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.

IITH offers&nbsp;<a href="https://iith.ac.in/academics/under-graduate/">undergraduate programs</a>&nbsp;in all the classical engineering disciplines, applied sciences, design, as well as several modern interdisciplinary areas. Students are given a flexibility to explore a broad set of areas, and potentially pursue a minor or double major in a discipline that is not their own. Students who wish to seek a deeper understanding of their own discipline are strongly encouraged to get involved in cutting-edge research with the help of a faculty to mentor them, and earn an Honors in their own field.

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.

IITH encourages and supports&nbsp;<a href="https://iic-iith.netlify.app/index.html" rel="noopener noreferrer" target="_blank">innovation and entrepreneurship</a>&nbsp;at every stage. The minor program in entrepreneurship is open to all students of IITH, and many of the courses are offered by industrialists who are entrepreneurs themselves. There are various&nbsp;<a href="https://iith.ac.in/research/centres-incubators/">innovation centers and incubators</a>&nbsp;that promote entrepreneurship and help mentor young innovators. IITH has been very successful in building tie-ups with leading academic institutions around the globe.

IITH enjoys a very special relationship with Japanese universities and industries that goes beyond academic and research collaborations. In fact, some of the iconic buildings in IITH campus will carry the signature of Japanese architecture.

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

Let A be a complex unital Banach algebra and M be a left A-module. Let Ʌ: M→ℂ be a map that is not necessarily linear. We establish conditions for Ʌ to be linear and of multiplicative kind, from its behavior on a small subset of M. We do not assume Ʌ to be continuous throughout. As an application, we give a characterization of weighted composition operators on the Hardy space H. © Instytut Matematyczny PAN. 2021.

World University root node

Department of Mathematics

Colloquium Mathematicum

A weaker gleason-kahane-zelazko theorem for modules and applications to hardy spaces

Conditionspectrum measures the computational stability of solving a linear system. In this paper, ten comparative results involving ε-conditionspectrum are presented. All these theorems generalize a well known eigenvalue theorem and simultaneously compare with an appropriate pseudospectra result. © 2019, Forum D'Analystes, Chennai.

Fulltext

Journal of Analysis

Some comparative results on eigenvalues, pseudospectra and conditionspectra

Let A be a complex Banach algebra with unit e. Let p be a non trivial idempotent element in A and ε> 0. For a∈ A, it is proved that the interior of the level set of (p, e- p) - ε pseudo spectrum of a is empty in the unbounded component of (p, e- p) resolvent set of a. An example is constructed to show that the condition ‘unbounded component’ can not be dropped. Further, it is proved this ‘unbounded component’ can be dropped in the case when A is B(X) where X is a complex uniformly convex Banach space. That is, if T∈ B(X) then interior of the level set of (p, I- p) - ε pseudo spectrum is empty in (p, I- p) resolvent set of T. © 2017, Forum D'Analystes, Chennai.

Level sets of (p, e- p) outer generalized pseudo spectrum

Let A be a complex unital Banach algebra, let be an element in it and let 0 < ϵ < 1.. In this article, we study the upper and lower hemicontinuity and joint continuity of the condition spectrum and its level set maps in appropriate settings. We emphasize that the empty interior of the -level set of a condition spectrum at a given plays a pivotal role in the continuity of the required maps at that point. Further, uniform continuity of the condition spectrum map is obtained in the domain of normal matrices. © 2019 Australian Mathematical Publishing Association Inc.

Journal of the Australian Mathematical Society

CONTINUITY of A CONDITION SPECTRUM and ITS LEVEL SETS

In this article, we consider the Schrödinger semigroup related to the Dunkl–Laplacian Δ μ (associated to finite reflection group G) on R n . We characterize the image of L2(Rn,eu2hμ(u)du) under the Schrödinger semigroup as a reproducing kernel Hilbert space. We define Dunkl–Sobolev space in L2(Rn,eu2hμ(u)du) and characterize it’s image under the Schrödinger semigroup associated to G=Z2n as a reproducing kernel Hilbert space up to equivalence of norms. Also we provide similar results for Schrödinger semigroup associated to Dunkl–Hermite operator. © 2017, Springer International Publishing AG, part of Springer Nature.

Journal of Pseudo-Differential Operators and Applications

On the images of Dunkl–Sobolev spaces under the Schrödinger semigroup associated to Dunkl operators

In this article, we consider the Schrödinger semigroup for the Laplacian Δ on Rn, and characterize the image of a Sobolev space in L2(Rn,eu2du) under this semigroup as weighted Bergman space (up to equivalence of norms). Also we have a similar characterization for Hermite Sobolev spaces under the Schrödinger semigroup associated to the Hermite operator H on Rn. © 2018 Walter de Gruyter GmbH, Berlin/Boston.

