This is an announcement for the paper "Every operator has almost-invariant subspaces" by Alexey I. Popov and Adi Tcaciuc.
Abstract: We show that any bounded operator $T$ on a separable, reflexive, infinite-dimensional Banach space $X$ admits a rank one perturbation which has an invariant subspace of infinite dimension and codimension. In the non-reflexive spaces, we show that the same is true for operators which have non-eigenvalues in the boundary of their spectrum. In the Hilbert space, our methods produce perturbations that are also small in norm, improving on an old result of Brown and Pearcy.
Archive classification: math.FA
Mathematics Subject Classification: 47A15 (Primary) 47A55 (Secondary)
Remarks: 11 pages
Submitted from: atcaciuc@ualberta.ca
The paper may be downloaded from the archive by web browser from URL
http://front.math.ucdavis.edu/1208.5831
or