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

Giulia Cavagnari, Giuseppe Savaré, Giacomo Enrico Sodini

TL;DR

This work develops a constructive, invariant extension theory for monotone operators and Lipschitz maps under groups of isometries, using self-dual Lagrangians and kernel averaging to produce maximal invariant extensions with controlled domains. It extends Kirszbraun–Valentine-type results to invariant Lipschitz maps and applies these ideas to $L^p$ spaces of random variables under measure-preserving transformations, connecting invariance with law invariance. The authors also establish explicit resolvent, Moreau–Yosida, and semigroup representations for invariant maximal dissipative operators, and provide a robust Borel-space framework for approximating couplings by measure-preserving maps, with potential impact on transport and dissipative evolution in Wasserstein spaces.

Abstract

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 Kirzsbraun-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 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

TL;DR

This work develops a constructive, invariant extension theory for monotone operators and Lipschitz maps under groups of isometries, using self-dual Lagrangians and kernel averaging to produce maximal invariant extensions with controlled domains. It extends Kirszbraun–Valentine-type results to invariant Lipschitz maps and applies these ideas to spaces of random variables under measure-preserving transformations, connecting invariance with law invariance. The authors also establish explicit resolvent, Moreau–Yosida, and semigroup representations for invariant maximal dissipative operators, and provide a robust Borel-space framework for approximating couplings by measure-preserving maps, with potential impact on transport and dissipative evolution in Wasserstein spaces.

Abstract

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 Kirzsbraun-Valentine extension Theorem. We then provide a relevant application to the case of monotone operators in -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.
Paper Structure (11 sections, 34 theorems, 151 equations)

This paper contains 11 sections, 34 theorems, 151 equations.

Table of Contents

  1. Introduction
  2. Extension of monotone operators and Lipschitz maps invariant by a group of isometries
  3. Maximal extensions of monotone operators by self-dual Lagrangians
  4. Extension of monotone operators invariant w.r.t. the action of a group of linear isomorphisms of XxX
  5. Extension of invariant dissipative operators in Hilbert spaces
  6. Extension of invariant Lipschitz maps in Hilbert spaces
  7. Borel partitions and almost optimal couplings
  8. Technical tools on standard Borel spaces and measure-preserving isomorphisms
  9. Approximation of couplings by using measure-preserving isomorphisms
  10. Monotone-dissipative operators and Lipschitz maps in L-spaces invariant by measure-preserving transformations
  11. An alternative proof of the extension theorem for G-invariant Lipschitz maps

Key Result

Theorem 2.1

Theorems & Definitions (81)

  • Theorem 2.1: Representation of monotone operators
  • proof
  • Theorem 2.2: Kernel averages and maximal monotone extensions BauWang2009
  • Theorem 2.3: A maximal monotone extension with minimal domain
  • proof
  • Definition 2.4
  • Proposition 2.5
  • proof
  • Theorem 2.6: $\mathsf G$-invariant maximal monotone extensions
  • proof
  • ...and 71 more