Toeplitz operators on Bergman spaces with exponential weights

TL;DR

This paper characterizes bounded and compact Toeplitz operators on weighted Bergman spaces with exponential weights, including criteria for Schatten class membership, advancing understanding of operator theory in complex analysis.

Contribution

It provides new characterizations of measures for boundedness and compactness of embeddings and Toeplitz operators on weighted Bergman spaces with exponential weights.

Findings

Characterization of measures for bounded/compact embeddings

Criteria for boundedness and compactness of Toeplitz operators

Schatten class characterization of Toeplitz operators

Abstract

In this paper, we focus on the weighted Bergman spaces $A_{\varphi}^{p}$ in $\mathbb{D}$ with $\varphi\in\mathcal{W}_{0}$. We first give characterizations of those finite positive Borel measures $\mu$ in $\mathbb{D}$ such that the embedding $A_{\varphi}^{p}\subset L_{\mu}^{q}$ is bounded or compact for $0<p,q<\infty$. Then we describe bounded or compact Toeplitz operators $T_{\mu}$ from one Bergman space $A_{\varphi}^{p}$ to another $A_{\varphi}^{q}$ for all possible $0<p,q<\infty$. Finally, we characterize Schatten class Toeplitz operators on $A_{\varphi}^{2}$.