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

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Author

Summary, in English

Let H be a Hilbert space of analytic functions on the open unit disc D such that the operator M. of multiplication with the identity function. defines a contraction operator. In terms of the reproducing kernel for H we will characterize the largest set Delta(H) subset of partial derivative D such that for each f, g is an element of H, g not equal 0 the meromorphic function f/g has nontangential limits a.e. on Delta( H). We will see that the question of whether or not Delta( H) has linear Lebesgue measure 0 is related to questions concerning the invariant subspace structure of M-zeta. We further associate with H a second set Sigma(H) subset of partial derivative D, which is defined in terms of the norm on H. For example, Sigma(H) has the property that vertical bar zeta(n)f vertical bar vertical bar -> 0 for all f is an element of H if and only if Sigma( H) has linear Lebesgue measure 0. It turns out that.( H). S( H) a. e., by which we mean that Delta(H) backslash Sigma(H) has linear Lebesgue measure 0. We will study conditions that imply that Delta(H) = Sigma(H) a.e.. As one corollary to our results we will show that if dim H/zeta H = 1 and if there is a c > 0 such that for all f is an element of H and all lambda is an element of D we have parallel to(1-(lambda) over bar zeta)/(zeta-lambda) f parallel to >= c parallel to f||, then Delta(H) = Sigma(H) a.e. and the following four conditions are equivalent: (1) parallel to zeta(n)f parallel to negated right arrow 0 for some f is an element of H, (2) parallel to zeta(n)f parallel to negated right arrow 0 for all f is an element of H, f not equal 0, (3).( H) has nonzero Lebesgue measure, (4) every nonzero invariant subspace M of M-zeta has index 1, i.e., satisfies dim M/zeta M= 1.

Publishing year

2007

Language

English

Pages

3369-3407

Publication/Series

Transactions of the American Mathematical Society

Volume

359

Issue

7

Document type

Journal article

Publisher

American Mathematical Society (AMS)

Topic

  • Mathematics

Keywords

  • index
  • invariant subspaces
  • nontangential limits
  • Hilbert space of analytic functions
  • contraction

Status

Published

ISBN/ISSN/Other

  • ISSN: 0002-9947