Nonlinear functions and the norm of the propagators of operational equations

We explore the relation between nonlinear functions and the norm functions of the propagators of operational equations in infinite-dimensional spaces. A general criterion is obtained to judge whether a given nonlinear function is the norm function of some C 0 -semigroup, the propagator of a wellposed Cauchy problem for an operational equation in an infinite-dimensional space, which gives a solution to an open problem posed in Lin et al. (2010) [6]. Moreover, applying our abstract results, we show the existence of two valuable C 0 -semigroups, which one has been seeking.