Commutants and Abelian Toeplitz Algebra and Operators with Sarason Toeplitz Product on Bergman and Fock Spaces

Abstract

We study by the function theory on Cartan domains, the Berezin-Toeplitz symbol calculus, the essential commutant of analytic Toeplitz operators and algebras associated with spherical isometries and on the Bergman space. The classification of reducing subspaces of a class of multiplication operators on the Bergman space, the Hardy space of the bidisk with the Totally Abelian Toeplitz operators and geometric invariants associated with their symbol curves are considered. The Hankel operators on Fock spaces and related Bergman kernel estimates and asymptotics for generalized Fock spaces are introduced . We obtain the localization, compactness and Sarason’s Toeplitz product problem in Bergman and Fock spaces .