Volume 34, Issue 2
Generalized Inverse Analysis on the Domain $\Omega(A, A ^+)$ in $B(E, F)$

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

Published online: 2018-07

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• Abstract

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.

• Keywords

Generalized inverse analysis, smooth diffeomorphism, smooth submanifold.

47B38, 15A29, 58A05

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@Article{ATA-34-127, author = {}, title = {Generalized Inverse Analysis on the Domain $\Omega(A, A ^+)$ in $B(E, F)$}, journal = {Analysis in Theory and Applications}, year = {2018}, volume = {34}, number = {2}, pages = {127--134}, abstract = {

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} }
TY - JOUR T1 - Generalized Inverse Analysis on the Domain $\Omega(A, A ^+)$ in $B(E, F)$ JO - Analysis in Theory and Applications VL - 2 SP - 127 EP - 134 PY - 2018 DA - 2018/07 SN - 34 DO - http://doi.org/10.4208/ata.2018.v34.n2.3 UR - https://global-sci.org/intro/article_detail/ata/12581.html KW - Generalized inverse analysis, smooth diffeomorphism, smooth submanifold. AB -

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.

Zhaofeng Ma. (1970). Generalized Inverse Analysis on the Domain $\Omega(A, A ^+)$ in $B(E, F)$. Analysis in Theory and Applications. 34 (2). 127-134. doi:10.4208/ata.2018.v34.n2.3
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