Essentiality of the spectrum of pseudodifferential operators in the Besicovitch space $B^2(\mathbf{R}^n)$

M. A. Shubin

Vestnik Moskovskogo universiteta. Matematika, mehanika, no. 5 (1982), pp. 56-59

Résumé

Let $\bar A$ be the closure of an almost periodic pseudo-differential operator $A$ in the Besicovitch Hilbert space of almost periodic functions $B^2(\mathbf R^n)$. The following theorem is proved. If $0\operatorname{dim} \operatorname{Ker}(\bar A-\lambda I)\infty$ for some $\lambda\in\mathbf C$ then the image $\operatorname{Im}(\bar A-\lambda I)$ is not closed in $B^2(\mathbf R^n)$. A corollary: If $A$ Is bounded, elliptic or hypoelliptic (from the Hörmander class) and $\bar A-\lambda I$ is Fredholm then it is invertible (i.e. possesses a bounded everywhere defined inverse operator). For a self-adjoint $A$ this is equivalent to the essentiality of the spectrum, i.e. the fact that every point of the spectrum is either non-isolated or is an eigenvalue of infinite multiplicity.