On expansive operators that are quasisimilar to the unilateral shift of finite multiplicity

An operator T on a Hilbert space H is called expansive, if ∥Tx∥⩾∥x∥ (x∈H). Expansive operators T quasisimilar to the unilateral shift of finite multiplicity N are studied. In particular, it is proved that I−T∗T is of trace class for such T and there exist invariant subspaces M of T such that the restriction T|M of T on M is similar to the unilateral shift of multiplicity 1. Such M span H, and the minimal number of such M which span H is N for N⩾2 and 2 for N=1.