Notes on Certain Star-Shift Invariant Subspaces

Abstract

Our work addresses the question: for which (infinite) Blaschke products B does the star-shift invariant subspace K b:= H 2(D) Θ BH 2(D) contain a (non-trivial) function with a non-trivial singular inner factor? In the case that the zeros of B have only finitely many accumulation points w 1,w 2, …, w n in T, a recent paper shows that, for an affirmative answer, there necessarily exist k, 1 ≤ k ≤ n, and a subsequence of the zeros of B that converges tangentially to w k on “both sides” of w k. One of the results in this article improves upon this theorem. And, currently, the only examples of Blaschke products in the literature that are shown to yield an affirmative answer are those that have a proper factor b that satisfies b(D) ≠ D. We produce many examples here that have no such factor.