Absolute equal distribution of the spectra of Hermitian matrices

If −∞<α<β<∞ let mid(α,x,β)=α if x<α, x if α x β, β if x>β. Let An=Bn+Pn where Bn and Pn are n×n Hermitian matrices. We show that if Pn F2=o(n) then, for any [α,β], (A) ∑i=1nF(mid(α,λi(An),β))−F(mid(α,λi(Bn),β))=o(n) if F C[α,β]. (Eigenvalues numbered in nondecreasing order.) We consider the special case where {Pn} are real Hankel matrices. We also show that if rank(Pn)=o(n) then (A) holds for every [α,β] and F C[α,β]. Combining these results yields a result concerning Cn=Bn+En+Rn, where En 2F=o(n) and rank(Rn)=o(n). We also consider the case where the conditions on {En} are stated in terms of Schatten p-norms. Finally, we show that if {Tn} are Hermitian Toeplitz matrices generated by f C[−π,π] with minimum mf and maximum Mf, (2(i−1)−n)π/n ξin (2i−n)π/n, 1 i n, and τn is a permutation of {1,2,…,n} such that f(ξτn(1),n) f(ξτn(2),n) f(ξτn(n),n), then ∑i=1nF(λi(Tn))−F(f(ξτn(i),n))=o(n) if F C[mf,Mf].