Commutant lifting, interpolation and toeplitz operators In several variables/ K D Deepak

Thesis (Ph.D) - Indian Statistical Institute, 2023

Includes bibliography

Introduction -- Partially isometric toeplitz operators on the polydisc -- Commutant lifting and interpolation on the polydisc --
Perturbations of analytic functions on the polydisc -- Commutant lifting and nevanlinna-pick interpolation on the unit ball

Guided by Prof. Jaydeb Sarkar

The purpose of this thesis is to examine some classical one variable Hilbert function
space theoretic results in the context of several complex variables and commuting tuples
of bounded linear operators on Hilbert spaces. More specifically, we will be interested
in the classical Sarason’s commutant lifting theorem on D, where
D = {z ∈ C : |z| < 1},
the open unit disc in C. A significant part of our discussion in this thesis will revolve
around the commutant lifting theorem in two different contexts, as well as its following
applications of independent importance. Another important object of study will be
Toeplitz operators on the polydisc D
n
, n ≥ 1.
It is worth noting that the operator theory, in terms of complexity and known as well
as unknown, is different for commuting tuples of contractions and commuting tuples of
row contractions, just like the theory of analytic functions differs from the open unit ball
to the open unit polydisc. From this perspective, we talk about the commutant lifting
theorem in the context of the open unit ball and the polydisc. As we will see in this
thesis, the latter scenario seems to be more interesting and challenging