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).[-]

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.[-]

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.[-]

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.[-]