Book Review: Analytic functional calculus and spectral decompositions
Raúl E. Curto · Bulletin of the American Mathematical Society · 1986
A linear transformation T acting on a finite-dimensional complex vector space SC can always be decomposed as T = D + N, where (i) D is diagonalizable and N is nilpotent; and (ii) DN = ND\ moreover, such a decomposition is unique with respect to the conditions (i) and (ii), and both D and N are indeed polynomials in T. When % is an infinite-dimensional Banach space, such a representation for a bounded operator T is no longer true, but an important class of transformations introduced and studied by N. Dunford [3] in the 1950s possesses a similar property.By definition, a spectral operator T acting on 3C is one for which there exists a spectral measure E (i.e., a homomorphism from the Boolean algebra of Borel subsets of the complex plane C into the Boolean algebra of projection operators on & such that E is bounded and E(C) = /) satisfying the following two properties: (1) TE(B) = E(B)T\ and (2) a(T\ E(B)sr ) c B, for all B (Borel) c C. Such an E is called a resolution of the identity for T, and is