The non-analytic growth bound of a \(C_{0}\)-semigroup and inhomogeneous Cauchy problems. (Q1414007)

This paper deals with the characterization of the non-analytic growth bound of a strongly continuous semigroup \((T(t))_{t\geq 0}\) on a Banach space \(X\) by means of Fourier multiplier properties of the resolvent of the generator \(A\) far from the real axis and also by the existence and uniqueness of mild solutions of the corresponding inhomogeneous Cauchy problem \[ u'(t)=Au(t)+f(t),\quad t\in \mathbb R , \] where the Carleman spectra of \(f\) and \(u\) satisfy \[ isp_c(f)\cup isp_c(u)\subset \rho(A). \] Here \(\rho(A)\) denotes the resolvent set of \(A\). These characterizations correspond to well-known results on hyperbolic semigroups.