Gamma-boundedness of $C_0$-semigroups and their $H^{\infty }$-functional calculi

Abstract

We discuss the notion of $\gamma$-$H^{\infty }$-bounded calculus, strong $\gamma$-$m$-$H^{\infty }$-bounded calculus on half-plane and weak-$\gamma$-Gomilko-Shi-Feng condition and give a connection between them. Then we state a characterization of generation of $\gamma$-bounded $C_0$-semigroup in $K$-convex space, which leads to a version of Gearhart-Prüss on $K$-convex space.