Spectral Theory of Linear Operators—and Spectral Systems in Banach Algebras, By

Let A be a bounded operator on a Banach space X. A scalar λ is in the spectrum of A if the operator A − λ is not invertible. Case closed. What more is there to say? As anyone with the slightest exposure to operator theory will testify, there is so much out there that no book could come close to being comprehensive. What authors do in such situations is choose a small area or topic of interest to the author and concentrate on that. The present book is no exception. The spectral theory of operators has its roots in the theory of matrices and in the theory of integral equations. In the early years of matrix theory, the terms “proper value”, “characteristic value”, “secular value”, and “latent root” were all used for what we now call an eigenvalue. Laguerre constructed the exponential function of a matrix, and Frobenius obtained expansions for the resolvent operator in the neighborhood of a pole. Sylvester constructed arbitrary functions of a matrix with distinct eigenvalues. This was generalized by Buchheim to the case of multiple eigenvalues. It was F. Reisz who extended these concepts to the space l. Dealing with compact operators on this space, he showed that the resolvent set is open, that the resolvent operator is analytic, and that the Cauchy integral theorem can be used in the case of a pole to obtain a projection operator commuting with the given operator. Wiener showed that Cauchy’s integral theorem and Taylor’s theorem remain valid for analytic functions with values in a complex Banach space. Nagumo extended some of the results of F. Riesz to Banach algebras. Hille applied similar ideas in the study of semi-groups. Gelfand developed the ideal theory of Banach algebras. He used the contour integral to obtain idempotents. The spectral mapping theory is due to Dunford. He introduced the concepts of continuous and residual spectrum. The concepts of ascent and decent of a bounded operator are also due to him. Fredholm studied integral equations. He gave a detailed representation of the resolvent as the quotient of two entire functions in terms of expansions in determinants. Schmidt used the method of approximating a compact operator by operators of finite rank in Hilbert space. Considerable work has been done by many authors concerning the computation and distribution of eigenvalues. The present monograph is an attempt to organize recent progress in certain areas of spectral theory. The aim is to present a survey of results concerning various types of spectra in a unified, axiomatic way. The setting is a Banach algebra A, and the generalized spectrum is defined to be σR(a) = {λ ∈ C : a − λ / ∈ R}, where R is a set of “nice” elements. He defines three sets of “nice” elements. Definition 1. Let A be a Banach algebra. A non-empty subset R of A is called a “regularity” if it satisfies the following conditions: (i) if a ∈ A and n ∈ N, then a ∈ R iff a ∈ R;