【作者】 王心军；

【导师】 李绍宽； 王勤；

【作者基本信息】 东华大学 ， 应用数学， 2005， 硕士

【摘要】 由V.Vloterrn创立的算子理论把定义在无限维空间上的函数作为研究对象。这个理论很快以一种实质性的方式渗透到许多极为重要的数学领域:如群论、微分方程、积分方程、实变函数、正交级数等。在这个理论中,它把经典的数学方法与现代数学方法以一个极其有效和十分协调的方式结合起来。 对线性算子谱的研究是算子理论中的一个重要课题。我们知道,在线性代数理论中,人们用了很大的篇幅研究,在微分方程与积分方程的理论中也着重讨论了特征值问题。这是因为其重要性在于:首先直接来自的需要,物理学与工程学的研究离不开矩阵特征值的研究;其次,通过对于特征值或者更一般,对于谱的研究,有助于了解算子的本身结构,从而可以刻划相应的方程的解的构造。 线性算子谱理论经过半个多世纪发展,有许多分支里面的问题都有了相对圆满系统的解决,其中一直被算子谱论学者所关注的单个算子RS和SR的共同性质就是一例。线性算子组谱理论的发展相对要缓慢得多,尽管线性算子组联合谱的概念早就出现过,但是真正引起人们的重视在1970年J.L.Taylor的工作之后。他用代数拓扑中的复形和同调的概念定义了联合谱,以后线性算子组联合谱的研究才蓬勃发展起来。但是,无论国内还是国外在算子组ab和ba的共同性质方面的研究一直都没有人涉及,直到李绍宽1992年在文献[1]中把交叉交更多还原

【Abstract】 The operator theory created by V.Vloterrn regards the function which be defined on the infinite dimensional space as a research object in that is linked, taking This theory have rapidly been permeating through a lot of very important domain of mathematics in a kind of substantive way . For instance, group theory、differential equation、 integral equation、 real variable function、 cross progression etc.. Among the theory, it combines the classical mathematics method with the modern mathematics method in a kind of extremely effective and very much coordinate mode.The linear operate spectrum is an important subject in the operate theory. As we all know, the linear algebra goes to great lengths to study the matrix eigenvalue, and differential and integratiag eguation also discusses the question of eigenvalue in detail. The significance of these studies lie in firstly the need of the physics and engineering;secondly, the need of further learning about oprate itself and a clear description of the structure of corresponding eguation through the study on eigenvalue, or the spectrum in a more general sense.Half a century’s development of the linear operate spectrum theoryhas witnessed the comprehensive and satisfactory establishment of many sub-theories within the linear operate spectrum theory , such as the sharing character of the single operate RS and SR which was paid great attention by many scholars. The n-tuples of operate theory, however was developed much more slowly. It was not until 1970 after the work of J.L.Taylor did the concept of n-nuples of operate that had been proposed for long became the focus of scholars. Mr. Taylor defined joint-spectrum with the idea of complex and homological in topology which lent a momentum to the concerned research. But the study on n-nuples had remained untouched upon until Professor Li Shaokuan introduced criss-cross commutativity nature to the field that broke fresh ground for the concerned study in an article . Meanwhile, Professor Li acquired the results of ab and ba on Taylor’s joint spectrum sp (ab)\(0)=sp(ba)\(0), and in another article he got the results on Taylor’s joint spectrum of nature sp (ab)\(0)=sp(ba)\(0). In 1997, Robin Harte concluded similar results about left-and-right joint spectrum in his article, which he further developed in 2001.This paper is mainly to provide the definition of the "Taylor-operators-group-spectrum" and the definition of the "e- joint pseudo-spectrum" about the tuples of operators, and it has discussed its some nature. We have also got some results on these kinds of definitions of the criss-cross operators group. Then it has discussed the (/?)- propertyabout the operators group and the common nature of the operators group ab and ba’s subscalar nature. Finally, under some terms, we have made some discussion about the problem of Taylor’s joint-spectrum on zero.Wang XinJun (Applied Mathematics) Supervised by Professor Li ShaoKuan更多还原