2021/03/09 by Marat V. Markin, Markin, Marat V.
Computer Science · Engineering · Mathematics · #47D03 #47D06 #47D60 #Advanced Mathematical Modeling in Engineering #Dynamical Systems (math.DS) #FOS: Mathematics #Functional Analysis (math.FA) #Primary 47B40 #Secondary 47B15 #Spectral Theory (math.SP) #Spectral Theory in Mathematical Physics #Stability and Controllability of Differential Equations
paper · pdf · doi:10.48550/arxiv.2103.05260
openalex publication_date 2021/03/09 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
We show that, for the C0-semigroups of scalar type spectral operators, a well-known necessary condition for the generation of eventually norm-continuous C0-semigroups, formulated exclusively in terms of the location of the spectrum of the semigroup's generator in the complex plane, is also sufficient and, in fact, characterizes the generators of immediately norm-continuous such semigroups. Combining characterizations of the immediate differentiability and the Gevrey ultradifferentiability of scalar type spectral C0-semigroups with the generation theorem, found earlier by the author, we arrive at respective characterizations of the generation of such semigroups. We further establish characterizations of the generation of eventually differentiable and immediately compact scalar type spectral C0-semigroups also in terms of the generator's spectrum and show that, for such semigroups, eventual compactness implies immediate. All the obtained results are instantly transferred to the C0-semigroups of normal operators.