Abstract: 

Isometric automorphisms of normed linear
spaces are characterized by suitable concavity properties of
powers of operators. Bounded selfadjoint operators in Hilbert
spaces are described by parallel concavity properties of the
exponential group. Unbounded infinitesimal generators of
${\cal C}_0$-groups of Hilbert space operators having concavity properties
are characterized as well.

1. Institute of Mathematics
   Jagiellonian University
   30-059 Kraków
   Poland