The Berezin and Garding Inequalities

Let ? be a convex function on ?, let ?(σ) be a pseudodifferential operator with symbol σ, let Λ_σ be the set of its eigenvalues, and let m(λ) be the multiplicity of an eigenvalue λ ∈ Λ_σ. Under certain natural assumptions about properties of pseudodifferential operators, we prove that (sum {_{lambda in Lambda _sigma } m(lambda )} varphi (lambda ) leqslant operatorname{Tr} L(varphi (sigma )) + R), where R is an error term of the same order as the remainder term in the Garding inequality.