$\displaystyle\left|\psi_S(t)\right>=U(t,t_0)\left|\psi_S(t_0)\right>$
$\displaystyle\left|\psi_H\right>=U^{\dagger} (t,t_0)\left|\psi_S(t)\right>=\left|\psi_S(t_0)\right>$
An observable $A_S$ in the Schrödinger representation transforms into:
$\displaystyle A_H=U^{\dagger}(t,t_0)A_SU(t,t_0)$
In general $A_H$ is not stationary even when $A_S$ doesn’t depend explicilty upon time.