Completely non unitary contractions and Analyticity

We study completely non-unitary contractions $T$ with finite dimensional defect spaces ${\mathcal{D}}_T$ and ${\mathcal{D}}_{T^{\ast }}$. We present a complete classification of all such contractions $T$ that satisfy a generalized property of Hardy shift operator: ${\mathcal{D}}_T\subseteq {\mathcal{D}}_{T^{\ast }}$. We show that $T$ is analytic if and only if it has no non-zero eigenvalue. Furthermore, we characterize the hyponormality of all those $T$.