Spectral inclusion relations for 𝑇,𝑇|𝑌 and 𝑇/𝑌

Tanahashi, Kôtarô; Yamagami, Shigeru

doi:10.1090/S0002-9939-1992-1098407-0

Spectral inclusion relations for $T,\ T\vert Y$ and $T/Y$
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by Kôtarô Tanahashi and Shigeru Yamagami

Proc. Amer. Math. Soc. 116 (1992), 763-768

DOI: https://doi.org/10.1090/S0002-9939-1992-1098407-0

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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) $\sigma (T) \subset \sigma (T|Y) \cup \sigma (T/Y)$, (2) $\sigma (T|Y) \subset \sigma (T) \cup \sigma (T/Y)$, and (3) $\sigma (T|Y) \subset \sigma (T) \cup \sigma (T/Y)$. 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.

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