Applications of Banach lattices to operator theory and stochastic processes

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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.**