Advances in Pure and Applied Mathematics

On the images of Sobolev spaces under the Schrödinger semigroup

In this article, we construct a commutative unital Banach algebra, in which the property ‖a2‖=‖a‖2 is true for the invertible elements, but cannot be extended to the whole algebra. © 2018 Académie des sciences

Comptes Rendus Mathematique

A characterizing property of commutative Banach algebras may not be sufficient only on the invertible elements [Une propriété caractérisant des algèbres de Banach commutatives qui ne peut être formulée sur les seuls éléments inversibles]

We study the spectral properties of positive absolutely minimum attaining operators defined on infinite dimensional complex Hilbert spaces and using this we derive a characterization theorem for such type of operators. We construct several examples and discuss some of the properties of this class. Also, we extend this characterization theorem for general absolutely minimum attaining operators by means of the polar decomposition theorem. © 2018 Elsevier Inc.

Journal of Mathematical Analysis and Applications

A characterization of absolutely minimum attaining operators

Let A be a complex unital Banach algebra. Since the set of invertible elements is open, there is an open ball around every invertible element. In this article, we investigate the Banach algebras for which the radius given by the Neumann series is optimal. &copy; 2018, Element D.O.O. All rights reserved.

Operators and Matrices

On the open ball centered at an invertible element of a banach Algebra

We prove that the minimum attaining property of a bounded linear operator on a Hilbert space H whose minimum modulus lies in the discrete spectrum, is stable under small compact perturbations. We also observe that given a bounded operator with strictly positive essential minimum modulus, the set of compact perturbations which fail to produce a minimum attaining operator is smaller than a nowhere dense set. In fact, it is a porous set in the ideal of all compact operators on H. Further, we try to extend these stability results to perturbations by all bounded linear operators with small norm and obtain subsequent results. © 2016 by the Tusi Mathematical Research Group.

Advances in Operator Theory

Perturbation of minimum attaining operators

For 0 < ε ≤ 1 and an element a of a complex unital Banach algebra A, we prove the following two topological properties about the level sets of the condition spectrum. (1) If ε = 1, then the 1-level set of the condition spectrum of a has an empty interior unless a is a scalar multiple of the unity. (2) If 0 < ε < 1, then the ε-level set of the condition spectrum of a has an empty interior in the unbounded component of the resolvent set of a. Further, we show that, if the Banach space X is complex uniformly convex or if X* is complex uniformly convex, then, for any operator T acting on X, the level set of the ε-condition spectrum of T has an empty interior. © 2017 by the Tusi Mathematical Research Group.

Annals of Functional Analysis

Level sets of the condition spectrum

In this article we prove a characterization theorem for positive absolutely minimum attaining operators of first type. Using this we derive a representation theorem for general absolutely minimum attaining operators of first type, which is similar to that of the singular value decomposition for compact operators. We also study their spectrum and discuss about several properties of such operators. © 2015 Elsevier Inc.

On the structure of absolutely minimum attaining operators

Let A be a complex commutative Banach algebra with unit 1 and δ &gt; 0. A linear map φ: A →C is said to be δ-almost multiplicative if |φ(ab) - φ (a) φ (b)| ≤ δ||a|| ||b|| for all a, b ε A. Let 0 &lt; e &lt; 1. The e-condition spectrum of an element a in A is defined by δe.(a) := {λ ε C:||λ-a||||λ- a||-1 ||≥ 1/e} with the convention that ||λ- a|| ||(λ - a)-1|| = ∞ when λ - a is not invertible. We prove the following results connecting these two notions: (1) If φ(1) = 1 and φ is δ-almost multiplicative, then φ(a) ε δe(a) for all a in A.then (2) If φis lenear and φ(a) εδe(a) for all a in A ,then φ-is δ almost multiplicative for some δ. The first result is analogous to the Gelfand theory and the last result is analogous to the classical Gleason-Kahane- Żelazko theorem.

Studia Mathematica

Almost multiplicative functions on commutative Banach algebras

We prove by elementary methods the following generalization of a theorem due to Gleason, Kahane, and Żelazko. Let A be a real algebra with unit 1 such that the spectrum of every element in A is bounded and let φ: A → ℂ be a linear map such that φ(1) = 1 and (φ(a))2 + (φ(b))2 ≠ 0 for all a, b in A satisfying ab = ba and a2 + b2 is invertible. Then φ(ab) = φ(a)φ(b) for all a, b in A. Similar results are proved for real and complex algebras using Ransford's concept of generalized spectrum. With these ideas, a sufficient condition for a linear transformation to be multiplicative is established in terms of generalized spectrum. Copyright © 2005 Hindawi Publishing Corporation.