On \(m\)-generalized invertible operators on Banach spaces
In the paper under review, the author provides an extension of the concepts left/right invertibility and \(m\)-left/right invertibility, introducing the notion of \textit{generalized inverses}: given a positive integer \(m\) and a bounded linear operator \(T\) on a complex Banach space \(X\), an \textit{\(m\)-left (resp., \(m\)-right) generalized inverse} for \(T\) is a bounded linear operator \(S\) on \(X\) such that \[ T \sum_{j=0}^m (-1)^j \binom{m}{j}S^{m-j}T^{m-j}=0 \quad \left( S \sum_{j=0}^m (-1)^j \binom{m}{j}T^{m-j}S^{m-j}=0 \right). \] Some structural and well-known results are obtained for this new class, extending the correspondents results in the classes generalized. Also, some cases in which a power of an \(m\)-left (resp., \(m\)-right) generalized invertible operator is again an \(m\)-left (resp., \(m\)-right) are investigated, generalizing some properties of \(2\)-partial isometries on Hilbert spaces.
scientific article; zbMATH DE number 6667757