A simple criterion for essential self-adjointness of Weyl pseudodifferential operators

Abstract

We prove a new criterion for the essential self-adjointness of pseudodifferential operators that does not involve ellipticity-type assumptions. For example, we show that self-adjointness holds in case the symbol is $C^{2d+3}$ with derivatives of order two and higher being uniformly bounded. These results also apply to hermitian operator-valued symbols on infinite-dimensional Hilbert spaces, which are important to applications in physics. Our method relies on a phase space differential calculus for quadratic forms on $L^2(\mathbb{R}^d)$, Calder\'on-Vaillancourt type theorems, and a recent self-adjointness result for Toeplitz operators on the Segal-Bargmann space.