Local Resolvents of Operators with One-Dimensional Self-Commutator

Abstract

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 bounded linear operator on %.The operator T is called hyponormal in case its self-commutator F* T -TT* = D is nonnegative.If the adjoint F* is a hyponormal operator, then F is called cohyponormal.If the self-commutator T*T -TT* is a one-dimensional operator, then either F or F* is hyponormal.It is not known at present whether every operator with a one-dimensional self-commutator has a nontrivial invariant subspace.In this note a result is described which increases the class of operators with one-dimensional selfcommutator that are known to have nontrivial invariant subspaces.Let F be a cohyponormal operator.The cartesian decomposition of F will be expressed T = H + U, where H and J are selfadjoint.For the purposes of this note it can be assumed that T is an irreducible operator.In this case the selfadjoint operators H and J are absolutely continuous [10, Theorem 3.2.1].Suppose H = / t dG, is the spectral resolution of H. Then there is a Borel set EH in the real line, determined up to a set of measure zero with the property $EHdGt = I and if ff dG, = I, for F C EH, then EH\Fis of Lebesgue measure zero.This set EH (really EH is an equivalence class of Borel sets) will be called the absolutely continuous support of H.It is known [10, Theorem 3.4.1]that if F = H + U is cohyponormal, then a(H) (the spectrum of H) is the projection on the x-axis of o(T).Similarly, a(J) is the projection of a(T) onto the j-axis.An operator with disconnected spectrum is known to have nontrivial invariant subspaces.It follows that every cohyponormal operator T = H + U, where a(H) or a(J) is not an interval, has a nontrivial invariant subspace.The main result in this note is the following:Theorem 1.Let T = H + U be an irreducible operator with one-dimensional self-commutator.Let EH be the absolutely continuous support of H and assume

Locations

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