Remark on Lomonosov’s lemma

Longstaff, W. E.

doi:10.1090/S0002-9939-1983-0695264-3

Remark on Lomonosov’s lemma
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by W. E. Longstaff PDF

Proc. Amer. Math. Soc. 88 (1983), 311-312 Request permission

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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