We study monotone operators in reflexive Banach spaces that are invariant with respect to a group of suitable isometric isomorphisms andwe showthat 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 Kirzsbraun-Valentine extension Theorem. We then provide a relevant application to the case of monotone operators in Lp-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 selfdual 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.

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

Savarè, Giuseppe
;
Sodini, Giacomo Enrico
In corso di stampa

Abstract

We study monotone operators in reflexive Banach spaces that are invariant with respect to a group of suitable isometric isomorphisms andwe showthat 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 Kirzsbraun-Valentine extension Theorem. We then provide a relevant application to the case of monotone operators in Lp-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 selfdual 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.
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Utilizza questo identificativo per citare o creare un link a questo documento: https://hdl.handle.net/11565/4061977
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