The spectrum of differential operators in $H^p$ spaces

Abstract

This paper is concerned with linear partial differential operators with constant coefficients in $H^p(\mathbf{R} ^n)$. In the case $0 \lt p\le1$, we establish some basic properties and the spectral mapping property, and determine completely the essential spectrum, point spectrum, approximate point spectrum, continuous spectrum, and residual spectrum of such differential operators. In the case $p \gt 2$, we show that the point spectrum of such differential operators in $L^p(\mathbf{R} ^n)$ is the empty set for $p\in(2,{2n\over n-1})$, but not for $p \gt {2n\over n-1}$ in general. Moreover, we make some remarks on the case $p \gt 1$ and give several examples.