Eventual domination for linear evolution equations

Abstract We consider two $$C_0$$ C 0 -semigroups on function spaces or, more generally, Banach lattices and give necessary sufficient conditions for the orbits of first semigroup to dominate second large times. As an important special case we $$L^2$$ L 2 -space self-adjoint operators A B which generate -semigroups; in this situation criteria existence a time $$t_1 \ge 0$$ t 1 ≥ such that $$e^{tB} e^{tA}$$ e tB tA all subsequent times $$t\ge t_1$$ . consequence our abstract theory, obtain many surprising insights into behaviour various fourth order differential operators.