Stability estimates for semigroups on Banach spaces

For a strongly continuous operator semigroup on a Banach space, we revisit a quantitative version of Datko's Theorem and the estimates for the constant $M$ satisfying the inequality $||T(t)|| ≤ M e^{\omega t}$, for all $t\ge0$, in terms of the norm of the convolution and other operators involved in Datko's Theorem. We use techniques recently developed by B. Helffer and J. Sjostrand for the Hilbert space case to estimate $M$ in terms of the norm of the resolvent of the generator of the semigroup in the right half-plane.