Essentially Commuting Hankel and Toeplitz Operators

We characterize when a Hankel operator and a Toeplitz operator have a compact commutator. Let dσ(w) be the normalized Lebesgue measure on the unit circle ∂D. The Hardy space H is the subspace of L(∂D, dσ), denoted by L, which is spanned by the space of analytic polynomials. So there is an orthogonal projection P from L onto the Hardy space H, the so-called Hardy projection. Let f be in L∞. The Toeplitz operator Tf and the Hankel operator Hf with symbol f are defined by Tfh = P (fh), and Hfh = P (Ufh), for h in H. Here U is the unitary operator on L defined by Uh(w) = w̄h̃(w). Clearly, H∗ f = Hf∗ , where f∗(w) = f(w̄). U is a unitary operator which maps H onto [H2]⊥ and has the following useful property: