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

DOI:
https://doi.org/10.1090/S0002-9947-07-04258-4

Published electronically:
February 12, 2007

MathSciNet review:
2299460

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Abstract | References | Similar Articles | Additional Information

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 $||\zeta ^{n}f|| \to 0$ 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 $||\frac {\zeta -\lambda }{1-\overline {\lambda }\zeta }f||\ge c||f||$, then $\Delta (\mathcal {H}) =\Sigma (\mathcal {H})$ a.e. and the following four conditions are equivalent: (1) $||\zeta ^{n} f||\nrightarrow 0$ for some $f \in \mathcal {H}$, (2) $||\zeta ^{n} f||\nrightarrow 0$ 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

MR Author ID:
215743

Email:
Richter@math.utk.edu

**Carl Sundberg**

Affiliation:
Department of Mathematics, University of Tennessee, Knoxville, Tennessee 37996-1300

Email:
Sundberg@math.utk.edu

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.

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© Copyright 2007
American Mathematical Society

The copyright for this article reverts to public domain 28 years after publication.