BOUNDED, COMPACT AND SCHATTEN CLASS WEIGHTED COMPOSITION OPERATORS BETWEEN WEIGHTED BERGMAN SPACES

Abstract

An analytic self-map <TEX>${\phi}$</TEX> of the open unit disk <TEX>$\mathbb{D}$</TEX> in the complex plane and an analytic map <TEX>${\psi}$</TEX> on <TEX>$\mathbb{D}$</TEX> induce the so-called weighted composition operator <TEX>$C_{{\phi},{\psi}}$</TEX>: <TEX>$H(\mathbb{D})\;{\rightarrow}\;H(\mathbb{D})$</TEX>, <TEX>$f{\mapsto} \;{\psi}\;(f\;o\;{\phi})$</TEX>, where H(<TEX>$\mathbb{D}$</TEX>) denotes the set of all analytic functions on <TEX>$\mathbb{D}$</TEX>. We study when such an operator acting between different weighted Bergman spaces is bounded, compact and Schatten class.