Exponential dichotomy and admissibility of linearized skew-product semiflows defined on a compact positively invariant subset of semiflows

We study exponential dichotomy of linear skew-product semiflows which come from linearizing skew-product semiflows on a compact positively invariant subset MM of semiflows and construct the relationship between continuous separation and exponential dichotomy under assumptions that skew-product semiflows are eventually strongly monotone. In addition, we deduce that the exponential dichotomy is trivial when MM is hyperbolically stable, and the hyperbolic instability of MM is the necessary condition of the state space admitting a trivial separation in another forms. Simultaneously, we list some conditions for hyperbolic stability and instability of MM. At last, we construct a sufficient and necessary condition for exponential dichotomy of linear skew-product semiflows in terms of the admissibility of the pair (B(R+,X),B0(R+,X))(B(R+,X),B0(R+,X)).