Baire category methods and the Invariant Subspace Problem
Given a separable Banach space $X$ of infinite dimension, one can consider on the space $\mathcal{B}(X)$ of bounded linear operators on $X$ several natural topologies which turn the closed unit ball $B_1(X)=\{T\in\mathcal{B}(X);||T||\le 1\}$ into a Polish space, i.e. a separable and completely metrizable space. In this talk, I will present some results concerning typical properties in the Baire Category sense of operators of $B_1(X)$ for these topologies when $X$ is a $\ell_p$-space. Our main motivation for this study is linked to the Invariant Subspace Problem, which asks for the existence of non-trivial invariant closed subspaces for bounded operators on Banach spaces. It is thus interesting to try to determine whether typical contractions on $\ell_p$-spaces have a non-trivial invariant subspace or not. The talk is based on joint work with Etienne Matheron and Quentin Menet.