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 KB≔H2(D)⊖BH2(D), 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 { fn}n=1∞ in KB such that the sequence of real measures {log | fn|dθ}n=1∞ converges weak-star to some nontrivial singular measure on ∂D? We show that, in order for this to hold, KB 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 KB 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.