Institute of Mathematics, University of Szczecin, Szczecin, Poland
E-mail: sklar@univ.szczecin.pl

Vitalii Marchenko

B. Verkin Institute for Low Temperature Physics and Engineering of the National
Academy of Sciences of Ukraine, 47 Nauky Ave., Kharkiv, 61103, Ukraine
E-mail: v.marchenko@ilt.kharkov.ua

Piotr Polak

Institute of Mathematics, University of Szczecin, Szczecin, Poland
E-mail: piotr.polak@usz.edu.pl

Received August 27, 2021, revised November 23, 2021.

Abstract

We consider special class of C_{0}-groups from [12], whose generators are unbounded, have pure point imaginary spectrum and corresponding dense
and minimal family of eigenvectors, which however does not form a Schauder
basis. We obtain two-sided estimates for norms of C_{0}-groups from this class and thus prove that these C_{0}-groups have linear growth. Moreover we show that C_{0}-groups from the considered class do not have any maximal asymptotics.
This means that the fastest growing orbits do not exist.