Jadav, Ganesh and G, Ramesh
(2018)
SPECTRAL THEORY OF ABSOLUTELY
MINIMUM ATTAINING POSITIVE
OPERATORS.
PhD thesis, Indian institute of technology Hyderabad.
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.
[error in script]
| IITH Creators: |
| IITH Creators | ORCiD |
|---|
| G, Ramesh | UNSPECIFIED |
|
| 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 |
| Last Modified: |
16 May 2019 09:20 |
| URI: |
http://raiithold.iith.ac.in/id/eprint/5193 |
| Publisher URL: |
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| Related URLs: |
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