Résumé:
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.
