Local analytic extensions of the resolvent
J. D. Gray · Pacific Journal of Mathematics · 1968
Consider an endomorphism T f (that is, a bounded, linear transformation) on a (complex) Banach space X to itself.As usual, let R(λ, T) = (λl -T) 1 be the resolvent of T at λ e p(T).Then it is known that the maximal set of holomorphism of the function λ -> R(λ, T) is the resolvent set p(T).However, it can happen that for some x e X, the X-valued function λ~> R(λ, T)x has analytic extensions into the spectrum σ{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. !• The localization of the spectrum* We begin by making the