $W_{\psi,\varphi}$ between vector-valued Bloch-type spaces $\mathcal{B}_{\alpha}\left( X\right)$ and $\mathcal{B}_{\beta}\left( Y\right)$ for $\alpha, \beta >0$ in terms of $\psi,\varphi$, their derivatives, and the $n$th power $\varphi^n$ of $\varphi$.
Öz Let $X$ and $Y$ be complex Banach spaces and $\mathbb{D}$ be the open unit disc in the complex plane $\mathbb{C}$. Let $\varphi$ be an analytic self-map of $\mathbb{D}$ and $\psi$ be an analytic operator-valued function from $\mathbb{D}$ into the space of all bounded linear operators from $X$ to $Y.$ The weighted composition operator $W_{\psi,\varphi}:\mathcal{H} \rightarrow\mathcal{H}(\mathbb{D}, Y)$ is defined by $$ W_{\psi,\varphi}( f)(z)=\psi(z)(f(\varphi(z))), \quad \quad (z\in\mathbb{D}, f\in \mathcal{H}),$$ where $\mathcal{H}$ is the space of all analytic $X$-valued functions on $\mathbb{D}$. In this paper we provide necessary and sufficient conditions for the boundedness and compactness of weighted composition operators $W_{\psi,\varphi}$ between vector-valued Bloch-type spaces $\mathcal{B}_{\alpha}\left( X\right)$ and $\mathcal{B}_{\beta}\left( Y\right)$ for $\alpha, \beta >0$ in terms of $\psi,\varphi$, their derivatives, and the $n$th power $\varphi^n$ of $\varphi$.