Applications of Banach lattices to operator theory and stochastic processes

本文的主题是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。该项目有几个方向。******1。Banach随机过程与向量格。与学生和合作者一起，我开发了一种基于巴拿赫格的鞅理论变体，用正算子代替了条件期望。Labuschagne, Grobler， Watson等人创建了一个关于向量格的平行理论。经典鞅理论的许多重要定理被推广到这个新的“无测度鞅理论”中。******目标：***-将伯克霍尔德型鞅不等式扩展到新的设置；***-发展Banach格的随机积分；***-研究正则鞅的空间。******2。Multinorms。多态空间理论是由Dales和Polyakov在巴拿赫代数问题的启发下提出的。多项项（或者更一般地说，p多项项）是范数的泛化，但它测量的不是向量的“大小”，而是有限向量序列的“大小”。泛函分析的许多概念可以用多项来表示。特别地，每一个巴拿赫格都有一个正则p多项。已知每一个多项式空间都可以表示为巴拿赫格的一个子空间。****目标：***-将表示定理扩展到p多项；***-确定所有p的Banach格之间的多界算子；***-找到在绝对求和算子理论中出现的几个特定多项在巴拿赫格的子空间中的具体表示。正算子的不变子空间。Banach格上的每一个正算子是否都有不变子空间是一个长期存在的开放问题。Enflo和Read在80年代发现了在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版本。******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。不相交齐次（DH）空间：这些是巴拿赫格，其中每两个不相交的规范化序列具有等价的子序列。大多数经典空间都是DH。******目标：***-研究DH空间中的互补不相交序列；***-确定可分离DH空间中每个不相交序列是否有补子序列；确定每个自反巴拿赫格是否包含闭合张成补的不相交序列。**** ****** 5。数学经济学的应用。在数学经济学中，向量格被用来模拟市场。***目标：研究数学经济学中有限生成子格及其在子市场中的应用

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.**