Abstract: In this talk, we will show that any bounded operator on a separable, reflexive, infinite-dimensional Banach space 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.

We will also show that if a (norm-closed) algebra A of operators on the Hilbert space has a non-trivial common almost-invariant subspace X (i.e., every member T of A can be perturbed by a finite-rank operator F_T so that X is invariant for T-F_T), then A admits a genuine non-trivial invariant subspace. Time permitting, we will talk about operators having many almost-invariant subspaces. Our principal result here is: if every projection from a masa produces an almost-invariant subspace for the operator T, then T=D+F where D is in the masa and F is finite-rank. This is a finite-rank version of a result of Johnson and Parrott from 1972.