Finite codimensional invariant subspaces in Hilbert spaces of analytic functions
(1994) In Journal of Functional Analysis 119(1). p.118 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 finitecodimensional 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 finitecodimensional 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 finitecodimensional invariant subspaces. The results in the paper are generalizations of those obtained in the Bergmanspace setting by the author [Trans. Amer. Math. Soc. 330 (1992), no. 2, 531544; MR1028755 (92f:47025)], by S. Axler [J. Reine Angew. Math. 336 (1982), 2644; MR0671320 (84b:30052)], and by Axler and the reviewer [Trans. Amer. Math. Soc. 306 (1988), no. 2, 805817; 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)
Please use this url to cite or link to this publication:
http://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
 00221236
 language
 English
 LU publication?
 no
 id
 819366afc9a445b7a4a35096c1dee3da (old id 1467270)
 alternative location
 http://ida.lub.lu.se/cgibin/elsevier_local?YYUM0050A00221236V0119I0184710019
 date added to LUP
 20090916 13:36:43
 date last changed
 20180107 10:22:01
@article{819366afc9a445b7a4a35096c1dee3da, 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 finitecodimensional 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 finitecodimensional invariant subspaces. The results in the paper are generalizations of those obtained in the Bergmanspace setting by the author [Trans. Amer. Math. Soc. 330 (1992), no. 2, 531544; MR1028755 (92f:47025)], by S. Axler [J. Reine Angew. Math. 336 (1982), 2644; MR0671320 (84b:30052)], and by Axler and the reviewer [Trans. Amer. Math. Soc. 306 (1988), no. 2, 805817; 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 = {00221236}, language = {eng}, number = {1}, pages = {118}, publisher = {Elsevier}, series = {Journal of Functional Analysis}, title = {Finite codimensional invariant subspaces in Hilbert spaces of analytic functions}, volume = {119}, year = {1994}, }