Generalized Weighted Composition Operators on Vector-Valued Weighted Bergman Space

TL;DR

This paper characterizes the conditions under which composition operators on vector-valued weighted Bergman spaces are isometric, unitary, Hermitian, or normal, and explores boundedness of generalized weighted composition operators.

Contribution

It provides necessary and sufficient conditions for various operator properties on vector-valued weighted Bergman spaces, extending the understanding of composition operators in this context.

Findings

Norm of $C_{\

$C_{\

Boundedness conditions for generalized weighted composition operators.

Abstract

In this research article the necessary and sufficient conditions for the norm of composition operator $C_{\Phi}$ on $\mathcal{A}_{\alpha}^2(H)$ to be one are obtained. Moreover, $C_{\Phi}$ is unitary on $\mathcal{A}_{\alpha}^2(H)$ if and only if it is co-isometry. The necessary and sufficient condition for Hermitian and normal composition operators on $\mathcal{A}_{\alpha}^2(H)$ are also explored. Also, the characterization for boundeness of generalized weighted composition operator is obtained under some condition on $\Phi.$