Concavity of Eigenvalue Sums and the Spectral Shift Function

Details Concavity of Eigenvalue Sums and the Spectral Shift Function Concavity of Eigenvalue Sums and the Spectral Shift Function

2001-12-26

arxiv.org/abs/math/0112279v1

J. Funct. Anal. 176 (2000) 100 - 114

Mathematics

Spectral Theory

Scientific paper

10.1006/jfan.2000.3620

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 $-\infty$ to $\lambda$ (from $\lambda$ to $+\infty$) is concave (convex) with respect to trace class perturbations. The case of relative trace class perturbations is also considered.

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