The spectrum of partial differential operators on $L^{p} (R^{n})$

Abstract:

The purpose of this paper is to prove that if the polynomial $P(\xi )$ associated with a partial differential operator $P$ on ${L^p}({R^n})$, with constant coefficients, has the growth property, $|P(\xi ){|^{ - 1}} = O(|\xi {|^{ - r}}),|\xi | \to \infty$ for some $r > 0$, then the spectrum of $P$ is either the whole complex plane or it is the numerical range of $P(\xi )$; and if $P(\xi )$ has some additional property (all the coefficients of $P(\xi )$ being real, for example), then the spectrum of $P$ is the numerical range for those $p$ sufficiently close to 2. It is further shown that the growth property alone is not sufficient to ensure that the spectrum of $P$ is the numerical range of $P(\xi )$.