Inequalities for imaginary parts of eigenvalues of Schatten–von Neumann operators

Let S_p be the p-Schatten–von Neumann ideal of operators in a Hilbert space, A∈ S_p(2 ≤ p< ∞) , A_R= (A+ A^∗) / 2 , A_I= (A- A^∗) / (2 i) and λ_k(A) (k= 1 , 2 ,..) are the eigenvalues of A. The asterisk means the adjointness. We derive the inequality ∑k=1∞|Imλk(A)-λk(AI)|p≤ηpNpp(AR), where η_p≤ 2 p is an explicitly defined constant dependent on p, only, and N_p(.) is the norm of S_p . In addition, assuming that A_I is a nuclear operator, we obtain an inequality for the series of the imaginary parts of the eigenvalues of A via the trace of A_I .