Composition Operators on Weighted Bergman and S^P Spaces

Abstract

Let $varphi$ be an analytic self-map of open unit disk $mathbb{D}$. The operator given by $(C_{varphi}f)(z)=f(varphi(z))$, for $z in mathbb{D}$ and  $f$ analytic on $mathbb{D}$ is called composition operator. For each $pgeq 1$, let $S^p$ be the space of analytic functions on $mathbb{D}$ whose derivatives belong to the Hardy space $H^p$.  For $alpha > -1$ and $p > 0$ the weighted Bergman space $A^{p}_{alpha}$ consists of all analytic functions in $L^{p}(mathbb{D}, dA_{alpha})$, where $dA_{alpha}$ is the normalized weighted area measure. In this presentation, we characterize boundedness and compactness of composition operators act between weighted Bergman $A_{alpha}^{p}$ and $S^q$ spaces, $1leq p,
q<infty$. Moreover, we give a lower bound for the essential norm of composition operator from  $A_{alpha}^{p}$ into $S^q$ spaces, $1leq pleq q$.