Semigroups and Scalar-Type Operators in Banach Spaces

The main result is that if {Tt : t ≥ 0} is a (C0)-semigroup of scalar-type operators (in Dunford′s sense) on a Banach space and if the resolutions of the identity Et for Tt are uniformly bounded in norm, then the generator is of scalar-type with resolution of the identity E for which Tt = ∫ exp(tλ) E(dλ) for t ≥ 0. In the case when either {Tt} is periodic or the Banach space is of special type and the spectra σ(Tt) are all contained in the unit circle, this sufficient condition can be relaxed significantly.