Quasi-Similarity of Weak Contractions

Abstract

Let T be a completely nonunitary (c.n.u.) weak contraction (in the sense of Sz.-Nagy and Foias). We show that T is quasi-similar to the direct sum of its CO part and C11 part. As a corollary, two c.n.u. weak contractions are quasi-similar to each other if and only if their CO parts and C11 parts are quasi-similar to each other, respectively. We also completely determine when c.n.u. weak contractions and CO contractions are quasi-similar to normal operators. Recall that a contraction T on the Hilbert space H is called a weak contraction if its spectrum a(T) does not fill the open unit disc D and 1 T* T is of finite trace. Contained in this class are all contractions T with finite defect index dTdim rank(1 T* T)'/2 and with a(T) =# D (cf. [9, p. 323]). Assume that T is a weak contraction which is also completely nonunitary (c.n.u.), that is, T has no nontrivial reducing subspace on which T is a unitary operator. For such a contraction, Sz.-Nagy and Foias obtained a C0-Cl1 decomposition and then found a variety of invariant subspaces which furnish its spectral decomposition (cf. [9, Chapter VIII]). In this note we are going to supplement other interesting properties of such contractions. We show that a c.n.u. weak contraction is quasi-similar to the direct sum of its C0 part and Cl part. Although the proof is not difficult, some of its interesting applications justify the elaboration here. An immediate corollary is that two such contractions are quasi-similar to each other if and only if their C0 parts are quasi-similar and their Cl parts are quasi-similar to each other. This is, in turn, used to show that two quasi-similar weak contractions have equal spectra. Another interesting consequence is that a c.n.u. weak contraction is quasi-similar to a normal operator if and only if its C0 part is. The latter can be shown to be equivalent to the condition that its minimal function is a Blaschke product with simple zeros, thus completely settling the question when a c.n.u. weak contraction is quasi-similar to a normal operator. Before we start to prove our main theorem, we provide some background work for our notations and terminology. The main reference is [9]. Presented to the Society September 9, 1976; received by the editors October 1, 1976. AMS (MOS) subject classifications (1970). Primary 47A45.