The characterization of Moore--Penrose inverse module maps and their continuity (Q1010998)

The author of present paper introduces the notion of Moore--Penrose inverse module map. A well-known result states that if \(T: E \to F\) is an adjointable mapping between Hilbert \(C^*\)-modules with closed range, then \(E=\ker(T)\oplus\text{ran}(T^*)\) and \(F=\ker(T)\oplus\text{ran}(T)\) (see [\textit{E.\,C.\thinspace Lance}, ``Hilbert \(C^*\)-modules'' (Lond.\ Math.\ Soc.\ Lect.\ Note Ser.\ 210; Camb.\ Univ.\ Pr.) (1995; Zbl 0822.46080)]). He uses this fact to show that \(T\) has a Moore--Penrose inverse if and only if \(\text{ran}(T)\) is a closed submodule in \(F\). He also provides a Douglas type factorization theorem and discusses the continuity of Moore--Penrose inverse module map depending upon a parameter.