This book presents a wide panorama of methods to investigate the spectral theory of bounded and unbounded block matrices of linear relations. It explains some conditions to prove some Frobenius-Schur decompositions for linear relations and characterizes the stability of the essential spectra of these linear relations. It focuses on the study of the Fredholm theory in both Banach and Hilbert spaces, the local spectral theory of multivalued linear operators and the block matrix of linear relations. As a pioneering literature discussing spectral theory of bounded and unbounded block matrices of linear relations, this book attempts to contribute to the scarce literature on the topic. It gathers the minimum needed background material which allows a relatively friendly access to the book. Detailed proofs of all theorems are exhibited with accuracy and clarity, allowing students to thoroughly familiarize themselves with all the basic concepts.

Autorentext

**Aymen Ammar is Professor at the Department of Mathematics, Faculty of Sciences of Sfax, University of Sfax, Tunisia. He is a co-author of the book, Spectral Theory of Multivalued Linear Operators, with Aref Jeribi. He has published more than 111 paper in several international journals. His areas of interest include spectral theory, matrix operators, transport theory, and linear relations.

Aref Jeribi is Professor in the Department of Mathematics, University of Sfax, Tunisia. He completed his Habilitation of Mathematics and Applications at the University of Sfax, Tunisia, in 2002, and defended his PhD thesis at the University of Corsica Pasquale Paoli, France, in 1998. His research interests include spectral theory, matrix operators, transport theory, Gribov operator, Bargmann space, fixed point theory, Riesz basis and linear relations. He is an author of 13 books, including Problems in Finite Element Methods Aubin Nitsche's Duality Process, Nodal Methods and Friedrichs Systems, (Springer, 2025), Perturbation Theory for Linear Operators: Denseness and Bases with Applications (Springer, 2021) and Spectral Theory and Applications of Linear Operators and Block Operator Matrices (Springer, 2015) and more than 234 research articles published in reputed journals and conference proceedings.