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 C 0 -semigroup T = { T ( t ) } t ≥ 0 generated by a (possible) unbounded A . Thus, we extend some known results given by Chicone and Latushkin, Daletskii and Krein, and Datko.