An extension of a theorem of R. Datko to the case of (non)uniform exponential stability of linear skew-product semiflows

In 1970 R. Datko proved that the trajectories of a C0C0-semigroup {T(t)}t≥0{T(t)}t≥0 on a Hilbert space X  , exhibit an exponential decay if and only if they stay in L2(R+,X)L2(R+,X). Datko's theorem was crucial in extending the Lyapunov operator equation to the case of autonomous systems x˙=Ax with unbounded A. Extensions have been done to the case of uniform exponential stability of evolution families and more recently of LSPS (linear skew-product semiflow). The aim of the present paper is to extend Datko's result to the general case of (non)uniform exponential stability of the LSPS that does not necessarily possess a uniform exponential growth.