On operators commuting with Toeplitz operators modulo the finite rank operators

Let X be a bounded linear operator on the Hardy space H2 of the unit disk. We show that if X−Tθ∗XTθ is of finite rank for every inner function θ, then X=Tϕ+F for some Toeplitz operator Tϕ and some finite rank operator F on H2. This solves a variant of an open question where the compactness replaces the finite rank conditions.