Asymptotic Toeplitz Operators and Coburn Type Theorem with Brown–Halmos Theorem on Hardy Spaces

Abstract

We study the Toeplitz and asymptotic Toeplitz operators on Hardy space of the multidisk. The commuting and products of Toeplitz operators on the polydisk are determined. The Coburn-Simonenko theorem for Toeplitz operators acting between Hardy type subspaces of different Banach function spaces with Toeplitness of composition operators in several variables and Toeplitz projections with essential commutants are explained. The operator theory, the Berger-Shaw theorem and a Coburn type theorem in the Hardy space and module over the bidisk are given. The pointwise multipliers of Orlicz function spaces and factorization are introduced. The Toeplitz operators with the density of analytic polynomials and Brown-Holmos theorem for a pair of abstract Hardy spaces are dealt with.