Compactification and decompactification by weights on Bergman spaces

TL;DR

This paper characterizes symbols for which weighted composition operators on Bergman spaces are compact, bounded, or Hilbert-Schmidt, based on the existence of suitable weights, advancing understanding of operator behavior in complex analysis.

Contribution

It provides new characterizations of weighted composition operators on Bergman spaces, focusing on compactness, boundedness, and Hilbert-Schmidt properties, through the existence of appropriate weights.

Findings

Characterization of symbols for compact weighted composition operators.

Conditions for bounded but non-compact operators.

Criteria for Hilbert-Schmidt operators on Bergman spaces.

Abstract

We characterize the symbols $\Phi$ for which there exists a weight w such that the weighted composition operator M w C $\Phi$ is compact on the weighted Bergman space B 2 $\alpha$. We also characterize the symbols for which there exists a weight w such that M w C $\Phi$ is bounded but not compact. We also investigate when there exists w such that M w C $\Phi$ is Hilbert-Schmidt on B 2 $\alpha$.