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: 2019
• 资助国家: 加拿大
• 起止时间: 2019-01-01 至 2020-12-31
• 项目状态: 已结题
• 来源: https://www.nserc-crsng.gc.ca/ase-oro/Details-Detailles_eng.asp?id=680279
• 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关于Banach Lattices及其应用的积极运营商的理论。它是功能分析的一般领域，并应用于随机过程和非负矩阵。项目中有几个方向。****** 1。 Banach和Vector Lattices中的随机过程。我与学生和合着者一起，基于Banach Lattices开发了Martingale理论的一种变体，有条件的期望被积极的运营商所取代。 Labuschagne，Grobler，Watson等人创建了一个关于向量晶格的平行理论。古典marting虫理论的许多重要定理已扩展到这种“无量程的马丁格理论”的新环境。 *****目标：**** - 将Burkholder型Martingale不平等扩展到新环境； **** - 在Banach Lattices中发展随机整合； **** - 调查常规Martingales的空间。****** 2。多媒体。多构型空间的理论是由Dales和Polyakov启动的，该理论是由Banach代数中的问题所激发的。多媒体（或更一般而言，p-multinorm）是对标准的概括，但它不是向量的“大小”，而是测量矢量有限序列的“大小”。许多功能分析的概念可以用多媒体表示。特别是，每个Banach晶格都有一个规范的P-Multinorm。众所周知，每个多构型空间都可以表示为Banach晶格的子空间。****目标：**** - 将表示理论扩展到P-Multinorms； *** - 确定所有P的Banach晶格之间的多曲子运算符； **** - 根据Banach Lattices的子空间找到具体的表示，用于在绝对求和操作员理论中出现的几种特定多物型。**** 3。正运算符的不变子空间。如果Banach晶格上的每个积极操作员都有一个不变的子空间，这是一个长期的开放问题。 ENFLO并在80年代阅读，发现了在Banach空间上没有不变子空间的操作员的示例。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的最新示例，构建了一个没有不变子空间的正操作员的示例。********** 4。分离均匀（DH）空间：这些是Banach晶格，其中两个分离归一化序列具有等效的子序列。大多数经典空间是DH。******目标：**** - 研究完整的DH空间中的脱节序列； **** - 确定单独的DH空间中的每个分离序列是否都有一个完整的子序列； **** - 确定每个反射性的Banach lattice是否包含一个封闭跨度的分离序列。申请数学经济学。在数学经济学中，使用向量晶格来建模市场。