Absztrakt: | A map $\rho:{\bbfN}_0={\bbfN}\cup\{0\}\rightarrow {\bbfR}^+$
is a norm-sequence if $\rho(0)=1$ and $\rho (m+n)\leq \rho(m)\rho(n)$
for $m,n\in{\bbfN}_0$. The main example of such a sequence is
$\rho (n)=\|T^n\|$ with non-nilpotent Banach or Hilbert space
operator $T$. These sequences can give a lot of information about
the operator $T$ itself. They were studied by {\it L. K\'erchy}
[Acta Sci. Math. 60, No. 3-4, 439-449 (1995; Zbl 0870.47016),
Proc. Am. Math. Soc. 127, No. 5, 1363-1370 (1999; Zbl 0914.47004),
and Integral Equations Oper. Theory 34, No. 4, 458-477 (1999)].
A norm-sequence $\rho$ is regular if there exists a certain gauge
function $p$ adjusted to $\rho$. The radius of a norm-sequence
$\rho $ is the limit $r(\rho)=\lim_{n\to\infty}\rho(n)^{1/n}$.
In the paper a necessary and sufficient condition for the regularity
of a norm-sequence is given and an example of a non-regular norm-sequence
with positive radius is provided. This solves a problem posed
by the first author in the negative.}
\RV{Andrzej So{\l}tysiak (Pozna\'n) |