On the Drazin inverse and generalized Drazin inverse of an operator and its applications

Abstract

This thesis aims to study the Drazin inverse of bounded linear operators in Banach spaces, focusing on its algebraic and spectral properties, as well as its applications in solving differential equations and linear systems. The thesis is divided into four chapters: • Chapter 1: Preliminaries

Covers fundamental concepts such as linear operators, spectrum, ascent and de- scent, and the Riesz projection.

• Chapter 2: Drazin Inverse Defines the Drazin inverse, conditions for its existence, its relation to the spectrum, and special cases. • Chapter 3: Generalized Drazin Inverse Involves operator decomposition, additional spectral properties, and key partial order relations. • Chapter 4: Applications

Uses the Drazin inverse to solve differential equations and demonstrates computa- tional methods with illustrative examples.

The thesis highlights the importance of the Drazin inverse in extending the concept of

invertibility and providing effective analytical tools for linear systems in applied mathe- matics.