Spectral theory for operators on a Banach space

Abstract

0. Introduction. The purpose of this paper is to study the spectral theory of a closed linear transformation T on a reflexive Banach space B. This will be done by means of certain vector-valued measures which are related to the transformation. (A set function m from the Borel sets of the complex plane to B will be called a vector-valued measure if the series EJ= j m (Si) converges to mr(Ui Si) for every sequence { Si I of disjoint Borel sets. The relevant properties of vector-valued measures are briefly derived in ?1(2). A vector-valued measure m will be called a T-measure if Tm(S) =fszdm(z) for all bounded Borel sets S. The properties of T-measures are studied in ?2. The results of ?2 are applied in ?3 to a class of transformations which have been called scalar-type transformations by Dunford [5], and which we call simply scalar transformations. A scalar operator as defined by Dunford is essentially one which admits a representation of the type t = fzdE(z), where E is a spectral measure. Unbounded scalar transformations have been studied by Taylor [16]. The main result of ?3 is Theorem 3.2, in which properties of the closures of certain sets of scalar transformations are derived. This theorem is actually a rather general spectral-type theorem, which has applications to several problems in the theory of linear transformations. As a corollary we obtain a well-known theorem, which might be called the spectral theorem for symmetric transformations, as given in Stone [15]. We also derive as a corollary the spectral theorem for self-adjoint transformations. Other corollaries of Theorem 3.2, which apply to results of Bade [2] and Halmos [9] are derived. In ?4 a functional calculus is developed for a class r of transformations T for which both T-measures and T*-measures exist in sufficient abundance. This is a very general functional calculus, so that correspondingly the usual theorems of functional calculus must be weakened if they are to remain true. There is a generalization of the concept of a T-measure introduced in ?5. Many theorems proved in ?2 have analogues which hold after the generalization. The new type of vector-valued measures have much the same relation