Extension of monotone operators and Lipschitz maps invariant for a group of isometries

Cavagnari, Giulia  ; Savaré, Giuseppe  ; Sodini, Giacomo Enrico 

Canadian journal of mathematics, Tome 77 (2025) no. 1, pp. 149-186

Résumé

We study monotone operators in reflexive Banach spaces that are invariant with respect to a group of suitable isometric isomorphisms, and we show that they always admit a maximal extension which preserves the same invariance. A similar result applies to Lipschitz maps in Hilbert spaces, thus providing an invariant version of Kirszbraun–Valentine extension theorem. We then provide a relevant application to the case of monotone operators in $L^{p}$-spaces of random variables which are invariant with respect to measure-preserving isomorphisms, proving that they always admit maximal dissipative extensions which are still invariant by measure-preserving isomorphisms. We also show that such operators are law invariant, a much stronger property which is also inherited by their resolvents, the Moreau–Yosida approximations, and the associated semigroup of contractions. These results combine explicit representation formulae for the maximal extension of a monotone operator based on self-dual Lagrangians and a refined study of measure-preserving maps in standard Borel spaces endowed with a nonatomic measure, with applications to the approximation of arbitrary couplings between measures by sequences of maps.

Détail

DOI : 10.4153/S0008414X23000846

Mots-clés : Extension of Lipschitz maps, dissipative/monotone operators, measure-preserving maps, invariance by law, optimal transport