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

Abstract

We prove a new criterion for essential self-adjointness of pseudodifferential operators, which does not involve ellipticity-type assumptions. Essential self-adjointness is proved for symbols in C2d+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 L2(Rd), Calderón-Vaillancourt type theorems and a recent self-adjointness result for Toeplitz operators.