Convex Operator Functions

Let (x) be a convex downwards function, A > 0 a selfadjoint operator in a Hilbert space H, P an orthogonal projector in H; suppose DA PH is dense in PH, and let Ap be the Friedrichs extension of the operator PAP defined on DA PH. The inequality tr (Ap) ≤ tr (PAP) is proved. An estimate for the Jacobi θ-function and a distant generalization of the Szasz inequality are obtained as corollaries. Bibliography: 3 items.