C 11 Contractions are Reflexive

Abstract

It is shown that a completely nonunitary Cl I contraction defined on a separable Hilbert space with finite defect indices is reflexive. In this note, only bounded linear operators defined on complex, separable Hilbert spaces will be considered. A contraction T (11 TII 0 and T*nx -+-> 0 for any x # 0. It is well known that such a contraction is quasi-similar to a unitary operator. Since unitary operators (even normal operators) are reflexive (cf. [3]), the question arises: Is the property of reflexivity preserved under the quasi-similarity? In other words, is a C11 contraction reflexive? In the present note we show that the answer is affirmative if the C11 contraction is completely nonunitary (c.n.u.) and has finite defect indices. We conjecture that the general case is also true. Recall that a contraction T is c.n.u. if there is no nontrivial reducing subspace on which T is unitary. The defect indices of T are, by definition, dT = dim[(1 T*T )1/29C ] , d7, = dim[(I TT*)1/29C]. If T is of class C11, then dT = dT*. In the following discussion we shall make use of the functional model for contractions developed by Sz.-Nazy and Foia? (cf. [4]). More specifically, if T is a c.n.u. contraction with dT = dT = n < oo, then T can be considered as defined on H = [Hn 2 2L, ] e {OTW @ Aw: w E H 2) by T(f @ g) = P(e'tf @ e"g) for f ED g E H, where Ln2 and Hn2 denote the standard Lebesgue and Hardy spaces of Cn-valued functions defined on the unit circle, EJT iS the characteristic function of T, A = (1 -_ *ET)1/2 and P denotes the (orthogonal) projection onto H. Any operator S in { T}', the commutant of T, has the form P[A ?], where A is a bounded analytic function while B and C are bounded measurable functions satisfying AeOT = OTAO and BET + CA = AA0 for some bounded analytic function A0 (cf. [5]). For an arbitrary operator T, { T}', { T}" and Alg T denote the commutant, double commutant and the weakly closed algebra generated by T and I, Presented to the Society, October 13, 1978; received by the editors October 24, 1978. AMS (MOS) subject classifications (1970). Primary 47A45; Secondary 47C05.