【作者】 赵旭；

【导师】 钟怀杰；

【作者基本信息】 福建师范大学 ， 基础数学， 2010， 硕士

【摘要】 设X是一个无限维Banach空间,B(X)是X上的有界线性算子全体构成的Banach代数,K(X)是B(X)中的紧算子理想,π：B(X)→C(X)表示由B(X)到Calkin代数C(X)：=B(X)/K(X)的典则商同态映射,于是产生了B(X)中的一个子集一根算子集I(X),其中每个T∈I(X),就定义为π(T)是Calkin代数C(X)中的Jacobson根者.已知I(X)形成了B(X)中的一个理想.本文主要就是围绕着根算子I(X)的等价特征和性质这一主题而展开的.全文共分为四章,第一章主要是介绍该课题研究的现状,本文所采用的研究思想方法及所得的主要结果.第二章主要是从一般代数A的角度出发来研究它的Jacobson根Rad(A)的性质与特征,完善了文献[25]中的定理1.4.14的证明,并从Z(A)(Z(A)={a∈A：ab-ba∈Rad(A)})的角度对Rad(A)进行研究,得到了Z(A)∩Q(A)=Rad(A)这一主要结果.在这章的最后我们推广了Rad(B(X))={O}这一命题,得到了相关的结论.在第三章中,一方面我们在第一章研究的基础上得到根算子相应的等价命题,另一方面我们从广义算子理想、算子半群及空间理想的角度出发来探讨根算子的性质特征,并探讨了I(X),R(X),Q(X)三者的包含关系.在这章的最后,我们应用Z(A)∩Q(A)=Rad(A)这一结果来研究遗传不可分解空间上的强不可约算子的性质.在第四章中,我们为了解决Gonzalez等在文献[35]中的问题2.10(设A为一算子理想,若Space(A)=F,是否A(?)I?其中I为根算子理想),我们定义了空间理想Z类可补奇异算子的概念,得到了一个与文献[35]中的问题2.10等价的命题,接着我们定义了与空间理想Z类可补奇异算子类似的弱空间理想Z类奇异算子及弱空间理想Z类余奇异算子的概念,并应用前面所得的结论分别探讨了它们根的性质.更多还原

【Abstract】 Let X be a an infinite dimensional Banach space, and B(X) is the Banach al-gebra formed by the whole of bounded linear operators on X. K(X) is the compact operator ideal in B(X).π:B(X)→C(X) represents the canonical quotient ho-momorphism from B(X) to C(X),so it generates a subset in B(X)-radical operators set(I(X)).For each T∈I(X),π(T) belongs to the Jacobson radical of C(X). we know I(X) forming a operator ideal in B(X). This thesis is organized around the equivalent characteristics and the nature of radical operators.The full-text is divided into four chapters. The first chapter mainly introduces the topic of the status quo, the approach to the study methods and main results of our research. The second chapter studies the nature of the general algebraic Jacobson radical. We complete the proof of the theorem 1.4.14 in the literature [25],and research the nature of Jacobson radical from the perspective of Z(A),so we obtain the result that Z(A)∩Q(A)=Rad(A).In the end of this chapter, we promote the proposition of Rad(B(X))={0}.In the third chapter,on the one hand, we discuss the inclusive relation among I(X)、R(X) and Q(X).On the other hand,we study the nature of radical operators from operator ideals、operator semigroups and space ideal.In the fourth chapter,we define the concept of space ideal Z complement singluar operators in order to slove the question 2.10 in literature[35](Let A is a operator ideal,if Space(A)=F,A(?)I?)and we obtain a equivalent proposition about question 2.10 in literature[35].Then,we define the concept of weak space ideal Z singluar operators and weak space ideal Z cosingluar operators. We research the nature of the two operators and the radical of the two operators respectively.更多还原