Hilbert-Schmidt composition-differentiation operators on the unit ball

TL;DR

This paper investigates the conditions under which composition-differentiation operators, defined via radial derivatives on Hardy and Bergman spaces of the unit ball and polydisk, are Hilbert-Schmidt, expanding understanding of operator theory in complex analysis.

Contribution

It introduces a new class of composition-differentiation operators using radial derivatives and characterizes when these operators are Hilbert-Schmidt.

Findings

Derived necessary and sufficient conditions for Hilbert-Schmidt property.

Extended operator analysis to Hardy and Bergman spaces.

Provided new insights into composition-differentiation operators.

Abstract

We use the notion of radial derivative to introduce composition-differentiation operators on the Hardy and Bergman spaces of the unit ball and the polydisk. We seek for necessary and sufficient conditions on the inducing functions to ensure that the composition-differentiation operator is Hilbert-Schmidt.