Singular Values of Compact Pseudodifferential Operators

This paper investigates the asymptotic decay of the singular values of compact operators arising from the Weyl correspondence. The motivating problem is to find sufficient conditions on a symbol which ensure that the corresponding operator has singular values with a prescribed rate of decay. The problem is approached by using a Gabor frame expansion of the symbol to construct an approximating finite rank operator. This establishes a variety of sufficient conditions for the associated operator to be in a particular Schatten class. In particular, an improvement of a sufficient condition of Daubechies for an operator to be trace-class is obtained. In addition, a new development and improvement of the Calderón–Vaillancourt theorem in the context of the Weyl correspondence is given. Additional results of this type are then obtained by interpolation.