On scalar type spectral operators, infinite differentiable and Gevrey ultradifferentiable \(C_0\)-semigroups (Q1777704)

Let \(X\) be a complex Banach space and let \(A\) be a scalar type spectral operator on \(X\). The author proves that a \(C_0\)-semigroup generated by \(A\) is infinite differentiable if and only if for arbitrary positive \(b\) there is a real \(a\) such that \(\text{Re}\, \lambda \leq a - b\ln| \text{Im}\;\lambda| \) for all \(\lambda \in \sigma(A)\). He also obtains characterizations of \(A\) as the infinitesimal generator of a Gevrey ultra-differentiable semigroup, in terms of the spectrum \(\sigma(A)\).