Journal of Operator Theory

Volume 73, Issue 2, Spring 2015 pp. 433-441.

An invariant subspace theorem and invariant subspaces of analytic reproducing kernel Hilbert spaces. I

Authors: Jaydeb Sarkar
Author institution:Indian Statistical Institute, Statistics and Mathematics Unit, 8th Mile, Mysore Road, Bangalore, 560059, India

Summary: Let $T$ be a $C_{\cdot 0}$-contraction on a Hilbert space $\mathcal{H}$ and $\mathcal{S}$ be a non-trivial closed subspace of $\mathcal{H}$. We prove that $\mathcal{S}$ is a $T$-invariant subspace of $\mathcal{H}$ if and only if there exists a Hilbert space $\mathcal{D}$ and a partially isometric operator $\Pi : H^2_{\mathcal{D}}(\mathbb{D}) \rightarrow \mathcal{H}$ such that $\Pi M_z = T \Pi$ and that $\mathcal{S} = \mbox{ran} \Pi$, or equivalently, \[P_{\mathcal{S}} = \Pi \Pi^*.\]As an application we completely classify the shift-invariant subspaces of analytic reproducing kernel Hilbert spaces over the unit disc. Our results also include the case of weighted Bergman spaces over the unit disk.

DOI: http://dx.doi.org/10.7900/jot.2014jan29.2042
Keywords: reproducing kernels, Hilbert modules, invariant subspaces, weighted Bergman spaces, Hardy space

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