# SPECTRAL THEORY OF ABSOLUTELY MINIMUM ATTAINING POSITIVE OPERATORS

Jadav, Ganesh and G, Ramesh (2018) SPECTRAL THEORY OF ABSOLUTELY MINIMUM ATTAINING POSITIVE OPERATORS. PhD thesis, Indian institute of technology Hyderabad.

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## Abstract

In this thesis our primary goal is to study the structure of absolutely minimum attaining operators. First we begin with studying spectral properties of absolutely minimum attaining positive operators and with the help of them we prove a spectral theorem for this class. Using the polar decomposition theorem we try to give a structure for general absolutely minimum attaining operators. Apart from this we also consider the minimum attaining operators and investigate for their perturbation properties. This thesis contains three chapters. In Chapter 1, we discuss about the class of minimum attaining operators and some of their basic properties. Using this we define absolutely minimum attaining operators, discuss some examples and list out some important basic properties of this class. We motivate our study of the structure of absolutely minimum attaining positive operators by the classical spectral theory of compact operators. We record some of the basic results and terminology from operator theory which will be useful for the further chapters. In Chapter 2, we study the spectral properties of absolutely minimum attaining positive operators defined on infinite dimensional complex Hilbert spaces. Using this we derive a spectral theorem for this class. We construct several examples and establish some important basic properties of this class such as the closed range property and finite dimensionality of the null space or the range space etc. Moreover, with the help of the polar decomposition theorem we give a possible structure for absolutely minimum attaining operators. Chapter 3 deals with the perturbation properties of minimum attaining operators. First we focus on the compact perturbations and prove that the minimum attaining property of a bounded operator whose minimum modulus lies in the discrete spectrum is stable under small compact perturbations. We 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 a very small set, in fact a porous set in the ideal of all compact operators on the given Hilbert space. Finally, we discuss the stability of minimum attaining viii property under perturbations by all bounded operators with small norm and obtain related results. At the end of the chapter we list a few problems based on our work.

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IITH Creators:
IITH CreatorsORCiD
G, RameshUNSPECIFIED
Item Type: Thesis (PhD)
Uncontrolled Keywords: minimum modulus, essential minimum modulus, minimum attaining operator, absolutely minimum attaining operator, diagonalizable operator, compact operator, spectral theorem, spectrum, essential spectrum, compact perturbation.
Subjects: Mathematics
Divisions: Department of Mathematics
Depositing User: Team Library
Date Deposited: 16 May 2019 09:18