Applications of Banach lattices to operator theory and stochastic processes

Banach 格在算子理论和随机过程中的应用

• 批准号: RGPIN-2015-04051
• 负责人: Troitsky, Vladimir
• 金额: $ 1.82万
• 依托单位: University of Alberta
• 依托单位国家: 加拿大
• 项目类别: Discovery Grants Program - Individual
• 财政年份: 2018
• 资助国家: 加拿大
• 起止时间: 2018-01-01 至 2019-12-31
• 项目状态: 已结题
• 来源: https://www.nserc-crsng.gc.ca/ase-oro/Details-Detailles_eng.asp?id=667165
• 关键词: Applications; Banach; lattices; operator; theory

The common theme of this proposal is the theory of positive operators on Banach lattices and its applications. It is in the general domain of Functional Analysis, with applications to stochastic processes and non-negative matrices. There are several directions in the project.******1. Stochastic processes in Banach and vector lattices. Together with students and co-authors, I have developed a variant of martingale theory based on Banach lattices, with conditional expectations replaced with positive operators. Labuschagne, Grobler, Watson, et al have created a somewhat parallel theory on vector lattices. Many important theorems of the classical martingale theory have been extended to this new setting of "measure-free martingale theory". ******Goals:***- extend Burkholder-type martingale inequalities to the new setting;***- develop stochastic integration in Banach lattices;***- investigate the space of regular martingales.******2. Multinorms. The theory of multinormed spaces was initiated by Dales and Polyakov, motivated by problems in Banach Algebras. A multinorm (or, more generally, a p-multinorm) is a generalization of a norm, but instead of the "size" of a vector, it measures the "size" of a finite sequence of vectors. Many concepts of Functional Analysis can be expressed in terms of multinorms. In particular, every Banach lattice has a canonical p-multinorm. It is known that every multinormed space can be represented as a subspace of a Banach lattice.****Goals:***- extend the representation theorem to p-multinorms; ***- identify multibounded operators between Banach lattices for all p;***- find concrete representations, in terms of subspaces of Banach lattices, for several specific multinorms that appear in the theory of absolutely summing operators.***3. Invariant subspaces of positive operators. It is a long-standing open problem whether every positive operator on a Banach lattice has an invariant subspace. Enflo and Read in the 80's found examples of operators with no invariant subspaces on Banach spaces. Recently, Sirotkin, Grivaux, and Roginskaya came up with modified variants of those examples.******Goal: based on the recent examples of Sirotkin, Grivaux, and Roginskaya, construct an example of a positive operator with no invariant subspaces.*** ***4. Disjointly homogeneous (DH) spaces: these are Banach lattices where every two disjoint normalized sequences have equivalent subsequences. Most classical spaces are DH.******Goals:***- study complemented disjoint sequences in DH spaces;***- determine whether every disjoint sequence in a separable DH space has a complemented subsequence;***- determine whether every reflexive Banach lattice contains a disjoint sequence whose closed span is complemented.**** ******5. Applications to Math Economics.  In Math Economics, vector lattices are used to model markets.***Goal: study finitely generated sublattices and their applications to submarkets in Math Economics.**

该提案的共同主题是 Banach 格上的正算子理论及其应用，它属于泛函分析的一般领域，适用于随机过程和非负矩阵。** 该项目有多个方向。 ****1. Banach 和向量格中的随机过程，我与学生和合著者一起开发了基于 Banach 格的鞅理论的变体，其中条件期望被替换为正算子。 Labuschagne、Grobler、Watson 等人创建了一种关于向量格的并行理论，经典鞅理论的许多重要定理已扩展到“无测度鞅理论”的新设置。 ***- 将伯克霍尔德型鞅不等式扩展到新的环境；***- 发展巴纳赫格中的随机积分；***- 研究正则鞅的空间。******2。多范数空间的概念是由 Dales 和 Polyakov 发起的，其动机是巴拿赫代数中的问题。多范数（或者更一般地说，p-多范数）是范数的推广，但它测量的不是向量的“大小”。有限向量序列的“大小”。泛函分析的许多概念可以用多范数来表达。特别是，每个 Banach 格都具有规范的 p-多范数。每个多范数空间都可以表示为 Banach 格子的子空间。****目标：***- 将表示定理扩展到 p-多范数；***- 识别所有 p 的 Banach 格子之间的多有界算子；*** - 找到绝对求和算子理论中出现的几个特定多范数的具体表示，以巴纳赫格子空间的形式表示。***3. 正算子的不变子空间 这是一个长期存在的问题。 Banach 格上的每个正算子是否都具有不变子空间这一悬而未决的问题 Enflo 和 Read 在 80 年代发现了 Banach 空间上没有不变子空间的算子示例。最近，Sirotkin、Grivaux 和 Roginskaya 提出了这些示例的修改变体。 ******目标：基于Sirotkin、Grivaux和Roginskaya最近的例子，构造一个没有不变式的正算子的例子*** ***4. 不相交齐次 (DH) 空间：这些是 Banach 格子，其中每两个不相交归一化序列都有等效的子序列。****** 目标：***- 研究。 DH 空间中的互补不相交序列；***- 确定可分离 DH 空间中的每个不相交序列是否都有互补子序列；***- 确定每个自反 Banach 格子是否包含一个不相交序列，其闭合跨度得到补充。**** ******5. 在数学经济学中，向量格用于对市场进行建模。***目标：研究有限生成的子格及其在子市场中的应用。数学经济学。**