Summary: This paper is concerned with linear nonautonomous differential equations in Banach spaces. We analyze spectral properties of the semigroup of operators \(T^h\), \(h \geq 0\) defined by the formula \((T^hv) (t) = X(t,t - h) v(t - h)\), where \(X(t,s)\) is the evolution operator of the underlying differential equation and \(v\) is an element of an appropriate function space. We present necessary and sufficient conditions for the exponential decay of all solutions or of some individual solutions of the underlying differential equation.
For the entire collection see [Zbl 0834.00036].