Weak Rolewicz's theorem in Hilbert spaces

Let $phi:mathbb{R}_+:=[0, infty) o mathbb{R}_+$ be a nondecreasing function which is positive on $(0, infty)$ and let $mathcal{U} ={U(t, s)}_{tge sge 0}$ be a positive strongly continuous periodic evolution family of bounded linear operators acting on a complex Hilbert space $H$. We prove that $mathcal{U}$ is uniformly exponentially stable if for each unit vector $xin H$, one has $$ int_0^infty phi(|langle U(t, 0)x, x angle|)dt