Asymptotic properties of matrix differential operators

For given matrices A(s) and B(s) whose entries are polynomials in s, the validity of the following implication is investigated: ylimt ¿ 8 A(D) y(t) = 0 limt ¿ 8 B(D) y(t) = 0. Here D denotes the differentiation operator and y stands for a sufficiently smooth vector valued function. Necessary and sufficient conditions on A(s) and B(s) for this implication to be true are given. A similar result is obtained in connection with an implication of the form y A(D) y(t) = 0, limt ¿ 8 B(D) y(t) = 0, C(D) y(t) is bounded limt ¿ 8 E(D) y(t) = 0.