Description: Let $P(partial/partial x)$ be an $mtimes n$ 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 $(S_t)_{tge0}subset L((calS(bbR^n))^m)$ if and only if $P(partial/partial x)$ satisfies the Petrovskiui correctness condition. Moreover, if it is the case, then $(S_t)_{tge0}$ 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.