Absolute equal distribution of the spectra of Hermitian matrices (Q1874680)

The author shows that if the \(n\times n\) Hermitian matrices \(C_n\), \(B_n\), \(R_n\) and \(E_n\) are related by \(C_n =B_n +R_n +E_n\), where rank\((R_n)=o(n)\) and either \(\|E_n \|_{S,p}^p =o(n)\) for some finite \(p\geq 1\) or else \(\|E_n \|_{S,\infty}=o(1)\), then, for all finite closed intervals \([\alpha ,\beta ]\) and all functions \(F\in C[\alpha ,\beta ]\), \[ \sum _{i=1}^n |F(\text{mid}(\alpha ,\lambda _i (C_n),\beta)) -F(\text{mid}(\alpha ,\lambda _i (B_n),\beta))|=o(n) \quad \text{as}\quad n\to \infty. \] Here the eigenvalues of a matrix \(A\) are labelled \(\lambda _1 (A)\leq \lambda _2 (A)\leq \dots\), its Schatten \(p\)-norm \(\|A\|_{S,p}\) is defined as usual as \(( \sum _{i=1}^n \lambda _i ^p (A)) ^{1/p}\), and mid\((\alpha ,x,\beta)\) is defined to be \(\alpha\) if \(x<\alpha\), \(\beta\) if \(x>\beta\) and \(x\) otherwise. Several related results are proved which generalize many other results in the literature, including some of the author [Linear Algebra Appl. 302-303, 155-162 (1999; Zbl 0947.15009)].