The problem of nonuniform exponential dichotomy of linear differential
systems in Banach spaces is discussed. It is established a connection
between the admissibility of a pair of certain function spaces which are translations
invariant, on one hand, and the nonuniform exponential dichotomy
of differential systems, on the other. Also, Some results due to Hartman,
Massera, Schäffer and Coppel are generalized as well.