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.

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