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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

A note concerning the index of the shift


Author: John R. Akeroyd
Journal: Proc. Amer. Math. Soc. 130 (2002), 3349-3354
MSC (2000): Primary 47A53, 47B20, 47B38; Secondary 30E10, 46E15
Published electronically: April 11, 2002
MathSciNet review: 1913014
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Abstract: Let $\mu$ be a finite, positive Borel measure with support in $\{z: \vert z\vert \leq 1\}$such that $P^2(\mu)$ - the closure of the polynomials in $L^2(\mu)$ - is irreducible and each point in $\mathbb{D} := \{z: \vert z\vert < 1\}$ is a bounded point evaluation for $P^2(\mu)$. We show that if $\mu(\partial{\mathbb{D}}) > 0$and there is a nontrivial subarc $\gamma$of $\partial{\mathbb{D}}$ such that

\begin{displaymath}\int_{\gamma}log(\mbox{\small {$\frac{d\mu}{dm}$ }})dm > -\infty,\end{displaymath}

then $\dim(\mathcal{M}\ominus z\mathcal{M}) = 1$ for each nontrivial closed invariant subspace $\mathcal{M}$ for the shift $M_z$ on $P^2(\mu)$.


References [Enhancements On Off] (What's this?)

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Additional Information

John R. Akeroyd
Affiliation: Department of Mathematics, University of Arkansas, Fayetteville, Arkansas 72701
Email: jakeroyd@comp.uark.edu

DOI: http://dx.doi.org/10.1090/S0002-9939-02-06464-X
PII: S 0002-9939(02)06464-X
Received by editor(s): April 17, 2001
Received by editor(s) in revised form: June 19, 2001
Published electronically: April 11, 2002
Communicated by: Joseph A. Ball
Article copyright: © Copyright 2002 American Mathematical Society