Anal. Theory Appl., 34 (2018), pp. 127-134.

Published online: 2018-07

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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 introduce an unbounded domain $\Omega(A, A^+)$ in $B(E, F)$ for $A\in B^+(E, F)$ and $A^+\in GI(A)$, and provide a necessary and sufficient condition for $T\in \Omega(A, A^+)$. 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 $T\in \Omega(A, A^+)$, where $I_F$ is the identity on $F$. Also we obtain the smooth $(C^∞)$ diffeomorphism $M_A(A^+, T)$ from $\Omega(A, A^+)$ onto itself with the fixed point $A$. Let $S =\{T\in \Omega(A, A^+): R(T)∩N(A^+) = \{0\}\}$, $M(X) =\{T\in B(E, F): TN(X) ⊂ R(X)\}$ 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: $S$ is a smooth submanifold in $B(E, F)$ and tangent to $M(X)$ at any $X\in S$. The theorem expands the smooth integrability of $\mathcal{F}$ at $A$ from a local neighborhoold at $A$ to the global unbounded domain $\Omega(A, A^+)$. It seems to be useful for developing global analysis and geomatrical method in differential equations.

}, issn = {1573-8175}, doi = {https://doi.org/10.4208/ata.2018.v34.n2.3}, url = {http://global-sci.org/intro/article_detail/ata/12581.html} }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 introduce an unbounded domain $\Omega(A, A^+)$ in $B(E, F)$ for $A\in B^+(E, F)$ and $A^+\in GI(A)$, and provide a necessary and sufficient condition for $T\in \Omega(A, A^+)$. 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 $T\in \Omega(A, A^+)$, where $I_F$ is the identity on $F$. Also we obtain the smooth $(C^∞)$ diffeomorphism $M_A(A^+, T)$ from $\Omega(A, A^+)$ onto itself with the fixed point $A$. Let $S =\{T\in \Omega(A, A^+): R(T)∩N(A^+) = \{0\}\}$, $M(X) =\{T\in B(E, F): TN(X) ⊂ R(X)\}$ 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: $S$ is a smooth submanifold in $B(E, F)$ and tangent to $M(X)$ at any $X\in S$. The theorem expands the smooth integrability of $\mathcal{F}$ at $A$ from a local neighborhoold at $A$ to the global unbounded domain $\Omega(A, A^+)$. It seems to be useful for developing global analysis and geomatrical method in differential equations.

*Analysis in Theory and Applications*.

*34*(2). 127-134. doi:10.4208/ata.2018.v34.n2.3