Local analytic extensions of the resolvent

Abstract

Consider an endomorphism Tf (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 ∈ ρ(T). Then it is known that the maximal set of holomorphism of the function λ → R(λ, T) is the resolvent set ρ(T). However, it can happen that for some x ∈ 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. © 1968 by Pacific Journal of Mathematics.