Banach 空间上套代数中的有限秩算子

Finite Rank Operators of Nest Algebras on Banach Spaces

研究了Ｂａｎａｃｈ空间上套代数中的有限秩算子，得到了有限秩算子的分解定理：Ｆ为ａｌｇＮ中的ｎ秩算子，当且仅当Ｆ可写成ａｌｇＮ中ｎ个一秩算子之和；及一秩算子的判断条件：Ｔ∈ａｌｇＮ，且Ｔ≠０，则Ｔ为一秩算子当且仅当下面的条件成立，若Ａ，Ｂ∈ａｌｇＮ且ＡＴＢ＝０，则ＡＴ＝０或ＴＢ＝０。

In this paper, the finite rank operators of nest algebras on Banach spaces were discussed.We obtain the decomposition theorem of finite rank operators:If F ∈algN,then F is a n rank operator iff F is the sum of n one rank operators which belong to alg;and we also obtain the condition for an operator to be rank one: T ∈algN,and T ≠0,then T is a one rank operator iff the following condition holds:if A,B ∈algN and ATB=0 ,then AT=0 or TB =0.