Convergence at the origin of integrated semigroups

Vincent Cachia ¹

¹ Department of Theoretical Physics University of Geneva quai Ernest-Ansermet 24 CH-1211 Genève 4, Switzerland

Studia Mathematica, Tome 187 (2008) no. 3, pp. 199-218

Résumé

We study a classification of $\kappa$-times integrated semigroups (for $\kappa>0$) by their (uniform) rate of convergence at the origin: $\|S(t)\|={\cal O}(t^\alpha)$ as $t\rightarrow 0$ (${0\leq\alpha\leq\kappa}$). By an improved generation theorem we characterize this behaviour by Hille–Yosida type estimates. Then we consider integrated semigroups with holomorphic extension and characterize their convergence at the origin, as well as the existence of boundary values, by estimates of the associated holomorphic semigroup. Various examples illustrate these results. The particular case $\alpha=\kappa$, which corresponds to the notions of Riesz means or tempered integrated semigroups, is of special interest; as an application, it leads to an integrated version of Euler's exponential formula.

Détail

DOI : 10.4064/sm187-3-1

Keywords: study classification kappa times integrated semigroups kappa their uniform rate convergence origin cal alpha rightarrow leq alpha leq kappa improved generation theorem characterize behaviour hille yosida type estimates consider integrated semigroups holomorphic extension characterize their convergence origin existence boundary values estimates associated holomorphic semigroup various examples illustrate these results particular alpha kappa which corresponds notions riesz means tempered integrated semigroups special interest application leads integrated version eulers exponential formula