Typical properties of contractions on l_p-spaces

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 interest being to determine whether typical contractions on these spaces have a non-trivial invariant subspace or not.