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• W2063972670 abstract "In the process of extending the spectral theory to ever more general linear operators, it was found that all types of operators known to perform a spectral decomposition of the underlying space possess the single valued extension property. It is then natural to ask whether this property is or is not an intrinsic element of the spectral decomposition. In order to answer this question we defined axiomatically the spectral decomposition of a continuous linear operator on a general Banach space. Then we obtained an affirmative answer. A minimal requirement imposed on operators by any spectral theory is the existence of proper invariant subspaces. Assuming this we did not require from the invariant subspaces any specific property, not even the “spectral inclusion property” of the very general v-spaces introduced by Bartle and Kariotis [3]. E. Bishop’s “duality theory of type 3” [4] for continuous linear operators on a reflexive Banach space comes close to the present formulation of the spectral decomposition. Bishop gives some sufficient conditions for an operator to admit a duality theory of type 3 [4], but he does not investigate for the properties of that operator. The spectral decomposition here developed entails some elements of an operational calculus. For T in the Banach algebra B(X) of b ounded linear operators acting on a Banach space X over the complex field C, we shall use the following notations: spectrum u(T), point spectrum u,(T), approximate point spectrum aa( T), local spectrum U(X) with x in X, the resolvent operator R(z; T) with a in the resolvent set p(T), and the local resolvent set p(x). For a set S, So indicates the interior, S the closure, SC the complement (in a given total set), a(S) the boundary, and d(x, S) the distance from a point z to S. N is reserved for the set of all positive integers." @default.
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• W2063972670 date "1978-11-01" @default.
• W2063972670 modified "2023-09-23" @default.
• W2063972670 title "Operators with spectral decomposition properties" @default.
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• W2063972670 doi "https://doi.org/10.1016/0022-247x(78)90266-4" @default.
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