Generalized inverses and similarity to partial isometries

We obtain some results related to the problems of Badea and Mbekhta (2005) [1] concerning the similarity to partial isometries using the generalized inverses. Especially, we involve the Moore–Penrose inverses. Also a characterization for such a similarity is given in the terms of dilations similar to unitary operators, which leads to a new criterion for the similarity to an isometry and to a quasinormal partial isometry.