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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

On the index and spectrum of differential operators on $ \mathbb{R}^{N}$


Author: Patrick J. Rabier
Journal: Proc. Amer. Math. Soc. 135 (2007), 3875-3885
MSC (2000): Primary 47A53, 47F05, 35J45
Published electronically: August 29, 2007
MathSciNet review: 2341938
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Abstract: If $ P(x,\partial )$ is an $ r\times r$ system of differential operators on $ \mathbb{R}^{N}$ having continuous coefficients with vanishing oscillation at infinity, the Cordes-Illner theory ensures that $ P(x,\partial )$ is Fredholm from $ (W^{m,p})^{r}$ to $ (L^{p})^{r}$ for all or no value $ p\in (1,\infty ).$ We prove that both the index (when defined) and the spectrum of $ P(x,\partial )$ are independent of $ p.$


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Additional Information

Patrick J. Rabier
Affiliation: Department of Mathematics, University of Pittsburgh, Pittsburgh, Pennsylvania 15260
Email: rabier@imap.pitt.edu

DOI: http://dx.doi.org/10.1090/S0002-9939-07-08896-X
PII: S 0002-9939(07)08896-X
Keywords: Fredholm operator, index, differential operator, system, spectrum.
Received by editor(s): January 14, 2006
Received by editor(s) in revised form: August 27, 2006
Published electronically: August 29, 2007
Communicated by: David S. Tartakoff
Article copyright: © Copyright 2007 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.