Composition and Volterra type operators on large Bergman spaces with rapidly decreasing weights

TL;DR

This paper characterizes the boundedness, compactness, and Schatten class properties of generalized Volterra-type operators on large Bergman spaces with rapidly decreasing weights, extending previous work on integration operators.

Contribution

It provides new characterizations of operator properties on large Bergman spaces using Berezin transforms and embedding theorems, generalizing earlier results.

Findings

Characterization of boundedness and compactness of Volterra operators

Criteria for Schatten class membership on large Bergman spaces

Extension of previous integration operator results

Abstract

We characterize boundedness, compactness and Schatten class properties of generalized Volterra-type integral operators acting between large Bergman spaces $A_\omega^p$ and $A_\omega^q$ for $0 <p, q\leq \infty$. To prove our characterizations, which involve Berezin-type integral transforms, we use the Littlewood-Paley formula of Constantin and Pel\'aez and corresponding embedding theorems. Our results generalize the work on integration operators of Pau and Pel\'{a}ez in J. Funct. Anal. 259 (2010), 2727--2756.