Computational Methods and Function Theory 4 (2004), No. 2, 461--474
Copyright Heldermann Verlag 2004
John R. Akeroyd
jakeroyd@uark.edu , Department of Mathematics, University of Arkansas, Fayetteville, AR 72701, U.S.A.
Kristi Karber
kkarber@uark.edu , Department of Mathematics, University of Arkansas, Fayetteville, AR 72701, U.S.A.
Our work addresses the question: for which (infinite) Blaschke products $B$ does the star-shift invariant subspace $K_B := H^2(\mathbb{D})\ominus BH^2(\mathbb{D})$ 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 $w_1, w_2, \ldots, w_n$ in $\mathbb{T}$, a recent paper shows that, for an affirmative answer, there necessarily exist $k$, $1\leq k\leq n$, 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 $b(\mathbb{D}) \neq \mathbb{D}$. We produce many examples here that have no such factor.
Keywords: Invariant subspace, backward shift, inner function.
MSC 2000: 30H05, 47B38, 49J20.
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