Convergence at the origin of integrated semigroups

Tom 187 / 2008

Vincent Cachia Studia Mathematica 187 (2008), 199-218 MSC: 47D62, 47D03. DOI: 10.4064/sm187-3-1


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.


  • Vincent CachiaDepartment of Theoretical Physics
    University of Geneva
    quai Ernest-Ansermet 24
    CH-1211 Genève 4, Switzerland

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