Kérchy László honlapja

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Cím:Criteria of regularity for norm-sequences. II.
Szerző:K\'erchy, L\'aszl\'o; M\"uller, Vladimir
Forrás:Acta Sci. Math. 65, No.1-2, 131-138 (1999).
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) 
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