Toeplitz Projections and Corona with Localization and Toeplitz Algebra on the Bergman Space

Abstract

We study the operator commuting and essential commutant of analytic Toeplitz operators module the compact operators and Toeplitz operators in several complex variables and on the Bergman space of the until ball. We show the ordered groups and some exact sequences and the commutator ideal of the Toeplitz algebras of spherical isometries and on the Bergman spaces of the unit ball in the unitary space. We give the lower bounds in the matrix, new estimate for the vector-valued and matrix-valued H^p, Corona problems in the disk and polydisk and the codimension one conjecture. We also give the Toeplitz Corona theorems for the polydisk, the unit ball and the Douglas property for free functions. We discuss the characterizations of Toeplitz and Hankel operators with Toeplitz projections and Dixmier traces on the unit ball of the unitary space. We determine the locatization, compactness and Toeplitz algebra on the Bergman and Fock spaces.