Lomonosov's theorem cannot be extended to chains of four operators

1998/09/18 by Vladimir G. Troitsky, Troitsky, Vladimir G.
Mathematics · #47A15 #FOS: Mathematics #Functional Analysis (math.FA) #math.FA #msc:47A15

paper · pdf · doi:10.48550/arxiv.math/9809100

5 pages, to appear in Proceedings of the AMS

arxiv created 1998/09/18 · arxiv updated 2009/11/30

Abstract

We show that the celebrated Lomonosov theorem cannot be improved by increasing the number of commuting operators. Specifically, we prove that if T is the operator on l₁ without a non-trivial closed invariant subspace constructed by C.J.Read, then there are three operators S₁, S₂ and K (non-multiples of the identity) such that T commutes with S₁, S₁ commutes with S₂, S₂ commutes with K, and K is compact. It is also shown that the commutant of T contains only series of T.

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