WEAK ROLEWICZ'S THEOREM IN HILBERT SPACES

Let phi : R+ := [0, infinity) -> R+ be a nondecreasing function which is positive on (0, infinity) and let U = {U(t, s)}(t >= s >= 0) be a positive strongly continuous periodic evolution family of bounded linear operators acting on a complex Hilbert space H. We prove that U is uniformly exponentially stable if for each unit vector x is an element of H, one has integral(infinity)(0) phi(vertical bar < U(t, 0)x, x >vertical bar)dt < infinity. The result seems to be new and it generalizes others of the same topic. Moreover, the proof is surprisingly simple.