On the Reflexivity of Contractions on Hilbert Space

Let T be a bounded operator on the Hilbert space Jf. We denote by W{T) the weakly closed algebra generated by T and / and, as usual, we denote by Alg Lat(T') the set of those operators X with the property that XM c M whenever M is an invariant subspace for T. Of course, we always have W{T) <= Alg Lat(r), and T is said to be reflexive if W{T) = Alg Lat(7). The letter S will always stand for the unilateral shift defined on the Hardy space H by SJ[X) = Xf{X), XeDJelP. The main purpose of the present note is to prove a reflexivity result (Theorem 1 below) which was conjectured by Wu [8]. As noted by Wu, this result contains all the reflexivity theorems in [8] as particular cases. In addition to proving Theorem 1, we shall see what its hypothesis means for the canonical model of the contraction T. We also prove a result on the invariance of reflexivity under quasisimilarities—the method of proof is related to that used in the proof of Theorem 1. The main result of this paper was obtained simultaneously and independently by the two authors.