Abstract. In [9] the shift index $\kappa(T)$ of a contraction $T$ acting on a Hilbert space is defined: $\kappa(T)$ is the supremum of $n$ such that $S_n$ can be injected into $T$, where $S_n$ is the unilateral shift of multiplicity $n$. In [11] the following question is posed: if $T$ is a $C_{10}$-contraction and its unitary asymptote is a reductive unitary operator, then $\kappa(T)=\infty $? In this paper, a positive answer to this question is given. A combination of the answer to this question with results of [11] gives that, for a $C_{10}$-contraction $T$, $\kappa(T) < \infty $ if and only if $T$ is a quasiaffine transform of $S_n$ for some finite $n$.

AMS Subject Classification (1991): 47A45

Keyword(s): contraction, unilateral shift, injection

Received January 19, 2011, and in final form January 20, 2012. (Registered under 6/2011.)