On Schatten-class perturbations of Toeplitz operators

摘要

Let D denote the unit ball or the unit polydisc in C-n with n >= 2. For 1 <= p <= 2n in the case of the ball and 1 <= p < infinity for the polydisc, we show that a bounded operator S on the Hardy space H-2(D) commutes with all analytic Toeplitz operators modulo the Schatten class S-p if and only if S = X + K with an analytic Toeplitz operator X and an operator K is an element of Sp. This partially answers a question of Guo and Wang [14]. For 1 <= p < infinity and a strictly pseudoconvex or bounded symmetric and circled domain D subset of C-n we show that a given operator S on H-2(D) is a Schatten-p-class perturbation of a Toeplitz operator if and only if T-theta*ST theta - S is an element of S-p for every inner function 9 on D. (C) 2016 Elsevier Inc. All rights reserved.