Groetschʹs representation of Moore–Penrose inverses and ill-posed problems in Hilbert -modules

The explicit representation lim ω → 0 + ( ω 1 + T ∗ T ) − 1 T ∗ for the Moore–Penrose inverse of an operator T between Hilbert spaces has been given by C.W. Groetsch (1975) [6]. We obtain his formula for the Moore–Penrose inverse of an unbounded operator between Hilbert C ∗ -modules. Ill-posed problems with unbounded operator between Hilbert C ∗ -modules are also discussed.