# Journal of Operator Theory

Volume 45, Issue 1, Winter 2001 pp. 81-110.

Asymptotics of subcoercive semigroups on nilpotent Lie groups**Authors**: Nick Dungey (1), A.F.M. ter Elst (2), Derek W. Robinson (3), and Adam Sikora (4)

**Author institution:**(1) Centre for Mathematics and its Applications, School of Mathematical Sciences, Australian National University, Canberra, ACT 0200, Australia

(2) Department of Mathematics and Computing Science, Eindhoven University of Technology, P.O. Box 513, 5600 MB Eindhoven, The Netherlands

(3) Centre for Mathematics and its Applications, School of Mathematical Sciences, Australian National University, Canberra, ACT 0200, Australia

(4) Centre for Mathematics and its Applications, School of Mathematical Sciences, Australian National University, Canberra, ACT 0200, Australia

**Summary:**One can associate asymptotic approximates $G_\infty$ and $H_\infty$ with each nilpotent Lie group $G$ and pure $m$-th order weighted subcoercive operator $H$ by a scaling limit. Then the semigroups $S$ and $S^{(\infty)}$ generated by $H$ and $H_\infty$, on the spaces $L_p(G)$, $p\in[1,\infty]$, satisfy $ \lim\limits_{t\to\infty}\|S_t-S^{(\infty)}_t\|_{p\to p}=0 $ if, and only if, $G=G_\infty$. If $G\neq G_\infty$ then $ \lim\limits_{t\to\infty}\|M_f(S_t-S^{(\infty)}_t)\|_{p\to p}=0 $ on the spaces $L_p(\gotg)$, where $\gotg$ denotes the Lie algebra of $G$, and $M_f$ denotes the operator of multiplication by any bounded function which vanishes at infinity.

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