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Transactions of the American Mathematical Society

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$ L\sp p$ spectra of pseudodifferential operators generating integrated semigroups

Author: Matthias Hieber
Journal: Trans. Amer. Math. Soc. 347 (1995), 4023-4035
MSC: Primary 47G30; Secondary 35P05, 35S05, 47D06
MathSciNet review: 1303120
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Abstract: Consider the $ {L^p}$-realization $ {\text{O}}{{\text{p}}_p}(a)$ of a pseudodifferential operator with symbol $ a \in S_{\rho ,0}^m$ having constant coefficients. We show that for a certain class of symbols the spectrum of $ {\text{O}}{{\text{p}}_p}(a)$ is independent of $ p$. This implies that $ {\text{O}}{{\text{p}}_p}(a)$ generates an $ N$-times integrated semigroup on $ {L^p}({\mathbb{R}^n})$ for a certain $ N$ if and only if $ \rho ({\text{O}}{{\text{p}}_p}(a)) \ne \emptyset $ and the numerical range of $ a$ is contained in a left half-plane. Our method allows us also to construct examples of operators generating integrated semigroups on $ {L^p}({\mathbb{R}^n})$ if and only if $ p$ is sufficiently close to $ 2$.

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