Analysis Seminar.
Date, time and Venue: April 2, 2025, 3.30 pm, Ramanujan Hall of
Mathematics department
Speaker: Mr. Nitin Tomar
Title: Operators associated with various domains in $\mathbb{C}^n$
Abstract: This talk explores operator theory on several important domains
in $\mathbb{C}^n$, focusing on dilation, decomposition and model theory.
We examine the interplay between operator theory and complex geometry,
particularly through spectral sets and distinguished varieties. The
domains of our interest include the polyannulus $\mathbb{A}_r^n$, bidisc
$\mathbb{D}^2$, biball $\mathbb{B}_2$, symmetrized bidisc $\mathbb{G}_2$
and pentablock $\mathbb{P}$. The first part examines operator theory on
the polyannulus $\A_r^n$, where we explore $ \mathbb{A}_r^n
$-contractions, $ \mathbb{A}_r^n $-unitaries, and $ \mathbb{A}_r^n
$-isometries, providing characterizations and decomposition results. We
also study rational dilation for certain classes of $ \mathbb{A}_r^n
$-contractions and identify minimal spectral sets for various classes of
operators related to annulus. In particular, we characterize the class of
$ \mathbb{A}_r $-contractions using a variety in the biball. The second
part focuses on the pentablock, where we define $ \mathbb{P}
$-contractions, $ \mathbb{P} $-unitaries, and $ \mathbb{P} $-isometries,
and develop canonical decomposition results for $\Pe$-contractions. We
establish conditions for dilating $ \mathbb{P} $-contractions to $
\mathbb{P} $-isometries and explore related operators in the biball and
symmetrized bidisc, identifying relations between these domains at
operator theoretic level. In the final part, we investigate distinguished
varieties in the bidisc and symmetrized bidisc, toral polynomials,
$\Gamma$-distinguished polynomials and operator pairs annihilated by such
polynomials. We provide necessary and sufficient conditions for dilating
toral contractions to toral unitaries and characterize $ \Gamma
$-distinguished $\Gamma$- contractions admitting a dilation to
$\Gamma$-distinguished $\Gamma$-unitaries.