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On the defect spectrum of an extension of a Banach space operator. (English) Zbl 0954.47002

Let \(T\) be a bounded linear operator on a Banach space \(X\). Given any extension of \(T\) to a Banach space \(Y\supset X\), the approximate point spectrum \(\sigma _{\pi } (T)\) is contained in the spectrum \(\sigma (S)\) and it is known that there exists an extension \(S\) of \(T\) to a suitable \(Y\supset X\) such that \(\sigma (S) = \sigma _{\pi }(T)\). The following three related problems concerning the defect spectrum of extensions are considered:
(1) characterize the sets \(F\) for which \(F=\sigma _{\delta }(S)\) for some extension \(S\) of \(T\) under the condition \(\sigma _{\delta }(S)\subset \sigma _{\delta }(T)\);
(2) the same problem under the condition \(\sigma (S) \subset \sigma (T)\);
(3) characterize pairs of sets \(F_1\) and \(F_2\) for which there exists an extension \(S\) of \(T\) such that \(\sigma _{\pi }(S)=F_1\) and \(\sigma _{\delta }(S) =F_2\).
Complete characterizations are given for all three problems.
Reviewer: P.Pták (Praha)

MSC:

47A10 Spectrum, resolvent
47A20 Dilations, extensions, compressions of linear operators
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References:

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