Boundedness and Compactness of Toeplitz operators with L^1 symbols on the Bergman space

TL;DR

This paper characterizes when Toeplitz operators with L^1 symbols are bounded or compact on the Bergman space, linking compactness to boundary behavior of the Berezin transform, extending previous results for special symbol classes.

Contribution

It provides a complete characterization of boundedness and compactness for Toeplitz operators with L^1 symbols on the Bergman space, generalizing earlier known cases.

Findings

Boundedness characterized by L^1 symbol properties.

Compactness determined by boundary behavior of the Berezin transform.

Extends results to general L^1 symbols beyond positive or bounded functions.

Abstract

We characterise the boundedness of a Toeplitz operator on the Bergman space with an L^1 symbol.We also prove that the compactness of a Toeplitz operator on the Bergman space with an L^1 symbol is completely determined by the boundary behaviour of itss Berezin transform. This result extends known results in the cases when the symbol is either a positive L^1 function, an L^\infty function or a general BMO^1 function.