Geodesic
Convex operator functions
Sbornik. Mathematics, Tome 17 (1972) no. 2, pp. 269-277 Cet article a éte moissonné depuis la source Math-Net.Ru

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Let $\varphi(x)$ be a convex downwards function, $A>0$ a selfadjoint operator in a Hilbert space $H$, $P$ an orthogonal projector in $H$; suppose $D_A\cap PH$ is dense in $PH$, and let $A_p$ be the Friedrichs extension of the operator $PAP$ defined on $D_A\cap PH$. The inequality $\mathrm{Sp}\varphi(A_p)\leqslant\mathrm{Sp}\varphi(PAP)$ is proved. An estimate for the Jacobi $\theta$-function and a distant generalization of the Szasz inequality are obtained as corollaries. Bibliography: 3 titles. 
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     author = {F. A. Berezin},
     title = {Convex operator functions},
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     language = {en},
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F. A. Berezin. Convex operator functions. Sbornik. Mathematics, Tome 17 (1972) no. 2, pp. 269-277. http://geodesic.mathdoc.fr/item/SM_1972_17_2_a7/

[1] M. A. Naimark, “Ob odnom predstavlenii additivnykh operatornykh funktsii mnozhestv”, DAN SSSR, 41 (1943), 373–375

[2] N. I. Akhiezer, I. M. Glazman, Teoriya lineinykh operatorov, Gostekhizdat, Moskva, 1957

[3] R. Bellman, Vvedenie v teoriyu matrits, Nauka, Moskva, 1969 | MR | Zbl