Numerical range of a sum of weighted composition operators on the weighted Bergman spaces

Document Type : Research Articles

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Department of Mathematics, Shiraz University of Technology, Shiraz, Iran.

Abstract

Suppose that $\varphi$ is an analytic self-map of the open unit disk $\mathbb{D}$ and $\psi$ is a bounded analytic function on $\mathbb{D}$. The weighted composition operator $C_{\psi,\varphi}$ is the operator on the weighted Bergman spaces $A_{\alpha}^{2}$ given by $C_{\psi,\varphi}f=\psi f\circ\varphi$ for any $f\in A_{\alpha}^{2}$. If $\varphi_1,\varphi_2,...,\varphi_n$ are linear fractional non-automorphisms and $\psi_{1},\psi_{2},...,\psi_{n} $ are bounded analytic functions, under some conditions, we obtain a subset of the numerical range of $C_{\psi_1,\varphi_1}+C_{\psi_2,\varphi_2}+...+C_{\psi_n,\varphi_n}$ on $A_{\alpha}^{2}$ and determine when zero lies in the interior of its numerical range.

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