Spectral Theory and Applications of Linear Operators and Block Operator Matrices Examining recent mathematical developments in the study of Fredholm operators spectral theory and block operator matrices with a rigorous treatment of classical Riesz theory of polynomially compact operators this volume covers both abstract and applied developments in the study of spectral theory. These topics are intimately related to the stability of underlying physical systems and play a crucial role in many branches of mathematics as well as numerous interdisciplinary applications. By studying classical Riesz theory of polynomially compact operators in order to establish the existence results of the second kind operator equations this volume will assist the reader working to describe the spectrum multiplicities and localization of the eigenvalues of polynomially compact operators Introduction Fredholm Operators and Riesz Theory Abstract Cauchy Problem Fredholm Theory Related to Some Measures Pertubation Results Essential Spectra of Linear Operators Essentia Pseudo spectra S Essential Spectra Essential Spectra of 2 X 2 Block Operator Matrices Essential Spectra of 3 X 3 Block Operator Matrices Applications in Mathematical Physics and Biology