Equivalences between nonuniform exponential dichotomy and admissibility

Relationship between exponential dichotomies and admissibility of function classes is a significant problem for hyperbolic dynamical systems. It was proved that a nonuniform exponential dichotomy implies several admissible pairs of function classes and conversely some admissible pairs were found to imply a nonuniform exponential dichotomy. In this paper we find an appropriate admissible pair of classes of Lyapunov bounded functions which is equivalent to the existence of nonuniform exponential dichotomy on half-lines R±R± separately, on both half-lines R±R± simultaneously, and on the whole line RR. Additionally, the maximal admissibility is proved in the case on both half-lines R±R± simultaneously.