Nonlinear functions and the norm of the propagators of operational equations (Q1948188)

In this short, but interesting paper, the authors study the relation between positive, nonlinear functions and norm functions of semigroups in infinite dimensional spaces. In the (single) main result, it states that, if \(g:[0, \infty )\mapsto [0, \infty )\) is a continuous function satisfying \(g(0)=1\) and \(g(t+s)\leq g(t)g(s)\) for \(t, s \geq 0\), then there exists a \(C_0\) semigroup \((T(t))_{t\geq 0}\) on an infinite-dimensional Banach space such that \(g(t)= \|T(t)\|\) for \(t\geq 0\). As applications, the authors show the existence of two valuable \(C_0\)-semigrous: one is non-nilpotent but has growth bound \(-\infty\) and the other has growth as slow as logarithmical.