Mathematics – Functional Analysis
Scientific paper
2009-10-06
Mathematics
Functional Analysis
Scientific paper
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.
No associations
LandOfFree
The Petrovskii correctness and semigroups of operators does not yet have a rating. At this time, there are no reviews or comments for this scientific paper.
If you have personal experience with The Petrovskii correctness and semigroups of operators, we encourage you to share that experience with our LandOfFree.com community. Your opinion is very important and The Petrovskii correctness and semigroups of operators will most certainly appreciate the feedback.
Profile ID: LFWR-SCP-O-35362