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- W2065565587 abstract "In this note we characterize certain types of spectral decomposition in terms of “universal” notions valid for any operator on a Banach space. To be precise, let X be a complex Banach space and let T be a bounded linear operator on X . If F is a closed set in the plane C, let X ( T, F ) consist of all y ∈ X satisfying thes identity where f :C F → X is analytic. It is then easy to see that X ( T, F ) is a T -invariant linear manifold in X . Moreover, if y ∈ X then is a compact subset of the spectrum σ( T ). Our aim is to give necessary and sufficient conditions for a decomposable or strongly decomposable operator in terms of X ( T, F ) and γ( y, T ). Recall that T is decomposable if whenever G 1 G 2 are open and cover C there exist T -invariant closed linear manifolds M 1 , M 2 with X = M 1 + M 2 and σ( T | M 1 ) ⊂ G i ( i = 1,2) (equivalently, σ( T | M i )⊂ Ḡ i , see [4, p. 57]). In this case, X ( T, F ) is norm closed if F is closed and each y in X has a unique maximally defined local resolvent satisfying (1.1) on C F y ; F y is called the local spectrum σ( y, T ) and coincides with γ( y, T ). Hence T has the single valued extension property (SVEP); i.e., zero is the only analytic function f : V → X satisfying ( z − T ) f(z) = 0 on V . If T is decomposable and the restriction T | X ( T, F ) is also decomposable for each closed F , then T is called strongly decomposable . We point out that Albrecht [2] has shown by example that not every decomposable operator is strongly decomposable, while Eschmeier [6]has given a simpler construction to show that this phenomenon occurs even in Hilbert space." @default.
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- W2065565587 date "1992-01-01" @default.
- W2065565587 modified "2023-10-18" @default.
- W2065565587 title "Universal notions characterizing spectral decompositions" @default.
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- W2065565587 doi "https://doi.org/10.1017/s0017089500008594" @default.
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