The Structure of C ₁.-Contractions

Abstract

We systematically exploit the operators intertwining a given contraction with an isometry or unitary operator. Given operators T on $\mathfrak{N}$ and T on $\mathfrak{N^\prime}$, we denote by$\mathfrak{N^\prime}$ g $(T, T^\prime)$the set of all intertwining operators; these are the bounded linear transformations$X: \mathfrak{N} \rightarrow \mathfrak{N^\prime}$ such that $XT=T^\prime X$. We also use the notation {T} = ℐ(T,T) for the commutant of T. Fix a contraction T on $\mathfrak{H}$ an isometry (resp., unitary operator) V on ℌ, and X∈ ℐ (T, V) such that ∥X∥ ≤ 1. The pair (X,V) is called an isometric (resp., unitary) asymptote of T if for every isometry (resp., unitary operator) V′, and every X′ ∈ ℐ(T, V′) with ∥X∥ ≤ 1, there exists a unique Y ∈ ℐ(V, V′) such that V′ = Y X and ∥Y′∥ ≤ 1.