An operator not satisfying Lomonosov's hypothesis

D.W Hadwin, E.A Nordgren, Heydar Radjavi, Peter Rosenthal
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
more » ... iant subspace. As Pearcy and Shields [7] pointed out (cf. Remark (a) at the end of this note), it is not clear that there are operators C for which there is no B satisfying the above hypothesis. Thus, it appears possible that Lomonosov's work implies that all operators have invariant subspaces. An obvious C to consider is the unilateral shift. Then the operators that commute with C are the analytic Toeplitz operators, so the question becomes: does any non-scalar analytic Toeplitz operator commute with a non-zero compact operator? Partial results suggesting that the answer to this question was negative were obtained by many authors: see [2, 4, II] and the references given there. Recently, however, Cowen [3] found an analytic Toeplitz operator that does commute with a non-zero compact operator. This example stimulated the present authors to investigate whether weighted shifts satisfy Lomonosov's hypothesis.
doi:10.1016/0022-1236(80)90073-7 fatcat:bm4nszvbe5ew7fi4oaszmzmrdm