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 lambda(k)(A)(k=1,2,& mldr;). We derive a new inequality for the series & sum;(infinity)(k=1)|lambda(k)(A)-z(k)|(2), where {z(k)} is a sequence of numbers satisfying the condition & sum;(k)|z(k)|(2)<infinity. That inequality is expressed via the self-commutator AA(& lowast;)-A(& lowast;)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 and L. Elsner [1] which is an infinite-dimensional analog of the Hoffman-Wielandt theorem on perturbations of normal matrices.