Spectra of Weighted Composition Operators on Analytic Function Spaces

Abstract

Let E be a complex Banach space with open unit ball $$B_E.$$ For analytic self-maps $$\varphi $$ of $$B_E$$ with $$\varphi (0) =0,$$ we investigate the spectra of weighted composition operators $$uC_\varphi $$ acting on a large class of spaces of analytic functions. This class contains, for example, weighted Banach spaces of $$H^\infty $$-type on $$B_E$$, weighted Bergman spaces $$A^p_\alpha ({\mathbb {B}}_N)$$ and Hardy spaces $$H^p({\mathbb {B}}_N).$$ We present a general approach for deducing new information about the spectrum and for estimating the essential spectral radius of $$uC_\varphi .$$