Indian Institute of Technology Hyderabad (IITH) is a premier institute of science and technology established in 2008. IITH has been consistently ranked in the&nbsp;<a href="https://iith.ac.in/reports/" rel="noopener noreferrer">top 10 institutes</a>&nbsp;in India for Engineering&nbsp;<a href="https://www.nirfindia.org/2021/EngineeringRanking.html" rel="noopener noreferrer" target="_blank">according to NIRF</a>&nbsp;making it one of the most coveted schools for science and technology in the country.

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The very foundation of IIT Hyderabad is based on&nbsp;<a href="https://iith.ac.in/research/">research and innovation</a>. The vibrant research culture is evident from the number of patents and publications that IITH produces consistently every year. IITH offers&nbsp;<a href="https://iith.ac.in/academics/programmes-offered/">graduate programs</a>&nbsp;at both a masters, and a doctoral level, in several diverse areas. There are separate programs for technology, design, science, and liberal arts. The&nbsp;<a href="https://iith.ac.in/academics/post-graduate/">MTech program</a>&nbsp;is offered in various modes and durations to cater to the ever-growing need of postgraduate level professionals in the country.

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IITH creates a unique holistic ecosystem for education that offers interactive learning, a very flexible academic structure, cutting-edge research, strong industry collaboration, and entrepreneurship. This is an environment which enables students and faculty to translate their dreams into realities.

IIT Hyderabad

A Toeplitz operator (Formula presented.), (Formula presented.), is a partial isometry if and only if there exist inner functions (Formula presented.) such that (Formula presented.) and (Formula presented.) depends on different variables and (Formula presented.). In particular, for (Formula presented.), along with new proof, this recovers a classical theorem of Brown and Douglas. We also prove that a partially isometric Toeplitz operator is hyponormal if and only if the corresponding symbol is an inner function in (Formula presented.). Moreover, partially isometric Toeplitz operators are always power partial isometry (following Halmos and Wallen), and hence, up to unitary equivalence, a partially isometric Toeplitz operator with symbol in (Formula presented.), (Formula presented.), is either a shift, or a co-shift, or a direct sum of truncated shifts. Along the way, we prove that (Formula presented.) is a shift whenever (Formula presented.) is inner in (Formula presented.). © 2022 The Authors. The publishing rights in this article are licensed to the London Mathematical Society under an exclusive licence.

Department of Mathematics

World University root node

Bulletin of the London Mathematical Society

Partially isometric Toeplitz operators on the polydisc

We discuss the reflexivity of hyperexpansions and their Cauchy dual operators. In particular, we show that any cyclic completely hyperexpansive operator is reflexive. We also establish the reflexivity of the Cauchy dual of an arbitrary 2-hyperexpansive operator. As a consequence, we deduce the reflexivity of the so-called Bergman-type operator, that is, a leftinvertible operator T satisfying the inequality TT∗+(T∗T)−1≤2IH. © 2021, Element D.O.O.. All rights reserved.

Operators and Matrices

The reflexivity of hyperexpansions and their cauchy dual operators

Tensor product of Fock spaces is analogous to the Hardy space over the unit polydisc. This plays an important role in the development of noncommutative operator theory and function theory in the sense of noncommutative polydomains and noncommutative varieties. In this paper we study joint invariant subspaces of tensor product of full Fock spaces and noncommutative varieties. We also obtain, in particular, by using techniques of noncommutative varieties, a classification of joint invariant subspaces of n-fold tensor products of Drury–Arveson spaces. © 2021, The Author(s), under exclusive licence to Springer Nature Switzerland AG.

Integral Equations and Operator Theory

Submodules in Polydomains and Noncommutative Varieties

This paper concerns a commutant lifting theorem and a Nevanlinna–Pick type interpolation result in the setting of multipliers from vector-valued Drury–Arveson space to a large class of vector-valued reproducing kernel Hilbert spaces over the unit ball in Cn. The special case of reproducing kernel Hilbert spaces includes all natural examples of Hilbert spaces like Hardy space, Bergman space and weighted Bergman spaces over the unit ball. © 2020, Springer Nature Switzerland AG.

Commutant Lifting and Nevanlinna–Pick Interpolation in Several Variables

Given a directed Cartesian product I of locally finite, leafless, rooted directed trees I1, . . ., Id of finite joint branching index, one may associate with I the Drury-Arveson-type C[z1, . . ., zd]-Hilbert module ℋca (I) of vector-valued holomorphic functions on the unit ball Bd in Cd, where a &gt; 0. The main result of this paper classifies all directed Cartesian products I for which the Hilbert modules ℋca (I ) are isomorphic in case a is an integer. Indeed, a careful analysis of these Hilbert modules allows us to prove that the cardinality of generations of I1, . . ., Id are complete invariants for ℋca (·) if ad ≠ 1. © THETA, 2019.

Journal of Operator Theory

Classification of Drury-Arveson-type Hilbert modules associated with certain directed graphs

Let T= (V, E) be a leafless, locally finite rooted directed tree. We associate with T a one parameter family of Dirichlet spaces Hq(q⩾1), which turn out to be Hilbert spaces of vector-valued holomorphic functions defined on the unit disc D in the complex plane. These spaces can be realized as reproducing kernel Hilbert spaces associated with the positive definite kernel (Formula Presented.) where V≺ denotes the set of branching vertices of T, nv denotes the depth of v∈ V in T, and P⟨eroot⟩, Pv(v∈V≺) are certain orthogonal projections. Further, we discuss the question of unitary equivalence of operators Mz(1) and Mz(2) of multiplication by z on Dirichlet spaces Hq associated with directed trees T1 and T2 respectively. © 2017, Springer International Publishing AG.

Dirichlet Spaces Associated with Locally Finite Rooted Directed Trees

We systematically develop the multivariable counterpart of the theory of weighted shifts on rooted directed trees. Capitalizing on the theory of products of directed graphs, we introduce and study the notion of multishifts on directed Cartesian products of rooted directed trees. This framework unifies the theory of weighted shifts on rooted directed trees and that of classical unilateral multishifts. Moreover, this setup brings out some new phenomena such as the appearance of a system of linear equations in the eigenvalue problem for the adjoint of a multishift. In the first half of the paper, we focus our attention mostly on the multivariable spectral theory and function theory including a finer analysis of various joint spectra and the wandering subspace property for multishifts. In the second half, we separate out two special classes of multishifts, which we refer to as torally balanced and spherically balanced multishifts. The classification of these two classes is closely related to toral and spherical polar decompositions of multishifts. Furthermore, we exhibit a family of spherically balanced multishifts on a d-fold directed Cartesian product T of rooted directed trees. These multishifts turn out to be multiplication d-tuples Mz,aon certain reproducing kernel Hilbert spaces ℋaof vector-valued holomorphic functions defined on the unit ball Bdin ℂd, which can be thought of as tree analogs of the multiplication d-tuples acting on the reproducing kernel Hilbert spaces associated with the kernels 1/(1-〈z,w〉)a(z,w ε Bd; a ε N): Indeed, the reproducing kernels associated with ℋaare certain operator linear combinations of 1/(1-〈z,w〉)a and multivariable hypergeometric functions2F1(dυ+ a + 1; 1; dυ+ 2;) defined on Bd× Bd, where dυdenotes the depth of a branching vertex υ in T. We also classify joint subnormal and joint hyponormal multishifts within the class of spherically balanced multishifts. © Instytut Matematyczny PAN, Warszawa 2017.