Abstract
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.
Highlights
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
In [22], the criteria of a scalar type spectral operator in a complex Banach space being a generator of a C0-semigroup and an analytic C0-semigroup were found
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 C0-semigroup are established
Summary
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. In [22], the criteria of a scalar type spectral operator in a complex Banach space being a generator of a C0-semigroup and an analytic C0-semigroup were found. 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 C0-semigroup are established. Let A be a scalar type spectral operator in a complex Banach space X and F (·) a complex-valued Borel measurable function on C (on σ (A)). Let I be an interval of the real axis, R, C∞(I, X) the set of all X-valued functions strongly infinite differentiable on I, and 0 ≤ β < ∞
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