Completely non-unitary contractions and analyticity

We study completely non-unitary contractions T with finite dimensional defect spaces DT and DT∗. We present a complete classification of all such contractions T that satisfy a generalized property of the Hardy shift operator: DT⊆DT∗. We show that T is analytic if and only if it has no non-zero eigenvalue. Furthermore, we characterize the hyponormality of all those T.