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Preprint (Vorabdruck) zugänglich unter
URN: urn:nbn:de:bsz:291-scidok-46359
URL: http://scidok.sulb.uni-saarland.de/volltexte/2012/4635/


Beurling-type representation of invariant subspaces in reproducing kernel Hilbert spaces

Barbian, Christoph

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Institut: Fachrichtung 6.1 - Mathematik
DDC-Sachgruppe: Mathematik
Dokumentart: Preprint (Vorabdruck)
Schriftenreihe: Preprint / Fachrichtung Mathematik, Universität des Saarlandes
Bandnummer: 167
Sprache: Englisch
Erstellungsjahr: 2006
Publikationsdatum: 09.03.2012
Kurzfassung auf Englisch: By Beurling´s theorem, the orthogonal projection onto an invariant subspace M of the Hardy space H^{2}(\mathbb{D}) on the complex unit disk can be represented as P_{M}=M_{\phi}M_{\phi}^{*} where \phi is a suitable multiplier of H^{2}(\mathbb{D}). This concept can be carried over to arbitrary Nevanlinna-Pick spaces but fails in more general settings. This paper introduces the notion of Beurling decomposability of subspaces. An invariant subspace M of a reproducing kernel space will be called Beurling decomposable if there exist (operator-valued) multipliers \phi_{1},\phi_{2} such that P_{M}=M_{\phi_{1}}M_{\phi_{1}}^{*}-M_{\phi_{2}}M_{\phi_{2}}^{*} and M=\textrm{ran}M_{\phi_{1}}. We characterize the finite-codimensional and the finite-rank Beurling-decomposable subspaces by means of the core function and the core operator. As an application, we show that in many analytic Hilbert modules \mathcal{H}, every fnite-codimensional submodule M can be written as M=\sum_{i=1}^{r}p_{i}\mathcal{H} with suitable polynomials p_{i}.
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