Invariant subspaces of positive operators

我的主要研究领域是有序Banach空间和正算子理论。正性是泛函分析的一个重要领域，在不同的科学学科中有许多应用，包括数理经济学，这是我的第二个研究领域。该理论最有趣的开放问题之一是不变子空间问题（ISP）： Banach格上的正算子T是否有非平凡的闭不变子空间？ 在所提出的研究中，我们希望采用一种不同的和新的方法来解决这个问题，这是使用集值映射的不动点定理。这种方法似乎可以在不假设T是拟幂零的情况下工作，因此它有望为这一问题带来新的见解。 到目前为止，在我的研究中，我一直在研究缺乏格结构的有序Banach空间中的问题。为此，我想研究后一类序Banach空间中正算子的ISP。此外，独立于ISP，我想通过考虑固体锥来进一步扩展自反锥的研究，并将这些类型的锥与Radon Nikodym性质联系起来。

My primary area of research is the theory of Ordered Banach spaces and positive operators. Positivity is an important field of Functional analysis that has many applications in different disciplines of sciences, including mathematical economics, which is my secondary area of research. One of the most interesting open problems of this theory, is the Invariant Subspace Problem(ISP): Does any positive operator T on a Banach lattice have a non trivial closed invariant subspace? In the proposed research we want to adopt a different and new approach to this problem, which is the use of fixed point theorems of set-valued maps. This approach seems that is able to work without the assumption that T is quasinilpotent, hence it is expected tol bring new insight to this problem. So far, in my research I have been studying problems in Ordered Banach spaces that lack lattice structure. In this vain, I want to study the ISP for positive operators in the latter class of Ordered Banach spaces. Moreover independently with the ISP, I want to further extend the study of reflexive cones by considering solid cones and relate these type of cones with the Radon Nikodym Property.