© Hindawi Publishing Corp. ON SCALAR TYPE SPECTRAL OPERATORS, INFINITE DIFFERENTIABLE AND GEVREY ULTRADIFFERENTIABLE C0-SEMIGROUPS

Necessary and sufficient conditions for a scalar type spectral operator in a Banach space to be a generator of an infinite differentiable or a Gevrey ultradifferentiable C0-semigroup are found, the latter formulated exclusively in terms of the operator’s spectrum. 2000 Mathematics Subject Classification: 47B40, 47D03, 47B15, 34G10. 1. Introduction. Despite what was said in the final remarks to [22], the author did decide to tackle the problems of the generation of infinite differentiable and Gevrey ultradifferentible C0-semigroups by a scalar type spectral operator in a complex Banach space. The more so as, in the former case, the task turned out to be more of a challenge than it seemed initially, the existence of a general characterization of infinite differ