Webbläsaren som du använder stöds inte av denna webbplats. Alla versioner av Internet Explorer stöds inte längre, av oss eller Microsoft (läs mer här: * https://www.microsoft.com/en-us/microsoft-365/windows/end-of-ie-support).

Var god och använd en modern webbläsare för att ta del av denna webbplats, som t.ex. nyaste versioner av Edge, Chrome, Firefox eller Safari osv.

Finite codimensional invariant subspaces in Hilbert spaces of analytic functions

In English

Författare

Summary, in English

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.

Publiceringsår

1994

Språk

Engelska

Sidor

1-18

Publikation/Tidskrift/Serie

Journal of Functional Analysis

Volym

119

Issue

1

Dokumenttyp

Artikel i tidskrift

Förlag

Elsevier

Ämne

  • Mathematics

Status

Published

ISBN/ISSN/Övrigt

  • ISSN: 0022-1236