Perturbations of Moore-Penrose metric generalized inverses of linear operators in Banach spaces
Let \(X\) be a real Banach space and \(K \subseteq X\). Let \({\mathcal P}_K\) denote the \textit{metric projection} of \(X\) into \(K\). \(K\) is said to be \textit{Chebyshev} if \({\mathcal P}_K(x)\) a singleton for each \(x \in X\). Let \(Y\) be another real Banach space and \(T:D(T) \subseteq X \rightarrow Y\) be a linear map. Let \(N(T)\) and \(R(T)\) denote the null space and the range space of \(T\), respectively. Suppose that \(cl N(T)\) and \(cl R(T)\) are subspaces that are also Chebyshev sets in \(X\) and \(Y\), respectively. If there exists a homogeneous operator \(T^M:D(T^M) \rightarrow D(T)\) such that \(TT^MT=T\) on \(D(T)\), \(T^MTT^M=T^M\) on \(D(T^M)\), \(T^MT=I-{\mathcal P}_{clN(T)}\) on \(D(T)\) and \(TT^M={\mathcal P}_{clR(T)}\) on \(D(T^M)\), then such an operator is unique and is called the Moore-Penrose metric generalized inverse of \(T\). The authors study perturbations of the Moore-Penrose metric generalized inverses and obtain error estimates. A perturbation formula for such a generalized inverse is given in a certain framework.
Moore-Penrose metric generalized inverse
scientific article; zbMATH DE number 6378951
scientific article