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An operator not satisfying Lomonosov's hypothesis
1980
Journal of Functional Analysis
An example is presented of a Hilbert space operator such that no non-scalar operator that commutes with it commutes with a non-zero compact operator. This shows that Lomonosov's invariant subspace theorem does not apply to every operator. The invariant subspace theorem of Lomonosov [6-S] includes the following assertion: if C is an operator such that CB = BC for an operator B that is not a multiple of the identity and that commutes with a non-zero compact operator, then C has a non-trivial
doi:10.1016/0022-1236(80)90073-7
fatcat:bm4nszvbe5ew7fi4oaszmzmrdm