Sarason Toeplitz Product Problem on Fock Spaces and Asymptotic Toeplitz Operators on Harmonically Weighted Bergman Spaces

Abstract

We give the products and algebraic properties of Hankel and Toeplitz operators on the Bergman space and on the polydisk with Hankel operators on weighted Fock spaces and related to Bergman kernel estimates, We characterize the commuting Toeplitz operators on the polydisk and operators which commute with analytic Toeplitz operators modulo the finite rank operators. Sarason, Schatten-calss perturbations and asymptotic behavior of eigenvalues of Toeplitz products problem and operators for a class of Fock spaces, on Hardy space and weighted analytic spaces are characterized. The approximation numbers and contact points of composition operators acting on weighted Dirichlet spaces and in several variable with Toeplitz operators on harmonically weighted Bergman spaces are discussed.