*(English)*Zbl 1058.34072

The author gives a new characterization of evolution families of operators in Banach spaces that have exponential dichotomy. The framework is similar to that of *N. V. Minh*, *F. Räbiger* and *R. Schnaubelt* [Integral Equations Oper. Theory 32, 332-353 (1998; Zbl 0977.34056)], based on the so-called “evolution semigroup” associated to an evolution family. In that paper, the exponential dichotomy of an evolution family $U(t,s)$ was characterized in terms of inversion properties of a suitably defined operator, ${I}_{Z}$, associated to the solutions of the integral equation

In the paper under consideration, the characterization is still given in terms of the operator ${I}_{Z}$ but a different characterizing property is considered which is based on the concepts of “exponentially dichotomous” and “quasi-exponentially dichotomous” operators, the latter being introduced by the author.

The new result allows the author to extend to a general Banach space results proved for finite-dimensional spaces by *A. Ben-Artzi*, *I. Gohberg* and *M. A. Kaashoek* [J. Dyn. Differ. Equations 5, 1-36 (1993; Zbl 0771.34011)].