The spectrum of partial differential operators on 𝐿^{𝑝}(𝑅ⁿ)
| Autor: | Franklin T. Iha, C. F. Schubert |
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| Rok vydání: | 1970 |
| Předmět: | |
| Zdroj: | Transactions of the American Mathematical Society. 152:215-226 |
| ISSN: | 1088-6850 0002-9947 |
| DOI: | 10.1090/s0002-9947-1970-0270211-2 |
| Popis: | The purpose of this paper is to prove that if the polynomial P ( ξ ) P(\xi ) associated with a partial differential operator P P on L p ( R n ) {L^p}({R^n}) , with constant coefficients, has the growth property, | P ( ξ ) | − 1 = O ( | ξ | − r ) , | ξ | → ∞ |P(\xi ){|^{ - 1}} = O(|\xi {|^{ - r}}),|\xi | \to \infty for some r > 0 r > 0 , then the spectrum of P P is either the whole complex plane or it is the numerical range of P ( ξ ) P(\xi ) ; and if P ( ξ ) P(\xi ) has some additional property (all the coefficients of P ( ξ ) P(\xi ) being real, for example), then the spectrum of P P is the numerical range for those p p sufficiently close to 2. It is further shown that the growth property alone is not sufficient to ensure that the spectrum of P P is the numerical range of P ( ξ ) P(\xi ) . |
| Databáze: | OpenAIRE |
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