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The iterative methods for computing the generalized inverse \(A^{(2)}_{T,S}\) 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 \(\mathcal{X}\) and \(\mathcal{Y}\) denote arbitrary Banach spaces and \(\mathcal{B}(\mathcal{X},\mathcal{Y})\) the set of all bounded operators. In one of their main results the authors define for \(A\in \mathcal{B}(\mathcal{X},\mathcal{Y})\) an approximating sequence \((X_{k})_{k}\) in \(\mathcal{B}(\mathcal{Y},\mathcal{X})\) and prove that \[ \| A_{(TS)}^{(2)} - X_{k}\| < (q^{p^{k}})\|(1-q)^{-1}\| X_{0}\| \] for a certain integer \(p\geq 2\) and \(q<1.\) Analogous results are obtained for the generalized Drazin inverse in Banach algebras. Examples are given for the matrix \[ A=\begin{pmatrix} 2 & 1 & 1\\ 0 & 2 & 0\\ 0 & 0 & 2\\ 0 & 0 & 0 \end{pmatrix} \in \mathcal{C}^{4\times 3} \] and a matrix from \(\mathcal{C}^{58\times 57}.\)

MSC:

65J10 Numerical solutions to equations with linear operators
65F20 Numerical solutions to overdetermined systems, pseudoinverses
47A05 General (adjoints, conjugates, products, inverses, domains, ranges, etc.)
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