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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
DOI: https://doi.org/10.1090/S0002-9939-1993-1128728-5
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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Article copyright: © Copyright 1993 American Mathematical Society