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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

The essential spectral radius of dominated positive operators


Author: Josep Martínez
Journal: Proc. Amer. Math. Soc. 118 (1993), 419-426
MSC: Primary 47B65; Secondary 47A10, 47A53
MathSciNet review: 1128728
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Abstract: Let $ E$ be an $ {\operatorname{AL} _p}$-space with $ 1 \leqslant p \leqslant \infty $. We prove that if a positive operator $ S \in \mathcal{L}(E)$ satisfies the Doeblin conditions and $ r(S) \leqslant 1$, then $ S$ is quasi-compact, i.e., $ {r_{\operatorname{ess} }}(S) < 1$. We then deduce the following result about the monotonicity of the essential spectral radius: Let $ S,\;T \in \mathcal{L}(E)$ be such that $ 0 \leqslant S \leqslant T$. If $ r(S) \leqslant 1$ and $ {r_{\operatorname{ess} }}(T) < 1$, then $ {r_{\operatorname{ess} }}(S) < 1$.


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DOI: http://dx.doi.org/10.1090/S0002-9939-1993-1128728-5
PII: S 0002-9939(1993)1128728-5
Article copyright: © Copyright 1993 American Mathematical Society