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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 $\clh$ and $\cls$ be a non-trivial closed subspace of $\clh$. We prove that $\cls$ is a $T$-invariant subspace of $\clh$ if and only if there exists a Hilbert space $\cld$ and a partially isometric operator $\Pi : H^2_{\cld}(\mathbb{D}) \raro \clh$ such that $\Pi M_z = T \Pi$ and that $\cls = \mbox{ran~} \Pi$, or equivalently, \[P_{\cls} = \Pi \Pi^*.\]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.

Keywords: reproducing kernels, Hilbert modules, invariant subspaces, weighted Bergman spaces, Hardy space

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