An inequality for eigenvalues of nuclear operators via traces and the generalized Hoffman–Wielandt theorem

Let A be a Hilbert-Schmidt operator, whose eigenvalues are λ_k(A)(k=1,2,…).We derivea new inequality for the series ∑_k=1^∞|λk(A)-zk|², where {z_k} is a sequence of numberssatisfying the condition∑_k|zk|²<∞. That inequality is expressedvia the self-commutator AA^∗-A^∗A. If A is a nuclear operator, we obtain an inequality for the eigenvalues via the trace and self-commutator. Our results are based on the generalization of the theorem of R. Bhatia andL. Elsner [1] which is an infinite-dimensional analog of the Hoffman–Wielandttheorem on perturbations of normal matrices.