John R. Akeroyd

Weak compactness in certain star-shift invariant subspaces

Abstract The context of much of the work in this paper is that of a backward-shift invariant subspace of the form K B ≔H 2 ( D )⊖BH 2 ( 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 { f n } n=1 ∞ in KB such that the sequence of real measures { log | f n |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.

A Note on Cyclic Vectors for the Shift

We strengthen a result in the literature that gives a necessary condition for the shift on the Hardy space H 2 ( G ) of some bounded, simply connected region G to be cyclic. Part of this involves a characterization, in terms of ‘ G , of what it means for G to be the image of under a conformal map whose boundary values are univalent a.e. m (normalized Lebesgue measure on ).

A note concerning the index of the shift

Let μ be a finite, positive Borel measure with support in {z: |z| 0 and there is a nontrivial subarc γ of ∂D such that log()dm > -∞, then dim(M ⊖ zM) = 1 for each nontrivial closed invariant subspace M for the shift M z on P 2 (μ).

Harmonic measures on complementary subregions of the disk

The author describes those curves Γ in D := {z : |z| < 1} that have the property that D|Γ is the disjoint union of two Dirichlet regions A and B and there exist points a in A and b in B such that ω(.,A a)|Γ ≡const ω (.,B b)|Γ ω(;A a) denotes harmonic measure on ∂A evaluated at a, and similarly for ω(.B b).

A Note on Harmonic Measure

Let Ω be a subregion of {z : |z| < 1} for which the Dirichlet problem is solvable, assume that 0 ∈ Ω and let ωΩ denote harmonic measure on ∂Ω for evaluation at 0. If E is a Borel subset of {z : |z| = 1} and ωΩ (E) > 0, then we find a simply connected region G, where 0 ∈ G ⊆ {z : |z| < 1}, ∂G ⊆ Ω ∪ E and ωG(E) > 0, such that U := G ∪ Ω has the property that wU and wΩ are boundedly equivalent on ∂U. We mention consequences of this in function theory.

A brief note on compactness of composition operators on

ABSTRACT Let be a univalent (analytic) mapping from the unit disk into itself. Under a certain regularity condition on the bundary of , we show that the composition operator is compact on the Hardy space if and only if

Notes on Certain Star-Shift Invariant Subspaces

Our work addresses the question: for which (infinite) Blaschke products B does the star-shift invariant subspace Kb:= H2(D) Θ BH2(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 w1,w2, …, wn 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 wk on “both sides” of wk. 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.