The Petrovskii correctness and semigroups of operators

Let $P(\partial/\partial x)$ be an $m\times n$ matrix whose entries are PDO on $\bbR^n$ with constant coefficients, and let $\calS(\bbR^n)$ be the space of infinitely differentiable rapidly decreasing functions on $\bbR^n$. It is proved that $P(\partial/\partial x)|_{(\calS(\bbR^n))^m}$ is the infinitesimal generator of a $(C_0)$-semigroup $(S_t)_{t\ge0}\subset L((\calS(\bbR^n))^m)$ if and only if $P(\partial/\partial x)$ satisfies the Petrovski\u\i correctness condition. Moreover, if it is the case, then $(S_t)_{t\ge0}$ is an exponential semigroup whose characteristic exponent is equal to the stability index of $P(\partial/\partial x)$. Similar statements are also proved for some other function spaces on $\bbR^n$, and for the space of tempered distributions.