Interpolation and duality in spaces of pseudocontinuable functions

Given an inner function theta on the unit disk, let K-theta(p) := H-p boolean AND theta(z) over bar(Hp) over bar be the associated starinvariant subspace of the Hardy space H-p. Also, we put K-*theta := K(theta)2 boolean AND BMO. Assuming that B = B-Z is an interpolating Blaschke product with zeros Z = {z(j)}, we characterize, for a number of smoothness classes X, the sequences of values W = {w(j)} such that the interpolation problem f vertical bar(Z) =W has a solution f in K-B(2) boolean AND X. Turning to the case of a general inner function theta, we further establish a non-duality relation between K-theta(1) and K-*theta. Namely, we prove that the latter space is properly contained in the dual of the former, unless theta is a finite Blaschke product. From this we derive an amusing non-interpolation result for functions in K-*B, with B = B-Z as above.