Weighted composition operators on weighted Bergman spaces induced by double weights

Abstract

In this paper, we investigate the boundedness, compactness, essential norm and the Schatten class of weighted composition operators $u{C}_{\phi }$ on Bergman type spaces $A_{\omega }^p$ with double weight $\omega$. Let $X=\{u\in H(D): uC_φ:A_ω^p\to A_ω^p \mbox{ is bounded}\}.$ For some regular weights $\omega$, we obtain that $X=H^{\infty }$ if and only if $\phi$ is a finite Blaschke product.