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).[-]

The Invariant Subspace Problem for (separable) Hilbert spaces is a long-standing open question that traces back to Jonhn Von Neumann's works in the fifties asking, in particular, if every bounded linear operator acting on an infinite dimensional separable Hilbert space has a non-trivial closed invariant subspace. Whereas there are well-known classes of bounded linear operators on Hilbert spaces that are known to have non-trivial, closed invariant subspaces (normal operators, compact operators, polynomially compact operators,...), the question of characterizing the lattice of the invariant subspaces of just a particular bounded linear operator is known to be extremely difficult and indeed, it may solve the Invariant Subspace Problem.

In this talk, we will focus on those concrete operators that may solve the Invariant Subspace Problem, presenting some of their main properties, exhibiting old and new examples and recent results about them obtained in collaboration with Prof. Carl Cowen (Indiana University-Purdue University).[-]

Bishop's operator arose in the fifties as possible candidates for being counterexamples to the Invariant Subspace Problem. Several authors addressed the problem of finding invariant subspaces for some of these operators; but still the general problem is open. In this talk, we shall discuss about recent results on the existence of invariant subspaces which are indeed spectral subspaces for Bishop operators, by providing an extension of a Theorem of Atzmon (Joint work with M. Monsalve-Lopez).[-]