Duality theorems for coinvariant subspaces of H1H1

Let θ   be an inner function satisfying the connected level set condition of B. Cohn, and let Kθ1 be the shift-coinvariant subspace of the Hardy space H1H1 generated by θ  . We describe the dual space to Kθ1 in terms of a bounded mean oscillation with respect to the Clark measure σασα of θ  . Namely, we prove that (Kθ1∩zH1)⁎=BMO(σα). The result yields a two-sided estimate for the operator norm of a finite Hankel matrix of size n×nn×n via BMO(μ2n)BMO(μ2n)-norm of its standard symbol, where μ2nμ2n is the Haar measure on the group {ξ∈C:ξ2n=1}{ξ∈C:ξ2n=1}.