摘要
设E和F是Banach空间,B(E,F)表示映E到F的有界线性算子全体.记T0+∈B(F,E)为To∈B(E,F)的一个广义逆.本文证明,每一个具有||T0+(T-T0)|| J<1的算子T∈B(E,F),B≡(I+T0+(T-T0))-1T0+是T的广义逆当且仅当(I-T0+T0)N(T)=N(T0),其中N(·)表示括弧中算子的零空间.这一结果改进了Nashed和Cheng的一个有用的定理,并进一步证明Nashed和Cheng的一个引理对半-Fredholm算子有效但一般未必成立。
Suppose that E and F are two Banach spaces and that B(E, F) is the space of all bounded linear operators from E to F. Let T0 ∈ B(E, F) with a generalized inverse T+0∈ B(F,E). This paper shows that, for every T ∈ B(E,F) with ||T+0(T - T0)|| < 1, B ≡ (I + T+0(T - To))-1T0+ is a generalized inverse of T if and only if (I - T0+T0)N(T) = N(T0), where N(-) stands for the null space of operator in the parenthesis. This result improves a useful theorem of Nashed and Cheng, and further shows that a lemma given by Nashed and Cheng is valid in the case that TO is semi-Fredholm operator, but not valid in general.
出处
《数学年刊（A辑）》
CSCD
北大核心
2003年第6期669-674,共6页
Chinese Annals of Mathematics
基金
国家自然科学基金(No.10271053)
国家教育部博士点基金资助的项目
