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

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.