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APRG Seminar

Title: Minimal dilations for commuting contractions and operator model
Speaker: Sourav Pal (IIT, Bombay)
Date: 28 April 2022
Time: 3 pm
Venue: LH-1, Mathematics Department

For commuting contractions $T_1,\ldots ,T_n$ acting on a Hilbert space $\mathcal{H}$ with $T=\prod_{i=1}^{n}T_i$, we find a necessary and sufficient condition under which $(T_1,\ldots ,T_n)$ dilates to commuting isometries $(V_1,\ldots ,V_n)$ or commuting unitaries $(U_1,\ldots ,U_n)$ acting on the minimal isometric dilation space or the minimal unitary dilation space of $T$ respectively, where $V=\prod_{i=1}^{n}V_i$ and $U=\prod_{i=1}^{n}U_i$ are the minimal isometric and the minimal unitary dilations of $T$ respectively. We construct both Schäffer and Sz. Nagy-Foias type isometric and unitary dilations for $(T_1,\ldots ,T_n)$. Also, a special minimal isometric dilation is constructed where the product $T$ is a $C_0$ contraction, that is $T^{*n}\to 0$ strongly as $n\to \infty$. As a consequence of these dilation theorems we obtain different functional models for $(T_1,\ldots ,T_n)$. When the product $T$ is a $C_0$ contraction, the dilation of $(T_1,\ldots ,T_n)$ leads to a natural factorization of $T$ in terms of compression of Toeplitz operators with linear analytic symbols.

Contact: +91 (80) 2293 2711, +91 (80) 2293 2265 ;     E-mail: chair.math[at]iisc[dot]ac[dot]in
Last updated: 08 Dec 2022