Analytic contractions, nontangential limits, and the index of invariant subspaces
(2007) In Transactions of the American Mathematical Society 359(7). p.33693407 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 Mzeta. 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 Mzeta. 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)/(zetalambda) 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 Mzeta has index 1, i.e., satisfies dim M/zeta M= 1. (Less)
Please use this url to cite or link to this publication:
http://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
 00029947
 language
 English
 LU publication?
 yes
 id
 401336d019b047c2a39491009c687c50 (old id 669503)
 alternative location
 http://www.ams.org/tran/200735907/S0002994707042584/S0002994707042584.pdf
 date added to LUP
 20071204 17:40:45
 date last changed
 20180529 10:57:37
@article{401336d019b047c2a39491009c687c50, 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 Mzeta. 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)/(zetalambda) 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 Mzeta has index 1, i.e., satisfies dim M/zeta M= 1.}, author = {Aleman, Alexandru and Richter, Stefan and Sundberg, Carl}, issn = {00029947}, keyword = {index,invariant subspaces,nontangential limits,Hilbert space of analytic functions,contraction}, language = {eng}, number = {7}, pages = {33693407}, 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}, }