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Spectral inclusion relations for $ T,\ T\vert Y$ and $ T/Y$

Authors: Kôtarô Tanahashi and Shigeru Yamagami
Journal: Proc. Amer. Math. Soc. 116 (1992), 763-768
MSC: Primary 47A10
MathSciNet review: 1098407
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Abstract: Let $ X$ be a complex Banach space and $ Y$ be an invariant subspace of a bounded linear operator $ T$ in $ X$. Then it is easy to prove that (1) $ \sigma (T) \subset \sigma (T\vert Y) \cup \sigma (T/Y)$, (2) $ \sigma (T\vert Y) \subset \sigma (T) \cup \sigma (T/Y)$, and (3) $ \sigma (T\vert Y) \subset \sigma (T) \cup \sigma (T/Y)$. In this paper, we first study these relations for unbounded linear operators: While (3) holds without any conditions, (1) and (2) do not hold in general. We shall make clear conditions on $ T$ that guarantee (1) and (2). Next we introduce the notion of extended spectrum for unbounded linear operators and prove similar results for the extended spectrum.

References [Enhancements On Off] (What's this?)

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Keywords: Spectrum, extended spectrum
Article copyright: © Copyright 1992 American Mathematical Society

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