On operators commuting with Toeplitz operators modulo the finite rank operators

Abstract

Let X be a bounded linear operator on the Hardy space H 2 of the unit disk. We show that if X−T θ ∗XT θ is of finite rank for every inner function θ, then X= T ϕ + F for some Toeplitz operator T ϕ and some finite rank operator F on H 2. This solves a variant of an open question where the compactness replaces the finite rank conditions.