## 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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