On contractions that are quasiaffine transforms of unilateral shifts

Abstract

It is known that if T is a contraction of class C10 and I−T * 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 * 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 * T is of trace class.Also, it is known that if a contraction T on a Hilbert space H is such that b λ (T )x ≥ δ x for every λ ∈ D, x ∈ H, with some δ > 0, and(here b λ is a Blaschke factor and S1 is the trace class of operators), then T is similar to an isometry. In this paper the converse for a finite multiplicity is proved: if T is a contraction and T is similar to an isometry of finite multiplicity, then T satisfies the above conditions.