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 C 0-semigroup are found, the latter formulated exclusively in terms of the operator's spectrum. 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 C 0-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-entiable C 0-semigroups [25] (see also [6, 26]) notwithstanding. In the latter case, such characterizations are not to be found in the plethora of the literature on the subject including such authoritative and exhaustive sources as [6, 9, 11, 15, 26, 28, 31]. In [22], the criteria of a scalar type spectral operator in a complex Banach space being a generator of a C 0-semigroup and an analytic C 0-semigroup were found. In the present paper, necessary and sufficient conditions for a scalar type spectral operator in a complex Banach space to be a generator of an infinite differentiable or a Gevrey ultradifferentiable C 0-semigroup are established. The main purpose is to show that such criteria, as well as those of [22], can be formulated exclusively in terms of the operator's spectrum, without any restrictions on its resolvent behavior. This fact distinguishes the case of scalar type spectral operators and makes the aformentioned results significantly more transparent and purely qualitative.