One-parameter semigroups in the convolution algebra of rapidly decreasing distributions

Jan Kisyński¹

¹

Colloquium Mathematicum, Tome 128 (2012) no. 1, pp. 49-68.

Résumé

The paper is devoted to infinitely differentiable one-parameter convolution semigroups in the convolution algebra ${\mathcal{O}}_C^{\prime }({\mathbb{R}}^n;M_{m\times m})$ of matrix valued rapidly decreasing distributions on ${\mathbb{R}}^n$. It is proved that $G\in {\mathcal{O}}_C^{\prime }({\mathbb{R}}^n;M_{m\times m})$ is the generating distribution of an i.d.c.s. if and only if the operator ${\partial }_t\otimes {\mathbb{1}}_{m\times m}-G\ast$ on ${\mathbb{R}}^{1+n}$ satisfies the Petrovskiĭ condition for forward evolution. Some consequences are discussed.

Détail

DOI : 10.4064/cm128-1-6

Mots-clés : paper devoted infinitely differentiable one parameter convolution semigroups convolution algebra mathcal prime mathbb times matrix valued rapidly decreasing distributions mathbb proved mathcal prime mathbb times generating distribution s only operator partial otimes mathbb times g * mathbb satisfies petrovski condition forward evolution consequences discussed