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Beurling-type Representation of Invariant Subspaces in Reproducing Kernel Hilbert Spaces

Authors:	Christoph Barbian

Institution:	1. Fachrichtung Mathematik, Universit?t des Saarlandes, Postfach 15 11 50, D-66041, Saarbrücken, Germany

Abstract:	By Beurling’s theorem, the orthogonal projection onto an invariant subspace M of the Hardy space $H^2({\mathbb{D}})$ on the unit disk can be represented as $P_M = M_\phi M_\phi^$ where Φ is an inner 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 decomposable subspaces. An invariant subspace M of a reproducing kernel Hilbert 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 $M = {\rm ran}\, M_{\phi_1}$ . We characterize the finite-codimensional and the finite-rank Beurling decomposable subspaces by means of their core function and core operator. As an application, we show that in many analytic Hilbert modules ${\mathcal{H}}$ , every finite-codimensional submodule M can be written as $M = \sum^r_ {i=1} p_i \cdot {\mathcal{H}}$ with suitable polynomials p _i.

Keywords:	Mathematics Subject Classification (2000)" target="_blank">Mathematics Subject Classification (2000) Primary 47B32 Secondary 47A13 47A15
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