Year: 2018
Analysis in Theory and Applications, Vol. 34 (2018), Iss. 2 : pp. 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∈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+∈GI(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 IF 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 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.
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Journal Article Details
Publisher Name: Global Science Press
Language: English
DOI: https://doi.org/10.4208/ata.2018.v34.n2.3
Analysis in Theory and Applications, Vol. 34 (2018), Iss. 2 : pp. 127–134
Published online: 2018-01
AMS Subject Headings: Global Science Press
Copyright: COPYRIGHT: © Global Science Press
Pages: 8
Keywords: Generalized inverse analysis smooth diffeomorphism smooth submanifold.