The generalized Volterra integral operator and Toeplitz operator on weighted Bergman spaces

TL;DR

This paper investigates the boundedness, compactness, and Schatten class membership of generalized Volterra and Toeplitz operators on weighted Bergman and Hardy spaces, providing new characterizations and extending operator theory in complex analysis.

Contribution

It introduces a generalized Toeplitz operator and characterizes Schatten class membership for these operators on various function spaces, extending previous results in operator theory.

Findings

Boundedness and compactness criteria established for generalized Volterra operators.

Schatten class membership characterized for both generalized Volterra and Toeplitz operators.

Applications include Schatten class characterizations on Hardy spaces.

Abstract

We study the boundedness and compactness of the generalized Volterra integral operator on weighted Bergman spaces with doubling weights on the unit disk. A generalized Toeplitz operator is defined and the boundedness, compactness and Schatten class of this operator are investigated on the Hilbert weighted Bergman space. As an application, Schatten class membership of generalized Volterra integral operators are also characterized. Finally, we also get the characterizations of Schatten class membership of generalized Toeplitz operator and generalized Volterra integral operators on the Hardy space $H^2$.