# 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.

**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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