On subnormal operators (Q1084314)

Let S be a pure subnormal operator such that \(C^*(S)\), the \(C^*\)- algebra generated by S, is generated by a unilateral shift U of multiplicity 1. We obtain conditions under which S is unitarily equivalent to \(\alpha +\beta U\), \(\alpha\) and \(\beta\) being scalars or S has \(C^*\)-spectral inclusion property. It is proved that if in addition, S has \(C^*\)-spectral inclusion property, then so does its dual T and \(C^*(T)\) is generated by a unilateral shift of multiplicity 1. Finally, a characterization of quasinormal operators among pure subnormal operators is obtained.