Sourav Pal

IIT Bombay

Sourav Pal is an Associate Professor in the Department of Mathematics at IIT Bombay. He obtained his Ph.D. from IISc, Bangalore (2012). Sourav’s interests include Functional Analysis, Operator Theory, and Several Complex Variables. He was an INSPIRE Faculty at the Stat-Math Unit, Indian Statistical Institute Delhi Centre, for a few months. Sourav was also a postdoctoral fellow at the Mathematics Department, Ben-Gurion University of the Negev, Israel, and Newcastle University, UK, from December 2011 to September 2014. He was selected as a Young Associate of the Indian Academy of Sciences in 2019.

Sourav Pal

Lectures by Fellows/Associates

C Pulla Rao, IIT, Tirupati

Distinguished Varieties In The Polydisc and Dilation of Commuting Matrices

For n ≥ 2, the polydisc is the following family of domains D n = {(z1, . . . , zn) : zi ∈ C, |zi | < 1, i = 1, . . . , n} . A distinguished variety in the polydisc D n is the intersection of an algebraic variety with D n such that the algebraic variety exits D n through the n-torus T n without intersecting any other part of the topological boundary of D n. We prove that for any natural number n ≥ 2, a distinguished variety in D n has complex-dimension 1. A tuple of commuting square matrices (A1, . . . , An) is said to have D n as a spectral set if kf(A1, . . . , An)k ≤ sup zi∈D |f(z1, . . . , zn)| = kfk∞,D n , for any polynomial f ∈ C[z1, . . . , zn]. We show that such a commuting tuple (A1, . . . , An) admits a commuting unitary dilation (U1, . . . , Un) with U = Qn i=1 Ui being the minimal unitary dilation of A = Qn i=1 Ai if and only if there is a distinguished variety in D n which is a spectral set for (A1, . . . , An).

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