(untitled) (Analysis in Theory and Applications, 2018)
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 AB^+(E, F). In this paper we introduce an unbounded domain Ω(A, A^+) in B(E, F) for AB^+(E, F) and A^+GI(A), and provide a necessary and sufficient condition for TΩ(A, A^+). Then several conditions equivalent to the following property are proved: B=A^+(I_F+(T−A)A^+)^−1 is the generalized inverse of T with R(B)=R(A^+) and N(B)=N(A^+), for TΩ(A, A^+), where I_F is the identity on F. Also we obtain the smooth (C^∞) diffeomorphism M_A(A^+, T) from Ω(A, A^+) onto itself with the fixed point A. Let S ={TΩ(A, A^+): R(T)∩N(A^+) = {0}}, M(X) ={TB(E, F): TN(X) ⊂ R(X)} for X ∈ B(E, F), and F = {M(X): ∀XB(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 XS. The theorem expands the smooth integrability of F at A from a local neighborhoold at A to the global unbounded domain Ω(A, A^+). It seems to be useful for developing global analysis and geomatrical method in differential equations.