Contraction operators quasisimilar to a unilateral shift

Abstract

Let U n {U_n} denote the unilaterial shift of finite multiplicity n n . It is shown that a contraction operator T T is quasisimilar to U n {U_n} if and only if T T is of Class C 1 {C_1} ., the canonical isometry V V associated with T T is pure and T T is n n -cyclic with analytically independent vectors. For this, the notions of operators of analytic type and analytic independence of vectors are introduced. A characterization of the cyclic vectors of the Backward Shift is also presented.