非交换域中多元算子组的数值不变量及相关代数结构

Mutivariable operator theory is an important branch in operator theory and the theoretical basis for multibody dynamical systems and quantum mechanics. In recent years, it has attracted many famous experts. Popescu firstly gave the definition of noncommutative domain associated with operator space and developed the operator theory on noncommutative domains in 2010. In this program, we mainly study numerical invariants of multi-tuples of operator in the noncommutative domain and the structrue and maximality of algebras generated by weighted left creation oerators on the full Fock space. Firstly, by use of Stingspring's and Arveson's dilation theory as well as the technique of block-operator matrix, we discuss curvature invariant and Euler characteristics through studying models of multi-tuples of operator and its dual in noncommutative domain to improve Popescu's results. Secondly, combining the characteristics of tensor product with properties of free holomorphic functions, we research the structrue of noncommutative Hardy algebra and its subalgebra. Furthermore, the maximality of subalgebra is considered, so as to get algebra invariant of the analytic operator algebra.

多变量算子论是现代算子论中重要的研究方向之一，并为多体动力系统、量子力学等的进一步研究提供数学的理论支撑，近年来吸引了许多专家的广泛重视。2010年，Popescu首次引入算子空间的非交换域概念并发展了非交换域上的算子论，同时提出了一些新问题。本项目主要研究非交换域中多元算子组的数值不变量和由Fock空间上的加权左生成算子组生成的算子代数结构。我们将使用分块算子矩阵技巧和Stingspring、Arveson扩张理论，探讨非交换域中的多元算子组及其对偶算子组的结构模型，建立非交换域中多元算子组的曲率不变式和Euler特征的表达式，延拓并发展Popescu的结果；运用自由全纯函数的性质，结合张量积的方法，探讨非交换域上的非交换Hardy代数及其子代数的结构和代数极大性问题，以期获得解析算子代数的代数不变量。

本项目的主要研究结果可分为两个方面。第一方面是关于算子乘积逆序律问题的研究。利用算子分块技巧给出两个闭值域算子乘积Moore-Penrose逆序律成立的充要条件的一个新刻画，并对{1,2,3}-逆逆序律问题进行探讨。在该方面发表论文1篇。第二方面是非交换域上多元算子组的结构表示问题的研究，利用交换行压缩算子的正规扩张及极小等距扩张，结合Possion核性质，给出交换纯行压缩在Fock空间上的表示；运用交换行压缩的正规扩张结合算子谱分析，得到交换的保迹行压缩的一个结构模型。在该方面完成论文2篇。