## Illinois Journal of Mathematics

### The spectrum of differential operators in $H\sp p$ spaces

#### Abstract

This paper is concerned with linear partial differential operators with constant coefficients in $H^p(\mathbf{R} ^n)$. In the case $0<p\le1$, 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 $p>2$, 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 $p>{2n\over n-1}$ in general. Moreover, we make some remarks on the case $p>1$ and give several examples.

#### Article information

Source
Illinois J. Math. Volume 49, Number 1 (2005), 45-62.

Dates
First available: 13 November 2009

Permanent link to this document
http://projecteuclid.org/euclid.ijm/1258138306

Mathematical Reviews number (MathSciNet)
MR2157368

Zentralblatt MATH identifier
1081.35061

#### Citation

Zheng, Quan; Li, Liangpan; Yao, Xiaohua; Fan, Dashan. The spectrum of differential operators in $H\sp p$ spaces. Illinois Journal of Mathematics 49 (2005), no. 1, 45--62. http://projecteuclid.org/euclid.ijm/1258138306.