Lethbridge Number Theory and Combinatorics Seminar: Alexey Popov

It a classical open problem in Operator Theory whether every bounded linear operator T on a Hilbert space H has a non-trivial invariant subspace (that is, a subspace Y of H such that TY is contained in Y; nontrivial means not {0} and not H). This is called the Invariant Subspace Problem; it is almost 100 years old.

In this talk we will show that any bounded operator on an infinite-dimensional Hilbert space admits a rank one perturbation which has an invariant subspace of infinite dimension and co-dimension. Moreover, the norm of the perturbation can be chosen as small as needed.

This is a joint work with Adi Tcaciuc.