Applications of Banach lattices to operator theory and stochastic processes

这个建议的共同主题是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。该项目有几个方向。* 1. Banach空间中的随机过程和向量格。与学生和合著者一起，我开发了一种基于Banach格的鞅理论变体，用正算子取代了条件期望。Labuschagne，Grobler，沃森等人建立了一个类似的向量格理论。经典鞅论中的许多重要定理被推广到“无测度鞅论”这一新的背景下。** 目标：*-将Burkholder型鞅不等式推广到新的情形;*-发展Banach格上的随机积分;*-研究正则鞅空间。** 2.多重标准。多范空间的理论是由Dales和Polyakov发起的，其动机是Banach代数中的问题。多项式（或更一般地，p-多项式）是范数的推广，但它不是向量的“大小”，而是测量有限向量序列的“大小”。泛函分析的许多概念可以用多项式来表示。特别地，每个Banach格都有一个标准p-多项式。众所周知，每个多赋范空间都可以表示为Banach格的子空间。目标：*-将表示定理扩展到p-多范数; *-识别所有p的Banach格之间的多有界算子;*-根据Banach格的子空间，为绝对和算子理论中出现的几个特定的多范数找到具体的表示。3.正算子的不变子空间。Banach格上的正算子是否都有不变子空间是一个长期存在的问题。Enflo和阅读在80年代发现的例子，运营商没有不变的子空间的Banach空间。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。不相交齐性（DH）空间：这些是Banach格，其中每两个不相交的规范化序列具有等价的连续性。大多数经典空间都是DH.*目的：*-研究DH空间中的可补不交序列;*-确定可分DH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格是否都包含闭跨距可补的不交序列。5.应用于数学经济学。 在数学经济学中，向量格被用来对市场进行建模。目标：研究量子生成子格及其在数学经济学中的子市场应用。**

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.**