On the index and spectrum of differential operators on \(\mathbb{R}^{N}\). (English) Zbl 1129.47034
Let \(P(x,\partial )\) be an \(r\times r\) system of differential operators having continuous coefficients with vanishing oscillation at infinity. Within the Banach algebra approach to the Fredholm theory for differential and pseudodifferential operators, it was proved that \(P(x,\partial)\) is Fredholm from \((W^{m,p})^r\) to \((L^p)^r\) for all or no values of \(p\in (1,\infty )\) [cf.H. O. Cordes, Appl. Anal. 2, 115–129 (1972; Zbl 0244.46080); R. Illner, Commun.Partial Differ.Equations 2, 359–393 (1977; Zbl 0352.47021); S.–H. Sun, Sci. Sin., Ser. A 27, 337–344 (1984; Zbl 0547.47033)].
The author proves in the present paper that both the index (when defined) and the spectrum of \(P(x,\partial)\) are independent of \(p\). His method makes use of the connection between the Fredholm property and the asymptotic behavior of solutions found earlier by the author [J. Differ.Equations 193, No. 2, 460–480 (2003; Zbl 1044.47035); erratum ibid. 237, No. 1, 257 (2007)].
The author proves in the present paper that both the index (when defined) and the spectrum of \(P(x,\partial)\) are independent of \(p\). His method makes use of the connection between the Fredholm property and the asymptotic behavior of solutions found earlier by the author [J. Differ.Equations 193, No. 2, 460–480 (2003; Zbl 1044.47035); erratum ibid. 237, No. 1, 257 (2007)].
Reviewer: Anatoly N. Kochubei (Kyïv)
MSC:
| 47F05 | General theory of partial differential operators |
| 35J45 | Systems of elliptic equations, general (MSC2000) |
| 47A53 | (Semi-) Fredholm operators; index theories |
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