Algebraic multiplicity of eigenvalues of linear operators
This book has three parts. Part~1 consists of three chapters and concerns the basics of finite-dimensional spectral theory. The exposition is clear and suitable even for an undergraduate audience. The main body of the book is Part~2 which consists of eight chapters and concerns a generalized concept of algebraic multiplicity. Part~3 is an illustration of some basic results on the topological degree and is aimed at illustrating some application to nonlinear problems of the results of Part~2. We now briefly illustrate the content of the Chapters 4--10 of Part~2. Chapter~4 introduces a concept of algebraic multiplicity \(\chi[{\mathfrak L},\lambda_{0}]\) of a family \({\mathfrak L}\in C^{r}(\Omega, {\mathcal{L}}(U,V))\) at any algebraic eigenvalue \(\lambda_{0}\in \Omega\). Here, \(\Omega\) denotes a connected subset of the real or complex numbers, \(U\) and \(V\) are Banach spaces, and \({\mathcal{L}}(U,V)\) denotes the space of bounded linear operators of \(U\) to \(V\). Chapter~5 describes an equivalent approach to the concept of multiplicity of Chapter~4 and analyzes the relations between the two types of multiplicity. In Chapter~6, the authors select some fundamental properties of the algebraic multiplicity which essentially determine it. In Chapter~7, the authors introduce the concept of algebraic multiplicity by exploiting the theory of Jordan chains, which extend the classical concept of chains of generalized eigenvectors. Such a concept of algebraic multiplicity is then shown to coincide with those introduced in Chapters~4 and 5. Here, the authors also introduce the concept of local Smith forms for a family \({\mathfrak L}\) at an eigenvalue \(\lambda_{0}\), which is a canonical form obtained by multiplying \({\mathfrak L}\) on the left and on the right by isomorphisms, and show that the existence of such a form at \(\lambda_{0}\) is equivalent for \(\lambda_{0}\) to be an algebraic eigenvalue. The concept of local Smith form has then been employed to analyze when two families as \({\mathfrak L}\) can be obtained one from the other by multiplication on the left and on the right by isomorphisms. Although part of the material is classical (the concept of local Smith form has been around since the 19th century), the exposition is due to the authors of the book. Chapter~8 considers some basic properties of analytic families of operators and then concentrates on the classical families of the form \(\lambda I-A\) for some bounded operator \(A\). Here, the purpose is to show that the concepts of algebraic multiplicity introduced in the previous chapters actually generalize the classical ones for \({\mathfrak L}(\lambda)=\lambda I-A\). The chapter ends with an analysis of the stability of the multiplicity of holomorphic families. In Chapter~9, the authors prove an asymptotic expansion for families \({\mathfrak L}\) of class \(C^{r}\) around an algebraic eigenvalue \(\lambda_{0}\) of a certain multiplicity \(\nu\leq r\), which is in the form of a finite Laurent expansion with operators of finite rank as coefficients, and show that under suitable assumptions the trace of the coefficient of order \(-1\) coincides with the multiplicity of \(\lambda_{0}\). Such a formula extends the classical one for holomorphic families in the finite-dimensional case. Chapter~10 is devoted to families of operators acting in \({\mathbb{C}}^{n}\) which are in the form of polynomials in \(\lambda\) with \(n\times n\)-matrix coefficients, and presents a theorem which can be considered a generalization of the Jordan Theorem. Almost all chapters have consistent sections of exercises. The body of references is overall certainly adequate. Concerning Part~3 however, one would have appreciated additional references to some recent contributions to nonlinear spectral theory such as those of the monograph of [\textit{J.\,Appell, E.\,De Pascale} and \textit{A.\,Vignoli}, ``Nonlinear spectral theory'' (de Gruyter Series in Nonlinear Analysis and Applications 10; Berlin:\ Walter de Gruyter) (2004; Zbl 1056.47001)]. The opinion of the reviewer is that the book under review is a good book.
real analytic and holomorphic families of operators
scientific article; zbMATH DE number 5162723