A family \(S(t, s)\) of bounded linear operators, \(0 \leq s \leq t < \infty\), is an evolution operator if \(S(t, t) = I,\) \(S(t, s)S(s, r) = S(t, r)\), \(0 \leq r \leq s \leq t\), and \(S(t, \cdot)u\) (resp. \(S(\cdot, s)u)\) is continuous in \([0, t]\) (resp. continuous in \([s, \infty)).\) The authors’ definition includes exponential growth. If \(T(t)\) is a strongly continuous semigroup then \(S(t, s) = T(t - s)\) is an evolution operator; in fact, evolution operators generalize semigroups for time-dependent evolution equations.
Generalizing a result of van Neerven for semigroups, the authors give conditions for uniform asymptotic stability \[ \|S(t, s)\|\leq C e^{- c(t - s)}, \quad c > 0, \] of a periodic evolution operator using the theory of Banach function spaces.