On the semi-continuity of generalized inverses in Banach algebras
Let \(A\) be a Banach algebra. An element \(b \in A\) is called a reflexive generalized inverse for \(a \in A\) if \(a=aba\) and \(b=bab\). A reflexive generalized inverse \(a^\dagger\) of a \(C^*\)-algebra \(A\) is called the Moore--Penrose inverse of \(a\) if \((aa^\dagger)^*=aa^\dagger\) and \((a^\dagger a)^*=a^\dagger a\). Utilizing the method of \textit{Q.--L.\ Huang} and \textit{J.--P.\ Ma} [Linear Algebra Appl.\ 389, 355--364 (2004; Zbl 1072.47012)], the authors give sufficient and necessary conditions for the lower semicontinuity of the reflexive generalized inverse as a set-valued mapping in a unital Banach algebra. They also investigate the continuity of the Moore--Penrose inverse in a \(C^*\)-algebra and obtain some new criteria in operator theory.
scientific article; zbMATH DE number 5077733