On operators which commute with analytic Toeplitz operators modulo the finite rank operators

Abstract

It is shown that an operator S S on the Hardy space H 2 ( D n ) H^2({\mathbb {D}}^n) (or H 2 ( B n ) H^2({\mathbb {B}}_n) ) commutes with all analytic Toeplitz operators modulo the finite rank operators if and only if S = T g + F S=T_g+F . Here F F is a finite rank operator, and in the case n = 1 n=1 , g g is a sum of a rational function and a bounded analytic function, and in the case n ≥ 2 n\geq 2 , g g is a bounded analytic function.