Volume 34, Issue 2
Generalized Inverse Analysis on the Domain Ω(A,A +) in B(E,F)

Zhaofeng Ma

10.4208/ata.2018.v34.n2.3

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

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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 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+)−1is the generalized inverse of T with R(B)=R(A +) and N(B)=N(A+), for T∈Ω(A,A+), where IFis 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 MA(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.

  • History

Published online: 2018-07

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