On quasi-similarity of contractions with finite defect indices (Q1112305)

Let \(T_ 1\) and \(T_ 2\) be \(C_{10}\) contractions on a Hilbert space whose defect indices \(d_{T_ 1}\), \(d_{T^*_ 1}\), \(d_{T_ 2}\) and \(d_{T^*_ 2}\) are finite and satisfy \(d_{T_ 1}-d_{T^*_ 1}=d_{T_ 2}-d_{T^*_ 2}=-k\) \((k=1,2,...)\). In this paper, the author obtains a necessary and sufficient condition in order that \(T_ 1\) and \(T_ 2\) be quasisimilar. The condition is in terms of the characteristic functions of \(T_ 1\) and \(T_ 2\). The result yields some more equivalent conditions for a contraction with finite defect indices to be quasisimilar to a unilateral shift (with finite multiplicity). All these are basically generalizations of the corresponding results for \(k=1\) obtained earlier by \textit{V. I. Vasyunin} and \textit{N. G. Makarov} [J. Soviet Math. 42, No.2, 1550-1561 (1988)].