TY - THES
T1 - Beurling-type representation of invariant subspaces in reproducing kernel Hilbert spaces
A1 - Barbian,Christoph
Y1 - 2011/07/05
N2 - By Beurling's theorem, the orthogonal projection onto a multiplier invariant subspace M of the Hardy space H^{2}(mathbb{D}) over the complex unit disk can be represented as P_{M}=M_{phi}M_{phi}^{*}, where phi is a suitable inner function. This result essentially remains true for arbitrary Nevanlinna-Pick spaces but fails in more general settings such as the Bergman space. We therefore introduce the notion of Beurling decomposability of subspaces: An invariant subspace M of a reproducing kernel space mathcal{H} is 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=mbox{ran}M_{phi_{1}}. Our aim is to characterize Beurling-decomposable subspaces by means of the core function and the core operator. More precisely, an invariant subspace M of mathcal{H} is Beurling decomposable precisely if its core function induces a completely bounded Schur multiplication on B(mathcal{H}), defined in an appropriate way. These Schur multiplications turn out to be left(mathcal{M}(mathcal{H}),overline{mathcal{M}(mathcal{H})}^{op}right)-module homomorphisms on B(mathcal{H}) (where mathcal{M}(mathcal{H}) denotes the multiplier algebra of mathcal{H}). This allows us, in formal analogy to the case of classical Schur multipliers and to the study of multipliers of the Fourier algebra A(G), to make use of the representation theory for completely bounded module homomorphisms. As an application, we show that, for the standard reproducing kernel Hilbert spaces over bounded symmetric domains, every finite-codimensional submodule M is Beurling decomposable and, in many concrete situations, can be represented as M=sum_{i=1}^{r}p_{i}mathcal{H} with suitable polynomials p_{i}. We thus extend well-known results of Ahern and Clark, Axler and Bourdon and Guo. Furthermore, we prove that, in vector-valued Hardy spaces over bounded symmetric domains, defect functions of Beurling decomposable subspaces have boundary values almost everywhere on the Shilov boundary of D and, moreover, that these boundary values are projections of constant rank. This is a complete generalization of results of Guo and of Greene, Richter and Sundberg. Finally, we characterize the Beurling decomposable subspaces of the Bergman space L_{a}^{2}(mathbb{D}). As a byproduct of the techiques developed in this paper, we obtain a new proof of the ';Wandering Subspace Theorem'; for the Bergman space.
KW - Hilbert-Kern
KW - Hardy-Raum
KW - Nevanlinna-Klasse
CY - Saarbrücken
PB - Saarländische Universitäts- und Landesbibliothek
AD - Postfach 151141, 66041 Saarbrücken
UR - http://scidok.sulb.uni-saarland.de/volltexte/2011/3853
ER -