Insights into the Invariant Subspace Problem for compact perturbations of normal operators

Despite its simplicity, apparently one of the most difficult questions in the theory of invariant subspaces in separable, infinite dimensional complex Hilbert spaces is the problem of the existence of non-trivial closed invariant subspaces for a compact perturbation of a self-adjoint operator. Livsic solved this problem for nuclear perturbations, Sahnovic for Hilbert-Schmidt perturbations, and Gohberg and Krein, Macaev, and Schwartz for the perturbation being in the von Neumann class. However, it is still an open question if every compact perturbation of a self-adjoint operator has a non-trivial closed invariant subspace. The situation is still even hopeless if one considers compact perturbations of a bit broader class of operators, namely normal operators.

In this talk we will address this latter question and show recent improvements regarding it. (Based on joint works with Javier González-Doña).