The Discussion On The Property Of The Criss-Cross Commuting Operators

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...