The Operator EquationAX−XB=C, Admissibility, and Asymptotic Behavior of Differential Equations

It is shown that there exists a natural relationship between the regular admissibility of a translation-invariant subspace ofBUC(R, E) (the space of uniformly continuous bounded functions onRwith values in a Banach spaceE), w.r.t. the differential equationu′(t)=Au(t)+f(t) (*), and the unique solvability of a special operator equation of Lyapunovʹs typeAX−X =−δ0, where is the restriction of the operator ≡d/dtonto andδ0(f)=f(0). This leads to some spectral conditions for the regular admissibility. Applications to questions of exponential dichotomy, exponential stability, the existence of periodic and almost periodic solutions of equation (*), as well as analogous questions for some nonlinear equations and functional differential equations are presented.