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.**

这一提议的共同主题是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，应用于随机过程和非负矩阵。这个项目有几个方向。*1.Banach和向量格中的随机过程。与学生和合著者一起，我开发了一种基于Banach格的鞅理论的变体，用正算子取代了条件期望。Labuschagne、Grobler、Watson等人在向量格上创建了一个有点类似的理论。经典鞅理论中的许多重要定理都被推广到这一新的“无测论”中。*目标：*-将Burkhold型鞅不等式推广到新的情形；*-在Banach格上发展随机积分；*-研究正则鞅空间。*2.多范数。多范空间理论是由Dales和Polyakov在Banach代数问题的启发下提出的。多范数(或者更广泛地说，p-多范数)是范数的推广，但它不是衡量向量的“大小”，而是衡量有限个向量序列的“大小”。泛函分析的许多概念都可以用多项式来表示。特别地，每个Banach格都有一个规范的p-多项式。众所周知，每个多范空间都可以表示为Banach格的一个子空间。*目标：*-将表示定理推广到p-多范数；*-确定所有p的Banach格之间的多有界算子；*-根据Banach格的子空间，找到绝对和算子理论中出现的几个特定多范数的具体表示。*3.正算子的不变子空间。Banach格上的每个正算子是否都有不变子空间是一个长期未解决的问题。Enflo和Read在80年代发现了Banach空间上没有不变子空间的算子的例子。最近，Sirotkin，Grivaux和Roginskaya提出了这些例子的修改变体。*目标：基于Sirotkin，Grivaux和Roginskaya最近的例子，构造一个没有不变子空间的正算子的例子。*4.不联合齐次空间：这是Banach格，其中每两个不相交的正规序列都有等价的子序列。大多数经典空间都是DH型的。*目标：*-研究DH型空间中的可补不交序列；*-确定可分的DH型空间中的每个不交序列是否都有可补子序列；*-确定每个自反Banach格是否包含闭跨距为可补的不交序列。*--研究有限生成的子格及其在数学经济学中的子市场中的应用。**