  We give necessary integral conditions and sufficient ones for the existence
of a general concept of $\mu$-dichotomy for evolution operators defined on the
half-line which includes as particular cases the well-known concepts of
nonuniform exponential dichotomy and nonuniform polynomial dichotomy, and also
contains new situations. Additionally, we consider an adapted notion of
Lyapunov function and use our results to obtain necessary and sufficient
conditions for the existence of nonuniform $\mu$-dichotomies using these
Lyapunov functions.
