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

Volume 128 / 2012

Jan Kisyński Colloquium Mathematicum 128 (2012), 49-68 MSC: Primary 35E15, 47D06, 46F99, 42B99. DOI: 10.4064/cm128-1-6

Abstract

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.

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