Analytic contractions, nontangential limits, and the index of invariant subspaces
(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)
Please use this url to cite or link to this publication:
https://lup.lub.lu.se/record/669503
- author
- Aleman, Alexandru LU ; Richter, Stefan and Sundberg, Carl LU
- organization
- publishing date
- 2007
- 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
- 2016-04-01 16:33:22
- date last changed
- 2022-03-30 08:41:23
@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}},
keywords = {{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}},
url = {{http://www.ams.org/tran/2007-359-07/S0002-9947-07-04258-4/S0002-9947-07-04258-4.pdf}},
volume = {{359}},
year = {{2007}},
}