Given a separable Banach space $X$ of infinite dimension, we can consider on the algebra $\mathcal{B}(X)$ of continuous linear operators on $X$ several natural topologies, which turn its closed unit ball $B_1(X)=\{T \in \mathcal{B}(X) ;|T| \leq 1}$ into a Polish space - that is to say, a separable and completely metrizable space. In this talk, I will present some results concerning the "typical" properties, in the Baire category sense, of operators of $B_1(X)$ for these topologies when $X$ is an $\ell_p$-space, with $1 \leq p<+\infty$. One motivation for this study is the Invariant Subspace Problem, which asks for the existence of non-trivial invariant closed subspaces for operators on Banach spaces. It is thus interesting to try to determine if a "typical" contraction on a space $\ell_p$ has a non-trivial invariant subspace (or not). I will present some recent results related to this question.

This talk will be based on joint work with Étienne Matheron (Université d'Artois, France) and Quentin Menet (UMONS, Belgium).