Quasi-isometric liftings for operators similar to contractions

Abstract

A class of quasi-isometric liftings for the operators T similar to contractions in Hilbert spaces $H$ is studied. These liftings are isometric operators on their ranges, and are naturally induced by T and an invertible intertwiner of T with a contraction. In the case when T is a quasicontraction, meaning that T is contractive on its range, we obtain a quasi-isometric lifting on a space $K\supset H$, which is isometric on $K\ominus H$. Some liftings with closed ranges, or even similar to quasinormal partial isometries are mentioned. Additionally, we study the isomorphic minimal quasi-isometric liftings for T, as well as the uniqueness property of such liftings. Our results show similarities with those from the isometric dilation theory for contractions, although our context is more general than that of the latter.