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

Rabier, Patrick

doi:10.1090/S0002-9939-07-08896-X

On the index and spectrum of differential operators on $\mathbb {R}^{N}$
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by Patrick J. Rabier

Proc. Amer. Math. Soc. 135 (2007), 3875-3885

DOI: https://doi.org/10.1090/S0002-9939-07-08896-X

Published electronically: August 29, 2007

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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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