Abstract: | 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 T
0∈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−T
0)‖<1, B ≡ (I + T
0
+(T−T
0))−1
T
0
+ is a generalized inverse of T if and only if (I−T
0
+
T
0)N(T) = N(T
0), where N(·) stands for the null space of the operator inside 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 where T
0 is a semi-Fredholm operator but not in general. |