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Hilbert direct integrals of monotone operators

Part of: Measures, integration, derivative, holomorphy Existence theories Variational problems in infinite-dimensional spaces

Published online by Cambridge University Press:  21 May 2024

Minh N. Bùi  and
Patrick L. Combettes
Minh N. Bùi
Affiliation:
Department of Mathematics and Scientific Computing, NAWI Graz, University of Graz, 8010 Graz, Austria e-mail: [email protected]
Patrick L. Combettes*
Affiliation:
Department of Mathematics, North Carolina State University, Raleigh, NC 27695, United States
*
e-mail: [email protected]
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Abstract

Finite Cartesian products of operators play a central role in monotone operator theory and its applications. Extending such products to arbitrary families of operators acting on different Hilbert spaces is an open problem, which we address by introducing the Hilbert direct integral of a family of monotone operators. The properties of this construct are studied, and conditions under which the direct integral inherits the properties of the factor operators are provided. The question of determining whether the Hilbert direct integral of a family of subdifferentials of convex functions is itself a subdifferential leads us to introducing the Hilbert direct integral of a family of functions. We establish explicit expressions for evaluating the Legendre conjugate, subdifferential, recession function, Moreau envelope, and proximity operator of such integrals. Next, we propose a duality framework for monotone inclusion problems involving integrals of linearly composed monotone operators and show its pertinence toward the development of numerical solution methods. Applications to inclusion and variational problems are discussed.

Keywords

Integration of set-valued mappingsmeasurable vector fieldmonotone operatoroptimizationvariational analysis

MSC classification

Primary: 49J53: Set-valued and variational analysis 46G10: Vector-valued measures and integration 58E30: Variational principles
Type
Article
Information
Canadian Journal of Mathematics , First View , pp. 1 - 32
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Copyright
© The Author(s), 2024. Published by Cambridge University Press on behalf of Canadian Mathematical Society

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Footnotes

Dedicated to the memory of Hédy Attouch

The work of P. L. Combettes was supported by the National Science Foundation under grant CCF-2211123.

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