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 G I(A) the set of generalized inverses of A ∈B^{+}(E,F). In this paper we introduce
an unbounded domain Ω(A,A^{+}) in B(E,F) for A∈B^{+}(E,F) and A^{+} ∈G I(A), and provide
a necessary and sufficient condition for T ∈ Ω(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∈Ω(A,A^{+}), where I_{F}is the identity on F. Also we obtain the smooth (C^{∞}) diffeomorphism MA(A^{+},T) from
Ω(A,A^{+}) onto itself with the fixed point A. Let S = {T ∈ Ω(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 Ω(A,A^{+}). It seems to be useful for developing global analysis and
geomatrical method in differential equations.

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 G I(A) the set of generalized inverses of A ∈B^{+}(E,F). In this paper we introduce
an unbounded domain Ω(A,A^{+}) in B(E,F) for A∈B^{+}(E,F) and A^{+} ∈G I(A), and provide
a necessary and sufficient condition for T ∈ Ω(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∈Ω(A,A^{+}), where I_{F}is the identity on F. Also we obtain the smooth (C^{∞}) diffeomorphism MA(A^{+},T) from
Ω(A,A^{+}) onto itself with the fixed point A. Let S = {T ∈ Ω(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 Ω(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