The Petrovskii correctness and semigroups of operators

TL;DR

This paper characterizes when matrix partial differential operators generate strongly continuous semigroups on spaces of smooth functions, linking Petrovskii correctness to the semigroup property and describing their exponential nature.

Contribution

It establishes a precise criterion, Petrovskii correctness, for matrix PDOs to generate $C_0$-semigroups on various function spaces, extending the understanding of operator semigroup theory.

Findings

Characterization of generators via Petrovskii correctness.

Identification of the semigroup as exponential with stability index.

Extension of results to tempered distributions and other function spaces.

Abstract

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.