Nonlinear functions and the norm of the propagators of operational equations

Abstract

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 C0-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 C0-semigroups, which one has been seeking.