Notes on the propagators of evolution equations (Q963123)

In this note, the authors study the following problem: given a nonnegative scalar function \(g\) defined on \(\mathbb R^+\) such that \(g(0)=0\) and \(g(t+s)\leq g(t)g(s)\) for \(t,s\geq 0\), is it always possible to find a \(C^0\)-semigroup \((T(t))_{t\geq 0}\) such that \(|T(t)|=g(t)\) for \(t\geq 0\)? The answer is no, a counterexample is provided even in a finite-dimensional space. Some discussions are also given for the norm continuity, compactness and differentiability in critical points.