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Consider an endomorphism T,
(that is, a bounded, linear transformation) on a (complex) Banach space X to itself.
As usual, let R(λ,T) = (λ1 − 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.
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