Boundedness and Compactness of products of Toeplitz operators on the Bergman Space

TL;DR

This paper proves Sarason's conjecture by establishing necessary and sufficient conditions for the boundedness and compactness of products of Toeplitz operators on the Bergman space, resolving a long-standing open problem.

Contribution

It provides a complete characterization of when products of Toeplitz operators are bounded or compact on the Bergman space, confirming Sarason's conjecture.

Findings

Necessary conditions for boundedness and compactness are also sufficient.

Resolved Sarason's conjecture on Toeplitz operator products.

Established a full criterion for operator boundedness and compactness.

Abstract

In a celebrated conjecture D.Sarason stated a necessary and sufficient condition on the symbols f, g in the Bergman space, L^2_a(\Delta) of the unit disk, \Delta, for the products T_{f}T_{\bar g} of associated Toeplitz operators to be bounded (respectively compact) on L^2_a(\Delta) . K. Stroethoff and D. Zheng proved that these conditions are necessary. We prove the sufficiency of these conditions, thus solvind Sarason's conjecture.