On triviality condition of one parameter group of isometries (Q2762993)

The author considers a strongly continuous one-parameter group \(T\) of isometries in a complex Banach space \(X.\) It is proved that the spectrum of the infinitesimal generator of the group \(T,\) acting almost transitively consists of one point (\(T\) is trivial) and that the property of almost transitivity is equivalent to some limit equality for such a group formerly found by J. Goldstein and B. Nagy. If such a group acts transitively, then \(X\) is a Hilbert space. Finally, it is shown that for separable rearrangement-invariant function spaces on \([0,1]\) the group \(T\) is trivial under some weaker limit equality.