Abstract. We consider contraction operators on Hilbert space with finite defect index. For those in $C_{\cdot0}$ we provide some new equivalent conditions, including one based on Fredholm index, for membership in the dual operator algebra classes ${\msbm A}_n$ or ${\msbm A}_{n, \aleph_0}$. For those in $C_{11}$, we give characterizations for membership in these classes including the size of scalar that can be compressed to a semi-invariant subspace and multiplicity of the unitary piece of the minimal coisometric extension.

AMS Subject Classification (1991): 47D27, 47A20

Received November 10, 1993 and in revised form November 1, 1994. (Registered under 5605/2009.)