Abstract. It is known from the Sz.-Nagy--Foias theory of operators that if $T$ is a Hilbert space contraction of class $C_{1\bullet }$ and if the unitary spectrum $\sigma(T)\cap{\partial }{\bf D}$ is of Lebesgue measure zero, then $T$ is a singular unitary operator. We extend this statement to polynomially bounded operators acting on arbitrary Banach spaces, presenting also its local version. It is shown how the method applied provides Katznelson--Tzafriri type theorems. One-parameter semigroups of Hilbert space contractions are also considered.

AMS Subject Classification (1991): 47A11, 47A10, 47D03

Received November 22, 1996 and in revised form May 5, 1997. (Registered under 5783/2009.)