Study On The Radical Operators And Related Issues

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.