The Lyapunov operator equation for the exponential dichotomy of one-parameter semigroups

We point out a Lyapunov-type operator equation that shows the existence of an exponential dichotomy for an arbitrary C0C0-semigroup T={T(t)}t≥0 generated by a (possible) unbounded AA. Thus, we extend some known results given by Chicone and Latushkin, Daletskii and Krein, and Datko.