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Universal notions characterizing spectral decompositions

Published online by Cambridge University Press:  18 May 2009

Ridgley Lange  and
Shengwang Wang
Ridgley Lange
Affiliation:
Department of Mathematics, Hampton University, Hampton, Virginia 23668, U.S.A.
Shengwang Wang
Affiliation:
Department of Mathematics, Nanjing University, Nanjing, Jiangsu 210008, Peoples Republic of China
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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 G1G2 are open and cover C there exist T-invariant closed linear manifolds M1, M2 with X= M1 + M2 and σ(T | M1) ⊂ Gi(i = 1,2) (equivalently, σ(T | Mi)⊂ Ḡi, see [4, p. 57]). In this case, X(T, F) is norm closed if Fis closed and each y in X has a unique maximally defined local resolvent satisfying (1.1) on C\Fy; Fy 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.

Type
Research Article
Information
Glasgow Mathematical Journal , Volume 34 , Issue 1 , January 1992 , pp. 109 - 116
Copyright
Copyright © Glasgow Mathematical Journal Trust 1992

References

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