Illinois Journal of Mathematics

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

Dashan Fan, Liangpan Li, Xiaohua Yao, and Quan Zheng

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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.

Article information

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

First available in Project Euclid: 13 November 2009

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 35P05: General topics in linear spectral theory
Secondary: 42B15: Multipliers 42B30: $H^p$-spaces 46E15: Banach spaces of continuous, differentiable or analytic functions 47F05: Partial differential operators [See also 35Pxx, 58Jxx] (should also be assigned at least one other classification number in section 47)


Zheng, Quan; Li, Liangpan; Yao, Xiaohua; Fan, Dashan. The spectrum of differential operators in $H^p$ spaces. Illinois J. Math. 49 (2005), no. 1, 45--62. doi:10.1215/ijm/1258138306.

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