This book analyzes the existence and uniqueness of a generalized algebraic m- tiplicity for a general one-parameter family L of bounded linear operators with Fredholm index zero at a value of the parameter ? whereL(? ) is non-invertible. 0 0 Precisely, given K?{R,C}, two Banach spaces U and V over K, an open subset ? ? K,andapoint ? ? ?, our admissible operator families are the maps 0 r L?C (? ,L(U,V)) (1) for some r? N, such that L(? )? Fred (U,V); 0 0 hereL(U,V) stands for the space of linear continuous operatorsfrom U to V,and Fred (U,V) is its subset consisting of all Fredholm operators of index zero. From 0 the point of view of its novelty, the main achievements of this book are reached in case K = R, since in the case K = C and r = 1, most of its contents are classic, except for the axiomatization theorem of the multiplicity.

This book brings together all the most important known results of research into the theory of algebraic multiplicities, from well-known classics like the Jordan Theorem to recent developments such as the uniqueness theorem and the construction of multiplicity for non-analytic families, which is presented in this monograph for the first time.

Part I (the first three chapters) is a classic course on finite-dimensional spectral theory; Part II (the next eight chapters) contains the most general results available about the existence and uniqueness of algebraic multiplicities for real non-analytic operator matrices and families; and Part III (the last chapter) transfers these results from linear to nonlinear analysis.

The text is as self-contained as possible. All the results are established in a finite-dimensional setting, if necessary. Furthermore, the structure and style of the book make it easy to access some of the most important and recent developments. Thus the material appeals to a broad audience, ranging from advanced undergraduates (in particular Part I) to graduates, postgraduates and reseachers who will enjoy the latest developments in the real non-analytic case (Part II).