This paper investigates the necessary and sufficient conditions under which a quasi-isometry T on a Hilbert space \({\cal H}\) admits a Wold-type decomposition in Shimorin’s sense. We establish a close connection between this decomposition and the kernel condition \(T^*T {\cal N} (T^*)\subset{\cal N} (T^*)\) , where \({\cal N}(T^*)\) is the kernel of the adjoint operator T* of T. Additionally, we discuss conditions related to certain cyclic and wandering subspaces, as well as the role of the Cauchy dual operator of T. Furthermore, we examine operators similar to contractions, that admit quasi-isometric liftings satisfying the kernel condition. This analysis leads to the identification of a special class of quasicontractions with such liftings, and on the other hand, to the construction of certain expansive quasi-isometric liftings Sα (0 < α < 1).
