A spectral theory of continuous functions and the Loomis-Arendt-Batty-Vu theory on the asymptotic behavior of solutions of evolution equations
The main purpose of the paper is to provide a simple and \(C_0\)-semigroup free approach to the Loomis-Arendt-Batty-Vu theory on the asymptotic behavior of solutions of evolution equations (see, e.g., [\textit{W. Arendt, C. J. K. Batty, M. Hieber, F. Neubrander}, Vector-valued Laplace Transforms and Cauchy Problems. Monographs in Mathematics. 96. (Basel): Birkhäuser. (2001; Zbl 0978.34001)]). First, a new approach to the concept of reduced spectrum on \(R\) or on \(R^+\) is presented. As a result, a new relation between the reduced spectrum and the spectrum of the differentiation operator is established. Gelfand's theorem is extended as the key tool for proving Loomis' theorem. A version of Loomis' theorem with an ergodic condition is then derived and used to obtain properties of the differentiation operator on the half-line. In the application of these results to stability of evolution equations, among other things, well-posedness assumption can be omitted. Reviewer's remark: In nearly the whole paper the author claims that his results are true without the assumption that functions considered are uniformly continuous. However, it is said in the final remark that uniform continuity is still crucial.
scientific article; zbMATH DE number 5577116
scientific article