Generalized Inverse Analysis on the Domain $Ω(A, A ^+)$ in $B(E, F)$ (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 $A\in B^+(E, F)$. In this paper we introducean unbounded domain $Ω(A, A^+)$ in $B(E, F)$ for $A\in B^+(E, F)$ and $A^+\in GI(A)$, and providea necessary and sufficient condition for $T\in Ω(A, A^+)$. Then several conditionsequivalent to the following property are proved: $B=A^+(I_F+(T−A)A^+)^{−1}$ is the generalizedinverse of $T$ with $R(B)=R(A^+)$ and $N(B)=N(A^+)$, for $T\in Ω(A, A^+)$, 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 $S =\{T\in Ω(A, A^+): R(T)∩N(A^+) = \{0\}\}$, $M(X) =\{T\in B(E, F): TN(X) ⊂ R(X)\}$ for $X ∈ B(E, \mathcal{F})$, and $\mathcal{F} = \{M(X): ∀X\in B(E, F)\}$. Using the diffeomorphism $M_A(A^+, T)$ we prove the following theorem: $S$ isa smooth submanifold in $B(E, F)$ and tangent to $M(X)$ at any $X\in S$. The theorem expandsthe smooth integrability of $\mathcal{F}$ at $A$ from a local neighborhoold at $A$ to the globalunbounded domain $Ω(A, A^+)$. It seems to be useful for developing global analysis andgeomatrical method in differential equations.
Keywords: Generalized inverse analysis; smooth diffeomorphism; smooth submanifold.