A characterization of the generators of analytic \(C_0\)-semigroups in the class of scalar type spectral operators (Q1773559)

Let \(A\) be a scalar type spectral operator in a complex Banach space \(X\) and let \[ e^{tA}:= \int_{\mathbb C} e^{t \lambda }d E_A(\lambda)\;d\lambda, \quad t >0, \] where \(E_A\) is the operator's resolution of the identity. The author proves that \(A\) is the infinitesimal generator of an analytic \(C_0\)-semigroup if and only if one of the equalities \[ E^{\{1\}} (A) = \bigcap_{t>0}R(e^{tA}), \quad E^{(1)}(A) = \bigcup_{t>0}R(e^{tA}) \] holds, where \(E^{\{1\}} (A) \) and \(E^{(1)}(A)\) are the first order Gevrey classes of the operator \(A\) of Roumie's and Beurling's types, respectively.