  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-Pr\"uss on $K$-convex space.
