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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

Analytic contractions, nontangential limits, and the index of invariant subspaces


Authors: Alexandru Aleman, Stefan Richter and Carl Sundberg
Journal: Trans. Amer. Math. Soc. 359 (2007), 3369-3407
MSC (2000): Primary 47B32, 46E22; Secondary 30H05, 46E20
Published electronically: February 12, 2007
MathSciNet review: 2299460
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Abstract: Let $ \mathcal{H}$ be a Hilbert space of analytic functions on the open unit disc $ \mathbb{D}$ such that the operator $ M_{\zeta }$ of multiplication with the identity function $ \zeta $ defines a contraction operator. In terms of the reproducing kernel for $ \mathcal{H}$ we will characterize the largest set $ \Delta (\mathcal{H}) \subseteq \partial \mathbb{D}$ such that for each $ f, g \in \mathcal{H}$, $ g \ne 0$ the meromorphic function $ f/g$ has nontangential limits a.e. on $ \Delta (\mathcal{H})$. We will see that the question of whether or not $ \Delta (\mathcal{H})$ has linear Lebesgue measure 0 is related to questions concerning the invariant subspace structure of $ M_{\zeta }$.

We further associate with $ \mathcal{H}$ a second set $ \Sigma (\mathcal{H}) \subseteq \partial \mathbb{D}$, which is defined in terms of the norm on $ \mathcal{H}$. For example, $ \Sigma (\mathcal{H})$ has the property that $ \vert\vert\zeta ^{n}f\vert\vert \to 0$ for all $ f \in \mathcal{H}$ if and only if $ \Sigma (\mathcal{H})$ has linear Lebesgue measure 0.

It turns out that $ \Delta (\mathcal{H}) \subseteq \Sigma (\mathcal{H})$ a.e., by which we mean that $ \Delta (\mathcal{H}) \setminus \Sigma (\mathcal{H})$ has linear Lebesgue measure 0. We will study conditions that imply that $ \Delta (\mathcal{H}) = \Sigma (\mathcal{H})$ a.e.. As one corollary to our results we will show that if dim $ \mathcal{H}/\zeta \mathcal{H} =1$ and if there is a $ c>0$ such that for all $ f \in \mathcal{H}$ and all $ \lambda \in \mathbb{D}$ we have $ \vert\vert\frac{\zeta -\lambda }{1-\overline{\lambda }\zeta }f\vert\vert\ge c\vert\vert f\vert\vert$, then $ \Delta (\mathcal{H}) =\Sigma (\mathcal{H})$ a.e. and the following four conditions are equivalent:

(1) $ \vert\vert\zeta ^{n} f\vert\vert\nrightarrow 0$ for some $ f \in \mathcal{H}$,

(2) $ \vert\vert\zeta ^{n} f\vert\vert\nrightarrow 0$ for all $ f \in \mathcal{H}$, $ f \ne 0$,

(3) $ \Delta (\mathcal{H})$ has nonzero Lebesgue measure,

(4) every nonzero invariant subspace $ \mathcal{M}$ of $ M_{\zeta }$ has index 1, i.e., satisfies dim $ \mathcal{M}/\zeta \mathcal{M} =1$.


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Additional Information

Alexandru Aleman
Affiliation: Department of Mathematics, Lund University, P.O. Box 118, S-221 00 Lund, Sweden
Email: Aleman@maths.lth.se

Stefan Richter
Affiliation: Department of Mathematics, University of Tennessee, Knoxville, Tennessee 37996-1300
Email: Richter@math.utk.edu

Carl Sundberg
Affiliation: Department of Mathematics, University of Tennessee, Knoxville, Tennessee 37996-1300
Email: Sundberg@math.utk.edu

DOI: http://dx.doi.org/10.1090/S0002-9947-07-04258-4
PII: S 0002-9947(07)04258-4
Keywords: Hilbert space of analytic functions, contraction, nontangential limits, invariant subspaces, index
Received by editor(s): July 11, 2005
Published electronically: February 12, 2007
Additional Notes: Part of this work was done while the second author visited Lund University. He would like to thank the Mathematics Department for its hospitality. Furthermore, work of the first author was supported by the Royal Swedish Academy of Sciences and work of the second and third authors was supported by the U. S. National Science Foundation.
Article copyright: © Copyright 2007 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.