Asymptotic behavior of discrete and continuous semigroups on Hilbert spaces

Let phi : [0, infinity) -> [0, infinity) be a nondecreasing function with phi(t) > 0 for all t > 0, H be a complex Hilbert space and let T be a bounded linear operator acting on H. Among our results is the fact that T is power stable (i.e. its spectral radius is less than 1) if [GRAPHICS] for all x E H with parallel to x parallel to <= 1. In the continuous case we prove that a strongly continuous uniformly bounded semigroup of operators acting on a Hilbert space H is spectrally stable (i.e. the spectrum of its infinitesimal generator lies in the open left half plane) if and only if for each x is an element of H and each mu is an element of R one has: [GRAPHICS]