Abstract. It is known that if $T$ is a contraction of class $C_{10}$ and $I-T^\ast T$ is of trace class, then $T$ is a quasiaffine transform of a unilateral shift. Also it is known that if the multiplicity of a unilateral shift is infinite, the converse is not true. In this paper the converse for a finite multiplicity is proved: if $T$ is a contraction and $T$ is a quasiaffine transform of a unilateral shift of finite multiplicity, then $I-T^\ast T$ is of trace class. As a consequence we obtain that if a contraction $T$ has finite multiplicity and its characteristic function has an outer left scalar multiple, then $I-T^\ast T$ is of trace class.

AMS Subject Classification (1991): 47A45

Received August 27, 2007, and in revised form September 3, 2008. (Registered under 6044/2009.)