Concavity of Eigenvalue Sums and the Spectral Shift Function

It is well known that the sum of negative (positive) eigenvalues of some finite Hermitian matrix V is concave (convex) with respect to V. Using the theory of the spectral shift function we generalize this property to self-adjoint operators on a separable Hilbert space with an arbitrary spectrum. More precisely, we prove that the spectral shift function integrated with respect to the spectral parameter from −∞ to λ (from λ to +∞) is concave (convex) with respect to trace class perturbations. The case of relative trace class perturbations is also considered.