Reducing Subspaces and Sarason Toeplitz Product Problem with Theorem of Brown–Halmos on Fock and Bergman Spaces

Abstract

The classification of reducing subspaces of a class of multiplication operators, analytic multipliers for a class of Toeplitz operators, tensor products of weighted shifts and a class of non-analytic Toeplitz operators on the Bergman space by the Hardy space of the Bidisk and polydisk are considered. We show the products products of Hankel and Toeplitz operators with Sarason’s Toeplitz product problem on the Bergman and a class of Fock spaces. We study the finite rank commentators, perturbation and semicommutators of Toeplitz operators with harmonic symbols and Bergman space. We give a theorem of Brown-Halmos type for Bergman space of Toeplitz operators and modulo finite rank operators.