Abstract. The regularity property of a norm-sequence $\rho(n)=\|T^n\| (n\in{\msbm N})$ ensures that the operator $T$ can be intertwined with an isometry $V$, which relation can be exploited to obtain a lot of information for $T$ itself, as it was shown in [1] and [2]. In [3] general sufficient conditions of regularity were provided. In the present note a necessary and sufficient condition of regularity is given. Applying this criterion a non-regular norm-sequence $\rho $ of positive radius is exhibited, settling the question, posed in [1] and [3], in the negative.

AMS Subject Classification (1991): 40A99, 47A99

Received September 30, 1998. (Registered under 2677/2009.)