Rational dilation on the polydisc and distinguished varieties
Prof. Sourav Pal, Indian Institute of Technology, Bombay,
Speaker 
Prof. Sourav Pal, Indian Institute of Technology, Bombay,


When 
Sep 07, 2022
from 03:00 PM to 04:00 PM 
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Abstract: We define spectral set, rational dilation and recall a brief literature of success of rational dilation on the unit disk $\mathbb D= \{ z\in \mathbb C : z<1 \}$, on an annulus, on the bidisk $\mathbb D^2$, symmetrized bidisc $\mathbb G_2$ and its failure on a triply connected domain and on the polydisk $\mathbb D^n$ for $n \geq 3$. We find a necessary and sufficient condition such that a tuple of commuting Hilbert space contractions $(T_1, \dots , T_n)$ having the closed polydisk $\mathbb D^n$ ($n>2$) as a spectral set admits a normal distinguishedboundary dilation on the minimal dilation space of the product $\prod_{i=1}^n T_i$ and show an explicit construction of such a dilation. We also show interaction of such a dilation with a class of algebraic curves called distinguished varieties when $T_1, \dots , T_n$ are commuting matrices.