On Beurling-Lax-Halmos theorem

Speaker: Monojit Bhattacharjee

Date: 31 March, 2015 (Tuesday)

Time: 09:15 pm – 10:15 pm

Venue: Lecture Hall I, Department of Mathematics



In this talk we will present to you the proof of the “Beurling-Lax-Halmos” theorem. Let T be a C_{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 partial isometric operator \Pi:H^2_{\mathcal{D}} \rightarrow \mathcal{H} such that \Pi \mathcal{M}_{z}= T \Pi and \mathcal{S} = ran \Pi. As a corollary of this theorem we will prove the “Beurling-Lax-Halmos” theorem. This theorem characterizes the shift-invariant subspaces of the vector-valued Hardy space.

Area: Functional Analysis, Operator Theory

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