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2022.01.26 Colin Petitjean, Université Gustave Eiffel

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).

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