Convolution Operators as Generators of One-Parameter Semigroups

Abstract

Let $G$ be an $m\times m$-matrix-valued rapidly decreasing distribution on $\mathbb R^{n}$ and let $E$ be one of the following locally convex vector spaces: $\fancyscript{S}(\mathbb R^{n};\mathbb C^{m})$, $\fancyscript{D}_{L^{2}}(\mathbb R^{n};\mathbb C^{m})$, $(\fancyscript{O}_{\mu })(\mathbb R^{n};\mathbb C^{m})$ where $\mu \in [0,\infty \mathclose [$, $\fancyscript{S}'(\mathbb R^{n};\mathbb C^{m})$. Then the convolution operator $(G\,*)|_{E}$ is equal to the infinitesimal generator of a one-parameter $(C_{0})$-semigroup of operators belonging to $L(E;E)$ if and only if the weak Petrovskiĭ condition is satisfied: