# 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

Abstract:

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