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 T0+ ∈B(E, F). This paper shows that, for every T∈B(E, F) with T0+(T-T0) <1, B(I + T0+(T-T0))-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 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 T0 is a semi-Fredholm operator but not in general.
MA Ji-pu
. A Rank Theorem of Operators between Banach Spaces[J]. Frontiers of Mathematics in China, 2006
, 1(1)
: 138
-143
.
DOI: 10.1007/s11464-005-0018-y