A class of linear operators on Bergman spaces

TL;DR

This paper investigates the boundedness and compactness of a specific linear operator on Bergman spaces, providing new necessary and sufficient conditions that improve upon previous results.

Contribution

It introduces weaker assumptions for the compactness criteria of the operator on Bergman spaces, advancing the theoretical understanding of these operators.

Findings

Established a necessary and sufficient condition for boundedness.

Derived a new criterion for compactness of the operator.

Weakened the assumptions compared to earlier studies.

Abstract

We study the boundedness of the linear operator $S$ on $L^{p}_{a}(dA_{\alpha})$ $(0<p<\infty)$. In particular, we obtain a sufficient and necessary condition for the compactness of the linear operator $S$ on $L^{p}_{a}(dA_{\alpha})$ $(1<p<\infty)$. Our results weaken the assumptions of earlier results of J. Miao and D. Zheng in a certain sense.