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

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}^{۲} given by C_{\psi,\varphi}f=\psi f\circ\varphi for any f\in A_{\alpha}^{۲}. If \varphi_۱,\varphi_۲,...,\varphi_n are linear fractional non-automorphisms and \psi_{۱},\psi_{۲},...,\psi_{n} are bounded analytic functions, under some conditions, we obtain a subset of the numerical range of C_{\psi_۱,\varphi_۱}+C_{\psi_۲,\varphi_۲}+...+C_{\psi_n,\varphi_n} on A_{\alpha}^{۲} and determine when zero lies in the interior of its numerical range.

کلیدواژه ها:

weighted Bergman spaces ، weighted composition operator ، numerical range