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

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

DOI: https://doi.org/10.1090/S0002-9939-1983-0695264-3

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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.

References

C. K. Fong, E. A. Nordgren, M. Radjabalipour, H. Radjavi, and P. Rosenthal, Extensions of Lomonosov’s invariant subspace theorem, Acta Sci. Math. (Szeged) 41 (1979), no. 1-2, 55–62. MR 534499
C. Laurie, E. Nordgren, H. Radjavi, and P. Rosenthal, On triangularization of algebras of operators, J. Reine Angew. Math. 327 (1981), 143–155. MR 631313, DOI 10.1016/S0024-3795(01)00288-9
V. I. Lomonosov, Invariant subspaces of the family of operators that commute with a completely continuous operator, Funkcional. Anal. i Priložen. 7 (1973), no. 3, 55–56 (Russian). MR 0420305
A. J. Michaels, Hilden’s simple proof of Lomonosov’s invariant subspace theorem, Adv. Math. 25 (1977), no. 1, 56–58. MR 500214, DOI 10.1016/0001-8708(77)90089-5
Eric Nordgren, Heydar Radjavi, and Peter Rosenthal, Operator algebras leaving compact operator ranges invariant, Michigan Math. J. 23 (1976), no. 4, 375–377 (1977). MR 458200
Carl Pearcy and Allen L. Shields, A survey of the Lomonosov technique in the theory of invariant subspaces, Topics in operator theory, Math. Surveys, No. 13, Amer. Math. Soc., Providence, R.I., 1974, pp. 219–229. MR 0355639
Heydar Radjavi and Peter Rosenthal, Invariant subspaces, Ergebnisse der Mathematik und ihrer Grenzgebiete, Band 77, Springer-Verlag, New York-Heidelberg, 1973. MR 0367682, DOI 10.1007/978-3-642-65574-6
Peter Rosenthal, Applications of Lomonosov’s lemma to non-self-adjoint operator algebras, Proc. Roy. Irish Acad. Sect. A 74 (1974), 271–281. Spectral Theory Symposium (Trinity College, Dublin, 1974). MR 361820