The Berezin and Garding Inequalities

Abstract

Let ϕ be a convex function on ℂ, let ℒ(σ) be a pseudodifferential operator with symbol σ, let Λσ be the set of its eigenvalues, and let m(λ) be the multiplicity of an eigenvalue λ ∈ Λσ. 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.