$L^ p$ spectra of pseudodifferential operators generating integrated semigroups

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 $2$.