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 $$C^{2d+3}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msup> <mml:mi>C</mml:mi> <mml:mrow> <mml:mn>2</mml:mn> <mml:mi>d</mml:mi> <mml:mo>+</mml:mo> <mml:mn>3</mml:mn> </mml:mrow> </mml:msup> </mml:math> 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 $$L^2(\mathbb {R}^d)$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:msup> <mml:mi>L</mml:mi> <mml:mn>2</mml:mn> </mml:msup> <mml:mrow> <mml:mo>(</mml:mo> <mml:msup> <mml:mrow> <mml:mi>R</mml:mi> </mml:mrow> <mml:mi>d</mml:mi> </mml:msup> <mml:mo>)</mml:mo> </mml:mrow> </mml:mrow> </mml:math> , Calderón-Vaillancourt type theorems and a recent self-adjointness result for Toeplitz operators.