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 $L^2({\mathbb{R}}^d)$, Calderón-Vaillancourt type theorems, and a recent self-adjointness result for Toeplitz operators on the Segal-Bargmann space.

6+3 pages. v3: improved presentation, new Appendix fixing an error in (Bauer et al. 2023. J. Funct. Anal. 284, 109778. arXiv:2202.04687)

References in corpus (1)

• Self-Adjointness of Toeplitz Operators on the Segal-Bargmann Space

Cited by in corpus (1)

• Heisenberg-smooth operators from the phase space perspective