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Local Resolvents of Operators with One-Dimensional Self-Commutator

1976
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Proceedings of the American Mathematical Society
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Let T = H + ¡J be an irreducible operator on a Hubert space with one-dimensional self-commutator. It is known that the selfadjoint operator H is absolutely continuous. Let EH denote the absolutely continuous support of H. In this note the following theorem is proven: Theorem. // there exists a real number p such that ess inf EH < p < ess sup EH and fE \t -p\ dt < oo, then the operator T has a nontrivial invariant subspace. Let % be a separable Hilbert space with inner product ( , ). Let F be a

doi:10.2307/2041377
fatcat:zb7tqajl55hsnm7umgrqnj5fhy