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

Aleman, Alexandru LU ; Richter, Stefan and Sundberg, Carl LU (2007) In Transactions of the American Mathematical Society 359(7). p.3369-3407
Abstract
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... (More)
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. (Less)
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author
organization
publishing date
type
Contribution to journal
publication status
published
subject
keywords
index, invariant subspaces, nontangential limits, Hilbert space of analytic functions, contraction
in
Transactions of the American Mathematical Society
volume
359
issue
7
pages
3369 - 3407
publisher
American Mathematical Society (AMS)
external identifiers
  • wos:000245149100018
  • scopus:77950941638
ISSN
0002-9947
language
English
LU publication?
yes
id
401336d0-19b0-47c2-a394-91009c687c50 (old id 669503)
alternative location
http://www.ams.org/tran/2007-359-07/S0002-9947-07-04258-4/S0002-9947-07-04258-4.pdf
date added to LUP
2007-12-04 17:40:45
date last changed
2017-08-20 04:27:03
@article{401336d0-19b0-47c2-a394-91009c687c50,
  abstract     = {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.},
  author       = {Aleman, Alexandru and Richter, Stefan and Sundberg, Carl},
  issn         = {0002-9947},
  keyword      = {index,invariant subspaces,nontangential limits,Hilbert space of analytic functions,contraction},
  language     = {eng},
  number       = {7},
  pages        = {3369--3407},
  publisher    = {American Mathematical Society (AMS)},
  series       = {Transactions of the American Mathematical Society},
  title        = {Analytic contractions, nontangential limits, and the index of invariant subspaces},
  volume       = {359},
  year         = {2007},
}