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Finite codimensional invariant subspaces in Hilbert spaces of analytic functions

Aleman, Alexandru LU (1994) In Journal of Functional Analysis 119(1). p.1-18
Abstract
Let $\scr H$ denote a Hilbert space consisting of functions analytic on a bounded, open, connected subset $\Omega$ of the complex plane. Given certain natural hypotheses on $\scr H$, the author characterizes the finite-codimensional subspaces of $\scr H$ that are invariant under multiplication by $z$, showing that all such subspaces have the form $(p\scr H)^-$, where $p$ is a polynomial whose zeros lie in the closure of $\Omega$ (a more precise description of $p$ is given in the paper). In addition to standard hypotheses on $\scr H$ (such as continuity of point evaluations for points in $\Omega$), the author requires that $M_z$ be subnormal and that the collection of multipliers of $\scr H$ contain all functions analytic on $\Omega$ and... (More)
Let $\scr H$ denote a Hilbert space consisting of functions analytic on a bounded, open, connected subset $\Omega$ of the complex plane. Given certain natural hypotheses on $\scr H$, the author characterizes the finite-codimensional subspaces of $\scr H$ that are invariant under multiplication by $z$, showing that all such subspaces have the form $(p\scr H)^-$, where $p$ is a polynomial whose zeros lie in the closure of $\Omega$ (a more precise description of $p$ is given in the paper). In addition to standard hypotheses on $\scr H$ (such as continuity of point evaluations for points in $\Omega$), the author requires that $M_z$ be subnormal and that the collection of multipliers of $\scr H$ contain all functions analytic on $\Omega$ and continuous on its closure. The final section of the paper contains a characterization of the Fredholm multiplication operators on $\scr H$, which is derived as a consequence of the author's description of finite-codimensional invariant subspaces. The results in the paper are generalizations of those obtained in the Bergman-space setting by the author [Trans. Amer. Math. Soc. 330 (1992), no. 2, 531--544; MR1028755 (92f:47025)], by S. Axler [J. Reine Angew. Math. 336 (1982), 26--44; MR0671320 (84b:30052)], and by Axler and the reviewer [Trans. Amer. Math. Soc. 306 (1988), no. 2, 805--817; MR0933319 (89f:46051)].



Hilbert spaces of analytic functions where multiplication by z is a subnormal operator with a rich commutant are considered. We determine the invariant subspaces with finite codimension and the Fredholm operators in the commutant. (Less)
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publishing date
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Contribution to journal
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published
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in
Journal of Functional Analysis
volume
119
issue
1
pages
1 - 18
publisher
Elsevier
external identifiers
  • Scopus:0001883328
ISSN
0022-1236
language
English
LU publication?
no
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819366af-c9a4-45b7-a4a3-5096c1dee3da (old id 1467270)
alternative location
http://ida.lub.lu.se/cgi-bin/elsevier_local?YYUM0050-A-00221236-V0119I01-84710019
date added to LUP
2009-09-16 13:36:43
date last changed
2016-10-13 04:31:47
@misc{819366af-c9a4-45b7-a4a3-5096c1dee3da,
  abstract     = {Let $\scr H$ denote a Hilbert space consisting of functions analytic on a bounded, open, connected subset $\Omega$ of the complex plane. Given certain natural hypotheses on $\scr H$, the author characterizes the finite-codimensional subspaces of $\scr H$ that are invariant under multiplication by $z$, showing that all such subspaces have the form $(p\scr H)^-$, where $p$ is a polynomial whose zeros lie in the closure of $\Omega$ (a more precise description of $p$ is given in the paper). In addition to standard hypotheses on $\scr H$ (such as continuity of point evaluations for points in $\Omega$), the author requires that $M_z$ be subnormal and that the collection of multipliers of $\scr H$ contain all functions analytic on $\Omega$ and continuous on its closure. The final section of the paper contains a characterization of the Fredholm multiplication operators on $\scr H$, which is derived as a consequence of the author's description of finite-codimensional invariant subspaces. The results in the paper are generalizations of those obtained in the Bergman-space setting by the author [Trans. Amer. Math. Soc. 330 (1992), no. 2, 531--544; MR1028755 (92f:47025)], by S. Axler [J. Reine Angew. Math. 336 (1982), 26--44; MR0671320 (84b:30052)], and by Axler and the reviewer [Trans. Amer. Math. Soc. 306 (1988), no. 2, 805--817; MR0933319 (89f:46051)].<br/><br>
<br/><br>
Hilbert spaces of analytic functions where multiplication by z is a subnormal operator with a rich commutant are considered. We determine the invariant subspaces with finite codimension and the Fredholm operators in the commutant.},
  author       = {Aleman, Alexandru},
  issn         = {0022-1236},
  language     = {eng},
  number       = {1},
  pages        = {1--18},
  publisher    = {ARRAY(0xb7ba818)},
  series       = {Journal of Functional Analysis},
  title        = {Finite codimensional invariant subspaces in Hilbert spaces of analytic functions},
  volume       = {119},
  year         = {1994},
}