On 02/28/2013 08:12 PM, David Bernier wrote: > On 02/28/2013 07:56 PM, David Bernier wrote: [...]
>> I was reading on problems considered "modulo a compact >> operator K". >> >> If T is a bounded operator on a separable, infinite >> dimensional Hilbert space H, >> >> can one show that for some compact operator K >> on H, T+K has a non-trivial invariant >> subspace? > [...] > > Ok. I think we can find a finite-rank K such that > T+K has some non-zero eigenvector, i.e. > one can construct K such that for some v in H, > > T+K maps v to a scalar multiple of v. [...]
Alexey I. Popov and Adi Tcaciuc have an arxiv pre-print implying that on a separable infinite-dimensional Hilbert space, any operator T can be perturbed by a rank-one operator K such that T + K has an invariant subspace of infinite dimension and codimension.