On the inverse Laplace transform for \(C_0\)-semigroups in UMD-spaces (Q1289287)

The authors show that for a \(C_{0}\)-semigroup \((T(t))_{t \geq 0}\) on a Banach space \(X\), the representation \(T(t)x = {1 \over 2 \pi i}\lim_{R \to \infty} \int_{a-iR}^{a+iR} e^{t \lambda} R(\lambda, A)x d \lambda\) holds for every \(x \in X\) if \(X\) is an UMD-space. They show by a counterexample that this does not hold in general.