Unimodular contractions in Hilbert space

Abstract

Let be a unitary operator on a Hubert space H. Then in particular, (i) is a contraction, i.e. || T 1; and (ii) The spectrum of is a subset of the unit circle, i.e. Sp (T)cC, where C denotes the set of complex numbers of absolute value one. Call an arbitrary operator a unimodular contraction if it satisfies conditions (i) and (ii) above. Then several questions immediately come to mind. Do there exist nonunitary unimodular contractions? If so, what is the nature of their spectra, e.g. what subsets of the unit circle arise as spectra of nonunitary unimodular contractions; when does the spectrum contain point, residual, or continuous spectrum? Under what conditions is a unimodular contraction unitary? What is the nature of operator algebras containing nonunitary unimodular contractions? In this paper examples are given of nonunitary unimodular contractions. It is shown (Theorem 2) that such exist with arbitrarily prescribed spectrum, which however can contain no residual spectrum. It is also shown (Theorem 1) that nonunitary unimodular contractions exist only in infinite von Neumann algebras. This result is applied to a mapping problem of operator algebras.