Let $\mathrm{X}$ be a topological space of holomorphic functions on the open unit disc $D$. The study of the geometry of a space $X$ is centered on the identification of the linear isometries on $\mathrm{X}$, and there is an obvious connection between weighted composition operators and isometries. This connection can be traced back to Banach himself and emphasized by Forelli, El-Gebeily, Wolfe, Kolaski, Cima, Wogen, Colonna and many others. A characterisation is given of all the linear isometries of Hol($\Omega$), the Fr´ echet space of all holomorphic functions on $\Omega$ when $\Omega$ is the unit disc or an annulus, endowed with one of the standard metrics. Further, the larger class of operators isometric when restricted to one of the defining seminorms is identified. This is a joint work with Lucas Oger and Jonathan Partington.[-]

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 classical von Neumann inequality provides a fundamental link between complex analysis and operator theory. It shows that for any contraction $T$ on a Hilbert space and any polynomial $p$, the operator norm of $p(T)$ satisfies
$\|p(T)\| \leq \sup _{|z| \leq 1}|p(z)|$
Whereas Andô extended this inequality to pairs of commuting contractions, the corresponding statement for triples of commuting contractions is false. However, it is still not known whether von Neumann's inequality for triples of commuting contractions holds up to a constant. I will talk about this question and about function theoretic upper bounds for $\|p(T)\|$.[-]