Positive Toeplitz operators on Large Bergman spaces in the unit disk

TL;DR

This paper characterizes the boundedness, compactness, and Schatten class membership of positive Toeplitz operators on weighted Bergman spaces with fast decreasing weights, using Berezin transforms and averaging functions.

Contribution

It provides new characterizations for Toeplitz operators on weighted Bergman spaces with fast decreasing weights, including Schatten class criteria.

Findings

Characterization of boundedness and compactness via Berezin transforms

Conditions for Toeplitz operators to belong to Schatten classes

Analysis of Toeplitz operators on weighted Bergman spaces with fast decreasing weights

Abstract

We study positive Toeplitz operators on the Bergman spaces having the fast decreasing weight functions in a certain class. We give the characterizations for the boundedness and compactness of Toeplitz operators in terms of their Berezin transforms and averaging functions. In particular, we show the equivalent conditions in order that Toeplitz operators on $A^2(\omega)$ belong to the Schatten class $S_p$, $0<p<\infty$.