## Some remarks about ‘‘Lipschitz-free operators’’

If *M* is a metric space, then the so-called Lipschitz-free space over* M*, usually denoted *F*(*M*), is a Banach space which is built around *M* in such a way that – *M *is isometric to a subset of* F*(*M*); – Lipschitz maps from *M* into any other Banach space *X* uniquely extend to bounded linear operators from *F*(*M*) into *X*. An interesting feature of Lipschitz-free spaces is that every Lipschitz map between two metric spaces *M* and *N* can be ‘‘linearised’’ in such a way that it becomes a bounded linear operator between the free spaces *F*(*M*) and *F*(*N*). We refer to these linearisations as ‘‘Lipschitz-free operators’’, or simply ‘‘Lipschitz operators’’. In this talk, we will study how the properties of the Lipschitz maps and their linearisations are related. After a few simple observations, we will mainly focus on some dynamical properties, compactness properties, and injectivity. This talk is based on ongoing works joint with Arafat Abbar (Marne-la-Vallée) and Clément Coine (Caen); Luis García-Lirola (Zaragoza) and Antonín Procházka (Besançon).