Hilbert空间上的算子动力系统

This project will study operator dynamics on Hilbert spaces, it is aimed at dynamic properties of operator classes induced by functional calculus. Papers on the dynamic properties of bounded linear operator can be dates back to the well known problem of invariant subspace.And their main research is about of chaos and transitivity for bounded linear operator.However, that is still at an initial stage the research for dynamic properties of operators induced by functional calculus,there will be a lot of problems to be solved.The chaos of operators,induced by functional calculus on Hilbert spaces, is not only to define operator classes,but also to give its applications on the classification and noncommutative geometry, such as to study the Lebesgue classes induced by noncommutative functional calculus on geometric module.Moreover, the research that is analytic functional calculus of operators on Hardy space could be made on double-disk and polydisk,such that we get relations between operator dynamics and the well known problem of invariant subspace.Therefore,that is inspired methods and ideas for this problem.

本项目在Hilbert空间上研究算子动力系统，旨在研究由函数演算给出的算子类动力学性质。对有界线性算子的动力学性质的研究可以追溯到著名的不变子空间问题，主要内容是有界线性算子的传递性和各类混沌的研究。但是，对函数演算给出的算子类动力学性质的研究尚在起步阶段，将有大量问题有待解决。Hilbert空间上由函数演算给出的算子类的混沌性质不仅可以定义算子类，还可以给出其在分类以及非交换几何里的应用，如几何模上非交换函数演算给出的Lebesgue类。另外，Hardy空间上算子的解析函数演算可以推广到双圆盘和多圆盘的情形，从而将算子类动力系统与不变子空间这样的公开问题相联系，因此为该问题提供新的思路和方法。

本项目在Hilbert空间上研究算子动力系统，旨在研究由函数演算给出的算子类动力学性质。首先，在Hardy空间上，我们给出了f(z)是Cowen-Douglas函数的充要条件，同时给出了其在混沌反问题里的应用。其次，对一个可分Hilbert空间上的有界线性算子T，我们给出了T的非交换函数演算，对正规算子而言我们的非交换的函数演算与经典的正规算子的函数演算是相一致的，利用非交换的函数演算得到了无限维可分Hilbert空间上不变子空间问题的一个充分条件，同时我们给出了Lebesgue类。最后我们研究了非交换的函数方程h(z) − h(z) = g(z), z ∈ ℂ并利用此方程我们给出了复矩阵的动力系统的完全分类。