Finite codimensional invariant subspaces in Hilbert spaces of analytic functions
(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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https://lup.lub.lu.se/record/1467270
- author
- Aleman, Alexandru LU
- publishing date
- 1994
- type
- Contribution to journal
- publication status
- published
- subject
- in
- Journal of Functional Analysis
- volume
- 119
- issue
- 1
- pages
- 1 - 18
- publisher
- Elsevier
- external identifiers
-
- scopus:0001883328
- ISSN
- 0022-1236
- DOI
- 10.1006/jfan.1994.1001
- language
- English
- LU publication?
- no
- id
- 819366af-c9a4-45b7-a4a3-5096c1dee3da (old id 1467270)
- date added to LUP
- 2016-04-04 09:30:18
- date last changed
- 2021-01-03 05:33:35
@article{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 = {{Elsevier}},
series = {{Journal of Functional Analysis}},
title = {{Finite codimensional invariant subspaces in Hilbert spaces of analytic functions}},
url = {{http://dx.doi.org/10.1006/jfan.1994.1001}},
doi = {{10.1006/jfan.1994.1001}},
volume = {{119}},
year = {{1994}},
}