A characterization of the growth bound of a semigroup via Fourier multipliers (Q2708557)

Let \(A\) be the generator of a strongly continuous semigroup \(T = (T_t)_{t \geq 0}\) on the Banach space \(X\). Its spectral bound \(s (A)\) equals \(\sup \{ \Re (\lambda) : \lambda \in \sigma (A) \}\) and its growth bound is defined by \(\omega_0 = \inf \{ \omega \in \mathbb R : \sup_{t \geq 0} e^{- \omega t} \|T_t \|< \infty \} \).NEWLINENEWLINENEWLINEDue to a famous result of Gearhart NEWLINE\[NEWLINE\omega_0 (T) = \inf \{ \mu > s (A) : \sup_{\Re (\lambda) \geq \mu} \|R (\lambda , A) \|< \infty \}NEWLINE\]NEWLINE holds in Hilbert spaces, where \(R (\lambda , A) = ( \lambda I - A)^{-1}\). In this paper the result above is generalized in the following form NEWLINE\[NEWLINE\omega_0 (T) = \inf \{ \mu > s (A) : \sup_{\alpha > \mu} \|R ( \alpha + i ., A) \|_{{\mathcal M}_p ({\mathcal L} (X))} < \infty \}.NEWLINE\]NEWLINE Here \({\mathcal M}_p ( {\mathcal L} (X))\) is the space of all \({\mathcal L} (X) \)--valued multipliers for \(L^p (\mathbb R , X)\); see the paper for details.NEWLINENEWLINEFor the entire collection see [Zbl 0957.00037].