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Simply invariant subspaces of H2 of some multiply connected regions

Aleman, Alexandru LU and Richter, Stefan (1996) In Integral Equations and Operator Theory 24(2). p.127-155
Abstract
The authors are concerned with the invariant subspaces of $H^2(\Omega)$, where $\Omega$ is a bounded, finitely connected, planar domain with an analytic boundary. A subspace of $H^2(\Omega)$ is said to be invariant if it is invariant under multiplication by $z$, and fully invariant if it is invariant under multiplication by all rational functions with poles off the closure of $\Omega$.



If $\Omega$ is simply connected, every invariant subspace is fully invariant; these subspaces are described by the famous theorem of A. Beurling from 1949. For fully invariant subspaces in the multiply connected case, several authors showed in the 1960s that an analogue of Beurling's theorem holds. The first description of invariant... (More)
The authors are concerned with the invariant subspaces of $H^2(\Omega)$, where $\Omega$ is a bounded, finitely connected, planar domain with an analytic boundary. A subspace of $H^2(\Omega)$ is said to be invariant if it is invariant under multiplication by $z$, and fully invariant if it is invariant under multiplication by all rational functions with poles off the closure of $\Omega$.



If $\Omega$ is simply connected, every invariant subspace is fully invariant; these subspaces are described by the famous theorem of A. Beurling from 1949. For fully invariant subspaces in the multiply connected case, several authors showed in the 1960s that an analogue of Beurling's theorem holds. The first description of invariant subspaces that are not fully invariant was obtained by H. L. Royden [Pacific J. Math. 134 (1988), no. 1, 151--172; MR0953505 (90a:46056)] and D. Hitt [Pacific J. Math. 134 (1988), no. 1, 101--120; MR0953502 (90a:46059)]; they treated the case where $\Omega$ is an annulus. Their work was extended to the general finitely connected case by D. V. Yakubovich [Zap. Nauchn. Sem. Leningrad. Otdel. Mat. Inst. Steklov. (LOMI) 178 (1989), Issled. Lineĭn. Oper. Teorii Funktsiĭ. 18, 166-183, 186--187; MR1037771 (91c:47061)]. The description of an invariant subspace that is not fully invariant involves a property of pseudocontinuation across one or more of the interior boundary components of $\Omega$.



In the present paper the authors present an alternative approach to the results of Royden, Hitt and Yakubovich. The main new ingredient is a norm equality, suggested by recent work in Bergman spaces, which the authors use in handling the case of an annulus. In addition, the authors study the operator of multiplication by $z$ on an invariant subspace, determining the spectrum and approximate point spectrum, and obtaining an upper bound for its cyclic multiplicity.



As the authors note, it would be desirable to sharpen several points in their description of invariant subspaces. They obtain partial results in this direction. In the erratum, the authors replace a flawed lemma (Lemma 2.3) from the original paper. (Less)
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author
publishing date
type
Contribution to journal
publication status
published
subject
in
Integral Equations and Operator Theory
volume
24
issue
2
pages
127 - 155
publisher
Springer
external identifiers
  • scopus:0030074899
ISSN
1420-8989
DOI
10.1007/BF01193457
language
English
LU publication?
no
id
50a202a1-4399-488f-afb4-3f65974d970d (old id 1467236)
date added to LUP
2009-09-16 13:22:15
date last changed
2017-01-01 04:31:52
@article{50a202a1-4399-488f-afb4-3f65974d970d,
  abstract     = {The authors are concerned with the invariant subspaces of $H^2(\Omega)$, where $\Omega$ is a bounded, finitely connected, planar domain with an analytic boundary. A subspace of $H^2(\Omega)$ is said to be invariant if it is invariant under multiplication by $z$, and fully invariant if it is invariant under multiplication by all rational functions with poles off the closure of $\Omega$. <br/><br>
<br/><br>
If $\Omega$ is simply connected, every invariant subspace is fully invariant; these subspaces are described by the famous theorem of A. Beurling from 1949. For fully invariant subspaces in the multiply connected case, several authors showed in the 1960s that an analogue of Beurling's theorem holds. The first description of invariant subspaces that are not fully invariant was obtained by H. L. Royden [Pacific J. Math. 134 (1988), no. 1, 151--172; MR0953505 (90a:46056)] and D. Hitt [Pacific J. Math. 134 (1988), no. 1, 101--120; MR0953502 (90a:46059)]; they treated the case where $\Omega$ is an annulus. Their work was extended to the general finitely connected case by D. V. Yakubovich [Zap. Nauchn. Sem. Leningrad. Otdel. Mat. Inst. Steklov. (LOMI) 178 (1989), Issled. Lineĭn. Oper. Teorii Funktsiĭ. 18, 166-183, 186--187; MR1037771 (91c:47061)]. The description of an invariant subspace that is not fully invariant involves a property of pseudocontinuation across one or more of the interior boundary components of $\Omega$. <br/><br>
<br/><br>
In the present paper the authors present an alternative approach to the results of Royden, Hitt and Yakubovich. The main new ingredient is a norm equality, suggested by recent work in Bergman spaces, which the authors use in handling the case of an annulus. In addition, the authors study the operator of multiplication by $z$ on an invariant subspace, determining the spectrum and approximate point spectrum, and obtaining an upper bound for its cyclic multiplicity. <br/><br>
<br/><br>
As the authors note, it would be desirable to sharpen several points in their description of invariant subspaces. They obtain partial results in this direction. In the erratum, the authors replace a flawed lemma (Lemma 2.3) from the original paper.},
  author       = {Aleman, Alexandru and Richter, Stefan},
  issn         = {1420-8989},
  language     = {eng},
  number       = {2},
  pages        = {127--155},
  publisher    = {Springer},
  series       = {Integral Equations and Operator Theory},
  title        = {Simply invariant subspaces of H2 of some multiply connected regions},
  url          = {http://dx.doi.org/10.1007/BF01193457},
  volume       = {24},
  year         = {1996},
}