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 $\partial_{t}\otimes \mathbb{1}_{m\times m}-G\,*$ on $\mathbb R^{1+n}$ satisfies the Petrovskiĭ condition for forward evolution. Some consequences are discussed.

Détail

DOI : 10.4064/cm128-1-6

Keywords: 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