A note on the propagators of second order linear differential equations in Hilbert spaces

Abstract:

The paper is concerned with the growth properties at infinity of the propagators $C( \cdot ),S( \cdot )$ of the equation $u''(t) + Bu’(t) + Au(t) = 0$, where $A,B$ are densely defined closed linear operators in a Hilbert space. We define ${\omega _0}(A,B) = \max \{ {\overline {\lim } _{t \to \infty }}{t^{ - 1}}\ln \left \| {C(t)} \right \|,{\overline {\lim } _{t \to \infty }}{t^{ - 1}}\ln \left \| {S’(t)} \right \|\}$, and give a criterion to judge whether ${\omega _0}(A,B) \leq b$ for a fixed $b \in R$.