Spectral Theory for Linear Operators: Demicompactness and Perturbation Theory

by Bilel Krichen

Overview

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.

Product Details

ISBN-13:	9781032936352
Publisher:	CRC Press
Publication date:	07/31/2025
Series:	Chapman & Hall/CRC Monographs and Research Notes in Mathematics
Pages:	344
Product dimensions:	6.12(w) x 9.19(h) x (d)

About the Author

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). 

Table of Contents

1 Basic Tools of Functional Analysis 2 Closed, Bounded, Compact and Weakly Compact Operators 3 Measures of Noncompactness MNCs 4 Fredholm and Semi-Fredholm Operators 5 Demicompact Linear Operators 6 Demicompact Nonlinear Operators 7 Demicompactness of Polynomial and Operator Matrices 8 Demicompactness and Fredholm Theory 9 Description of Essential Spectra Under Demicompactness 10 Fredholm Operator Matrices Under Demicompactness 11 Relatively Pseudo Weakly Demicompact Operators 12 Some Results on B-Fredholm and Demicompact Operators 13 Demicompactness and Restrictions 14 Generalized Weak Demicompactness and MNSS 15 Applications to Mathematical Physics Problems