Geodesic
The Berezin and G\aa rding Inequalities
Funkcionalʹnyj analiz i ego priloženiâ, Tome 39 (2005) no. 4, pp. 69-77

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Let $\varphi$ be a convex function on $\mathbb{C}$, let $\mathcal{L}(\sigma)$ be a pseudodifferential operator with symbol $\sigma$, let $\Lambda_\sigma$ be the set of its eigenvalues, and let $m(\lambda)$ be the multiplicity of an eigenvalue $\lambda\in\Lambda_\sigma$. Under certain natural assumptions about properties of pseudodifferential operators, we prove that $\sum_{\lambda\in\Lambda_\sigma}m(\lambda)\varphi(\lambda)\le\operatorname{Re}\operatorname{Tr}\mathcal{L}(\varphi(\sigma))+R$, where $R$ is an error term of the same order as the remainder term in the Gårding inequality.
Keywords: convex function, operator inequality. 
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     author = {Yu. G. Safarov},
     title = {The {Berezin} and {G\aa} rding {Inequalities}},
     journal = {Funkcionalʹnyj analiz i ego prilo\v{z}eni\^a},
     pages = {69--77},
     publisher = {mathdoc},
     volume = {39},
     number = {4},
     year = {2005},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/FAA_2005_39_4_a5/}
}
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Yu. G. Safarov. The Berezin and G\aa rding Inequalities. Funkcionalʹnyj analiz i ego priloženiâ, Tome 39 (2005) no. 4, pp. 69-77. http://geodesic.mathdoc.fr/item/FAA_2005_39_4_a5/