Variation of discrete spectra (Q1095367)

This paper is devoted to the extension to self-adjoint operators \(A, B\) and \(C=B-A\) in Hilbert space, with \(C\) compact, of the previously known property of Hermitian matrices according to which their eigenvalues \(\alpha_ j\), \(\beta_ j\) and \(\gamma_ j\) (repeated according to multiplicity) can be enumerated in such a way that for any real valued convex function \(\Phi\) on \({\mathbb{R}}\), \[ \sum_{j}\Phi (\beta_ j- \alpha_ j)\leq \sum_{k}\Phi (\gamma_ k). \] The extension to infinite dimension requires in addition that \(\Phi\geq 0\) with \(\Phi (0)=0\), and that the enumeration of the \(\alpha_ j\) and \(\beta_ j\) include, besides the discrete eigenvalues repeated according to multiplicity, some boundary points (possibly in infinite number) of the common essential spectrum of \(A\) and \(B\). The proof relies on analytic perturbation theory for the family of operators \(A+tC\), \(t\in {\mathbb{R}}\).