Quasicontractive n-step liftings for operators similar to contractions

The paper is devoted to operators \(T\) acting on a Hilbert space \(\mathcal{H}\), that are similar to contractions. In this setting, we investigate quasi-isometric liftings induced by certain intermediate quasicontractive liftings \(Q\) for \(T\), which satisfy the condition \(Q^*Q \mathcal{H}=\mathcal{H}\). Such operators \(Q\) are generated by \(T\) and an operator \(A\), which gives the similarity of \(T\) to a contraction, this meaning that \(A\) is invertible and \(A \geq T^*AT \geq T^*T\). In this setting, we introduce the \(n\)-step liftings of \(T\) with respect to \(A\), which lead to a natural quasi-isometric lifting for \(T\). The aim of this paper is to show that the step-by-step construction of the minimal isometric lifting of a contraction can be adapted to this framework, in order to obtain a minimal quasi-isometric lifting of \(T\).