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).[-]

For any countable group, and also for any locally compact second countable, compactly generated topological group, $G$, there exists a "universal" hypercyclic representation on a Hilbert space, in the sense that it simultaneously models every possible ergodic probability measure preserving free action of $G$. I will discuss the original proof of this theorem (a joint work with Benjy Weiss) and then, at the end of the talk, say some words about the development of this idea and its applications as expounded in a subsequent work of Sophie Grivaux.[-]

Let us say that a continuous linear operator $T$ acting on some Polish topological vector space is ergodic if it admits an ergodic probability measure with full support. This talk will be centred in the following question: how can we see that an operator is or is not ergodic? More precisely, I will try (if I'm able to manage my time) to talk about two “positive" results and one “negative" result. The first positive result says that if the operator $T$ acts on a reflexive Banach space and satisfies a strong form of frequent hypercyclicity, then $T$ is ergodic. The second positive result is the well-known criterion for ergodicity relying on the perfect spanning property for unimodular eigenvectors, of which I will outline a “soft" Baire category proof. The negative result will be stated in terms of a parameter measuring the maximal frequency with which (generically) the orbit of a hypercyclic vector for $T$ can visit a ball centred at 0. The talk is based on joint work with Sophie Grivaux.[-]

Most classical local properties of a Banach spaces (for example type or cotype, UMD), and most of the more recent questions at the intersection with geometric group theory are defined in terms of the boundedness of vector-valued operators between Lp spaces or their subspaces. It was in fact proved by Hernandez in the early 1980s that this is the case of any property that is stable by Lp direct sums and finite representability. His result can be seen as one direction of a bipolar theorem for a non-linear duality between Banach spaces and operators. I will present the other direction and describe the bipolar of any class of operators for this duality. The talk will be based on my preprint arxiv:2101.07666.[-]

We describe the idempotent Fourier multipliers that act contractively on $H^{p}$ spaces of the $d$-dimensional torus $\mathbb{T}^{d}$ for $d \geq 1$ and $1 \leq p \leq \infty$. When $p$ is not an even integer, such multipliers are just restrictions of contractive idempotent multipliers on $L^{p}$ spaces, which in turn can be described by suitably combining results of Rudin and Andô. When $p=2(n+1)$, with $n$ a positive integer, contractivity depends in an interesting geometric way on $n, d$, and the dimension of the set of frequencies associated with the multiplier. Our results allow us to construct a linear operator that is densely defined on $H^{p}\left(\mathbb{T}^{\infty}\right)$ for every $1 \leq p \leq \infty$ and that extends to a bounded operator if and only if $p=2,4, \ldots, 2(n+1)$. The talk is based on joint work with Ole Fredrik Brevig and Joaquim Ortega-Cerdà.[-]

Patrick Gérard and I introduced the cubic Szegö equation around ten years ago as a toy model of a totally non dispersive degenerate Hamiltonian equation. Despite of the fact that it is a complete integrable system, we proved that this equation develops some cascades phenomena. Namely, for a dense set of smooth initial data, the Szegö solutions have unbounded high Sobolev trajectories, detecting transfer of energy from low to high frequencies. However, this dense set has empty interior and a lot of questions remain opened to understand turbulence phenomena. Among others, we would like to understand how interactions of Fourier coefficients interfere on it. In a recent work, Biasi and Evnin explore the phenomenon of turbulence on a one parameter family of equations which goes from the cubic Szegö equation to what they call the 'truncated Szegö equation'. In this latter, most of the Fourier mode couplings are eliminated. However, they prove the existence of unbounded trajectories for simple rational initial data. In this talk, I will explain how, paradoxically, the turbulence phenomena may be promoted by adding a damping term. Those results are closely related to an inverse spectral theorem we proved on the Hankel operators.[-]