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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

The spectrum of partial differential operators on $ L\sp{p}\,(R\sp{n})$


Authors: Franklin T. Iha and C. F. Schubert
Journal: Trans. Amer. Math. Soc. 152 (1970), 215-226
MSC: Primary 47.65; Secondary 35.00
MathSciNet review: 0270211
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Abstract: The purpose of this paper is to prove that if the polynomial $ P(\xi )$ associated with a partial differential operator $ P$ on $ {L^p}({R^n})$, with constant coefficients, has the growth property, $ \vert P(\xi ){\vert^{ - 1}} = O(\vert\xi {\vert^{ - r}}),\vert\xi \vert \to \infty $ for some $ r > 0$, then the spectrum of $ P$ is either the whole complex plane or it is the numerical range of $ P(\xi )$; and if $ P(\xi )$ has some additional property (all the coefficients of $ P(\xi )$ being real, for example), then the spectrum of $ P$ is the numerical range for those $ p$ sufficiently close to 2. It is further shown that the growth property alone is not sufficient to ensure that the spectrum of $ P$ is the numerical range of $ P(\xi )$.


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DOI: http://dx.doi.org/10.1090/S0002-9947-1970-0270211-2
PII: S 0002-9947(1970)0270211-2
Keywords: Spectrum of differential operators, partial differential operators
Article copyright: © Copyright 1970 American Mathematical Society