Finite codimensional invariant subspaces in Hilbert spaces of analytic functions

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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)].

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.

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Research areas and keywords

Subject classification (UKÄ) – MANDATORY

  • Mathematics
Original languageEnglish
Pages (from-to)1-18
JournalJournal of Functional Analysis
Volume119
Issue number1
Publication statusPublished - 1994
Publication categoryResearch
Peer-reviewedYes
Externally publishedYes