The Petrovskii correctness and semigroups of operators

Let $P(\mathrm{\partial }\mathrm{/}\mathrm{\partial }x)$ be an $m\times n$ matrix whose entries are PDO on ${\text{bbR}}^n$ with constant coefficients, and let $\text{calS}({\text{bbR}}^n)$ be the space of infinitely differentiable rapidly decreasing functions on ${\text{bbR}}^n$. It is proved that $P(\mathrm{\partial }\mathrm{/}\mathrm{\partial }x){\mathrm{\mid }}_{(\text{calS}({\text{bbR}}^n){)}^m}$ is the infinitesimal generator of a $(C_0)$-semigroup $(S_t{)}_{t\ge 0}\subset L((\text{calS}({\text{bbR}}^n){)}^m)$ if and only if $P(\mathrm{\partial }\mathrm{/}\mathrm{\partial }x)$ satisfies the Petrovski\u\i correctness condition. Moreover, if it is the case, then $(S_t{)}_{t\ge 0}$ is an exponential semigroup whose characteristic exponent is equal to the stability index of $P(\mathrm{\partial }\mathrm{/}\mathrm{\partial }x)$. Similar statements are also proved for some other function spaces on ${\text{bbR}}^n$, and for the space of tempered distributions.