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

Voir la notice de l'article provenant de la source Math-Net.Ru

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. 
@article{FAA_2005_39_4_a5,
     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/}
}
TY  - JOUR
AU  - Yu. G. Safarov
TI  - The Berezin and G\aa rding Inequalities
JO  - Funkcionalʹnyj analiz i ego priloženiâ
PY  - 2005
SP  - 69
EP  - 77
VL  - 39
IS  - 4
PB  - mathdoc
UR  - http://geodesic.mathdoc.fr/item/FAA_2005_39_4_a5/
LA  - ru
ID  - FAA_2005_39_4_a5
ER  - 
%0 Journal Article
%A Yu. G. Safarov
%T The Berezin and G\aa rding Inequalities
%J Funkcionalʹnyj analiz i ego priloženiâ
%D 2005
%P 69-77
%V 39
%N 4
%I mathdoc
%U http://geodesic.mathdoc.fr/item/FAA_2005_39_4_a5/
%G ru
%F FAA_2005_39_4_a5
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/

[1] Berezin F. A., “Vypuklye funktsii ot operatorov”, Matem. sb., 88 (1972), 268–276 | MR | Zbl

[2] Berezin F. A., “Kovariantnye i kontravariantnye simvoly operatorov”, Izv. AN SSSR, ser. matem., 36 (1972), 1134–1167 | MR | Zbl

[3] Dimassi M., Sjöstrand J., Spectral Asymptotics in the Semiclassical Limit, Cambridge University Press, 1999 | MR | Zbl

[4] Evans W. D., Lewis R. T., Siedentop H., Solovej J. P., “Counting eigenvalues using coherent states with an application to Dirac and Schrödinger operators in the semiclassical limit”, Ark. Mat., 34 (1996), 265–283 | DOI | MR | Zbl

[5] Ekeland I., Temam R., Convex Analysis and Variational Problems, Stud. Math. Appl., 1, Elsevier, North-Holland, 1976 | MR | Zbl

[6] Giles J. R., Convex Analysis with Application in the Differentiation of Convex Functions, Res. Notes Math., 58, Pitman, 1982 | MR | Zbl

[7] Khërmander L., Analiz lineinykh differentsialnykh operatorov s chastnymi proizvodnymi, t. 3, Mir, M., 1987 | MR

[8] Laptev A., Safarov Yu., “A generalization of the Berezin–Lieb inequality for convex functions”, Contemporary Mathematical Physics, Amer. Math. Soc. Transl. Ser. 2, 175, Amer. Math. Soc., Providence, R.I., 1996, 69–80 | MR

[9] Laptev A., Safarov Yu., “Szegő type limit theorems”, J. Funct. Anal., 138 (1996), 544–559 | DOI | MR | Zbl

[10] Safarov Y., “Pseudodifferential operators and linear connections”, Proc. London Math. Soc., 74 (1997), 379–417 | DOI | MR

[11] Safarov Yu., Vassiliev D., The Asymptotic Distribution of Eigenvalues of Partial Differential Operators, Transl. Math. Monographs, 155, Amer. Math. Soc., Providence, R.I., 1996 | DOI | MR | Zbl

[12] Tataru D., “On the Fefferman–Phong inequality and related problems”, Comm. Partial Differential Equations, 27:11–12 (2002), 2101–2138 | DOI | MR | Zbl