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Remark on Lomonosov's lemma


Author: W. E. Longstaff
Journal: Proc. Amer. Math. Soc. 88 (1983), 311-312
MSC: Primary 47A15
DOI: https://doi.org/10.1090/S0002-9939-1983-0695264-3
MathSciNet review: 695264
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Abstract: The famous lemma of V. I. Lomonosov states that if $ \mathfrak{A}$ is a transitive algebra of operators acting on a complex, infinite-dimensional Banach space $ X$ and $ K$ is a nonzero compact operator on $ X$, then there is an $ A \in \mathfrak{A}$ such that 1 is an eigenvalue of AK. Lomonosov's proof uses Schauder's fixed point theorem. A proof, using only elementary techniques, is given for the case where $ K$ has finite-rank.


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DOI: https://doi.org/10.1090/S0002-9939-1983-0695264-3
Article copyright: © Copyright 1983 American Mathematical Society

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