On Schatten-class perturbations of Toeplitz operators

Let D denote the unit ball or the unit polydisc in Cn with n≥2. For 1≤p≤2n in the case of the ball and 1≤p<∞ for the polydisc, we show that a bounded operator S on the Hardy space H2(D) commutes with all analytic Toeplitz operators modulo the Schatten class Sp if and only if S=X+K with an analytic Toeplitz operator X and an operator K∈Sp. This partially answers a question of Guo and Wang [14]. For 1≤p<∞ and a strictly pseudoconvex or bounded symmetric and circled domain D⊂Cn, we show that a given operator S on H2(D) is a Schatten-p-class perturbation of a Toeplitz operator if and only if Tθ⁎STθ−S∈Sp for every inner function θ on D.