Asymptotic behaviour of -semigroups and γ-boundedness of the resolvent

Let X be a Banach space with type p and cotype q, and let A be the infinitesimal generator of a C 0 -semigroup on X. We show that if the resolvent of A has a γ-bounded analytic extension to C + , then A has negative fractional growth bound of order 1 p − 1 q .