Generalized Inverse Analysis on the Domain Ω(A,A+) in B(E,F)

Let B(E,F) be the set of all bounded linear operators from a Banach space E into another Banach space F,B~+(E, F) the set of all double splitting operators in B(E, F)and GI(A) the set of generalized inverses of A ∈ B~+(E, F). In this paper we introduce an unbounded domain ?(A, A~+) in B(E, F) for A ∈ B~+(E, F) and A~+∈GI(A), and provide a necessary and sufficient condition for T ∈ ?(A, A~+). Then several conditions equivalent to the following property are proved: B = A+(IF+(T-A)A~+)~(-1) is the generalized inverse of T with R(B)=R(A~+) and N(B)=N(A~+), for T∈?(A, A~+), where IF is the identity on F. Also we obtain the smooth(C~∞) diffeomorphism M_A(A~+,T) from ?(A,A~+) onto itself with the fixed point A. Let S = {T ∈ ?(A, A~+) : R(T)∩ N(A~+) ={0}}, M(X) = {T ∈ B(E,F) : TN(X) ? R(X)} for X ∈ B(E,F)}, and F = {M(X) : ?X ∈B(E, F)}. Using the diffeomorphism M_A(A~+,T) we prove the following theorem: S is a smooth submanifold in B(E,F) and tangent to M(X) at any X ∈ S. The theorem expands the smooth integrability of F at A from a local neighborhoold at A to the global unbounded domain ?(A, A~+). It seems to be useful for developing global analysis and geomatrical method in differential equations.