Local analytic extensions of the resolvent

Abstract

Consider an endomorphism T f (that is, a bounded, linear transformation) on a (complex) Banach space X to itself. As usual, let R(, T) = (l -T) 1 be the resolvent of T at e p(T). Then it is known that the maximal set of holomorphism of the function -> R(, T) is the resolvent set p(T). However, it can happen that for some x e X, the X-valued function ~> R(, T)x has analytic extensions into the spectrum {T) of T. Using this fact we shall, in 1, localize the concept of the spectrum of an operator. In sections 2, 3 and 4 we investigate, quite thoroughly, the structural properties of this concept. Finally, in 5, the results of the previous sections will be utilized to construct a local operational calculus which will then be applied to the study of abstract functional equations.