单调算子的希尔伯特直接积分

Minh N. Bùi, Patrick L. Combettes
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引用次数: 0

摘要

算子的有限笛卡儿积在单调算子理论及其应用中发挥着核心作用。将这种乘积扩展到作用于不同希尔伯特空间的任意算子族是一个未决问题,我们通过引入单调算子族的希尔伯特直接积分来解决这个问题。我们研究了这一构造的性质,并提供了直接积分继承因子算子性质的条件。确定凸函数子微分族的希尔伯特直接积分本身是否是子微分的问题,引导我们引入函数族的希尔伯特直接积分。我们建立了评估此类积分的 Legendre 共轭、次微分、衰退函数、莫罗包络和邻近算子的明确表达式。接下来,我们为涉及线性组合单调算子积分的单调包含问题提出了一个对偶框架,并展示了其对数值求解方法发展的相关性。我们还讨论了包容和变分问题的应用。
本文章由计算机程序翻译,如有差异,请以英文原文为准。
Hilbert direct integrals of monotone operators

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.

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