Weak compactness in certain star-shift invariant subspaces

The context of much of the work in this paper is that of a backward-shift invariant subspace of the form , where B is some infinite Blaschke product. We address (but do not fully answer) the question: For which B can one find a (convergent) sequence in K_B such that the sequence of real measures converges weak-star to some nontrivial singular measure on ? We show that, in order for this to hold, K_B must contain functions with nontrivial singular inner factors. And in a rather special setting, we show that this is also sufficient. Much of the paper is devoted to finding conditions (on B) that guarantee that K_B has no functions with nontrivial singular inner factors. Our primary result in this direction is based on the “geometry” of the zero set of B.