Vol. 27, No. 2, 1968

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Local analytic extensions of the resolvent

Jack D. Gray

Vol. 27 (1968), No. 2, 305–324

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.

Mathematical Subject Classification
Primary: 47.30
Received: 21 March 1967
Published: 1 November 1968
Jack D. Gray