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The iterative methods for computing the generalized inverse ${A}_{T,S}^{\left(2\right)}$ 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 ${A}_{\left(l\right)}^{-1},$ or the Moore-Penrose inverse ${A}_{\left(MN\right)}^{+}$ and many others are the generalized inverse ${A}_{\left(TS\right)}^{2}$, the (2) inverse with prescribed range T and null-space S. Let $𝒳$ and $𝒴$ denote arbitrary Banach spaces and $ℬ\left(𝒳,𝒴\right)$ the set of all bounded operators. In one of their main results the authors define for $A\in ℬ\left(𝒳,𝒴\right)$ an approximating sequence ${\left({X}_{k}\right)}_{k}$ in $ℬ\left(𝒴,𝒳\right)$ and prove that

$\parallel {A}_{\left(TS\right)}^{\left(2\right)}-{X}_{k}\parallel <\left({q}^{{p}^{k}}\right)\parallel {\left(1-q\right)}^{-1}\parallel {X}_{0}\parallel$

for a certain integer $p\ge 2$ and $q<1·$ Analogous results are obtained for the generalized Drazin inverse in Banach algebras. Examples are given for the matrix

$A=\left(\begin{array}{ccc}2& 1& 1\\ 0& 2& 0\\ 0& 0& 2\\ 0& 0& 0\end{array}\right)\in {𝒞}^{4×3}$

and a matrix from ${𝒞}^{58×57}·$

##### MSC:
 65J10 Equations with linear operators (numerical methods) 65F20 Overdetermined systems, pseudoinverses (numerical linear algebra) 47A05 General theory of linear operators