A generalization of the monodromy operator for non-periodic linear differential equations (Q1355028)

Let \({dx\over dt}=A(t)x\) be a system of ordinary differential equations in a complex Banach space \(X\), where the real variable \(t\) belongs to an interval \([a,\infty)\) for some fixed \(a\in\mathbb{R}\cup-\infty\). In the paper the notion of a characteristic operator for such systems is introduced. The authors establish the formula relating the spectrum of the characteristic operator to the general exponent of the equations and describe conditions for stability or exponential dichotomy in terms of this operator. It is shown that the characteristic operator generalizes the monodromy operator for periodic equations. The perturbation results which extend previous ones from \textit{W. A. Coppel} [Lect. Notes Math. 629 (1978; Zbl 0376.34001)] and \textit{Yu. L. Daleckii} and \textit{M. G. Krein} [Am. Math. Soc. VI, (1974; Zbl 0286.34094)] is obtained.