Extension of monotone operators and Lipschitz maps invariant for a group of isometries
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arXiv
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| Format: | Preprint |
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2023
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| _version_ | 1866912236211011584 |
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| author | Cavagnari, Giulia Savaré, Giuseppe Sodini, Giacomo Enrico |
| author_facet | Cavagnari, Giulia Savaré, Giuseppe Sodini, Giacomo Enrico |
| contents | 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. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2305_04678 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Extension of monotone operators and Lipschitz maps invariant for a group of isometries Cavagnari, Giulia Savaré, Giuseppe Sodini, Giacomo Enrico Functional Analysis Optimization and Control Primary: 47B44, 37A40. Secondary: 54C20, 49Q22 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. |
| title | Extension of monotone operators and Lipschitz maps invariant for a group of isometries |
| topic | Functional Analysis Optimization and Control Primary: 47B44, 37A40. Secondary: 54C20, 49Q22 |
| url | https://arxiv.org/abs/2305.04678 |