Spectral Theory for Linear Operators

This book focuses on spectral theory for linear operators involving bounded or unbounded demicompact linear operators acting on Banach spaces. This class played an important rule in the theory of perturbation. More precisely, it contributed in the construction of several classes of stability of essential spectra for bounded or unbounded linear operators. We should emphasize that this book is the first one dealing with the demicompactness concept and its relation with Fredholm theory for bounded and unbounded linear operators as well as block operator matrices acting on Banach spaces. Researchers, as well as graduate students in applicable analysis, will find that this book constitutes a useful survey of the fundamental principles of the subject. Nevertheless, the reader is assumed to be, at least, familiar with some related sections concerning notions like the compact, Fredholm operators, the basic tools of the weak topology, the concept of measures of weak noncompactness, etc. Otherwise, the reader is urged to consult the recommended literature in order to benefit fully from this book.

Features -

• First book dealing with demicompactness theory and its relation with Fredholm theory for bounded and unbounded linear operators as well as block operator matrices acting on Banach spaces.

• Self-contained coverage of classical and more recent classes of perturbations involving the concept of demicompactness.

• Offers a useful survey of the fundamental principles of spectral theory.

• Provides applications for problem arising in physics and which are modeled by integral or partial differential equations.

Bilel Krichen is a Professor in the department of mathematics at the University of Sfax, Tunisia. He completed both his PhD in Mathematics in 2011 and his Habilitation in Mathematics in 2016 at the same institution. His research focuses on key areas of functional analysis, including the interplay between operator theory and fixed point theory, with applications to transport equations and other mathematical models. Prof. Dr. Krichen has been an active member of the mathematical community, notably serving as the vice president of the Tunisian Association of Applied Mathematics and Industry (ATMAI) from 2019 to 2022. His involvement with the association began in 2015, reflecting his commitment to fostering the development and application of mathematical sciences in Tunisia and beyond. A prolific scholar, Prof. Krichen has authored 88 research papers and supervised 9 graduate theses, mentoring the next generation of mathematicians. He is the author of the book Spectral Properties and Fixed Point Theory for Block Operator Matrix: Fixed Point Theory, Spectral Properties for Block Operator Matrix and Applications to Transport Equation (Scholar's Press 2014) and he is the co-author of the monograph Nonlinear Functional Analysis in Banach Spaces and Banach Algebras: Fixed Point Theory under Weak Topology for Nonlinear Operators and Block Operator Matrices with Applications (Monographs and Research Notes in Mathematics, CRC Press Taylor and Francis, Boca Raton, 2015).