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

Abstract

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}.