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

Let \(A\) $A$ be a Hilbert-Schmidt operator, whose eigenvalues are \(\lambda_k(A)(k=1,2 , \ldots )\) ${\lambda }_k(A)(k=1,2,\dots )$ .We derivea new inequality for the series \(\sum^{\infty}_{k=1}|\lambda_k(A)-z_k|^2\) ${\sum }_{k=1}^{\infty }{|{\lambda }_k(A)-z_k|}^2$ , where \(\{z_k\}\) $\{z_k\}$ is a sequence of numberssatisfying the condition \(\sum_k |z_k|^2<{\infty}\) ${\sum }_k{|z_k|}^2<\infty$ . That inequality is expressedvia the self-commutator \(AA^*-A^*A\) $A{A}^{\ast }-A^{\ast }A$ . If \(A\) $A$ 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.