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The iterative methods for computing the generalized inverse A T,S (2) of the bounded linear operator between Banach spaces. (English) Zbl 1175.65063

The authors start by noting that many well known generalized inverses such as the Bott-Duffin inverse A (l) -1 , or the Moore-Penrose inverse A (MN) + and many others are the generalized inverse A (TS) 2 , the (2) inverse with prescribed range T and null-space S. Let 𝒳 and 𝒴 denote arbitrary Banach spaces and (𝒳,𝒴) the set of all bounded operators. In one of their main results the authors define for A(𝒳,𝒴) an approximating sequence (X k ) k in (𝒴,𝒳) and prove that

A (TS) (2) -X k <(q p k )(1-q) -1 X 0

for a certain integer p2 and q<1· Analogous results are obtained for the generalized Drazin inverse in Banach algebras. Examples are given for the matrix

A=211020002000𝒞 4×3

and a matrix from 𝒞 58×57 ·

MSC:
65J10Equations with linear operators (numerical methods)
65F20Overdetermined systems, pseudoinverses (numerical linear algebra)
47A05General theory of linear operators
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