Operator functions and the study of the spectrum. Paper from the 29th Brazilian mathematics colloquium -- 29\(^{\text o}\) Colóquio Brasileiro de Matemática, Rio de Janeiro, Brazil, July 22 -- August 2, 2013 (Q2848162)

The purpose of the monograph under review is to study certain decomposition theorems for operators in finite-dimensional or infinite-dimensional spaces. After recalling the Jordan canonical form (real and complex cases) in the first chapter, the author devotes the second chapter to some preliminaries which are necessary for the comprehension of the text: the fixed point theorem, normed spaces of continuous linear mappings, and analytic functions from open subsets of the complex plane into Banach spaces (Cauchy-Goursat theorem, Cauchy integral formula, and a few consequences).NEWLINENEWLINE The third and main chapter, which depends strongly on results about analytic functions discussed in the second chapter, is devoted to operator functions. Among the main results, one may mention an invariant decomposition theorem for operators in Banach spaces and a linear version of the stable manifold theorem for hyperbolic isomorphisms. Moreover, as an application of previous results encountered in the text, a program for the calculus of generalized eigenspaces is furnished.NEWLINENEWLINE In the fourth chapter, which contains basic results on Hilbert spaces (Riesz representation theorem, adjoint operators), a generalized version of J. von Neumann's ergodic theorem is established.NEWLINENEWLINE Finally, in the fifth chapter, a few applications to the study of isolated periodic points of operator functions are presented. The open mapping theorem and the Hahn-Banach theorem are assumed to be known and used in the text.