An inequality for imaginary parts of eigenvalues of non-compact operators with Hilbert-Schmidt Hermitian components

Abstract

Let \(A\) be a bounded linear operator in a complex separable Hilbert space, \(A^*\) be its adjoint one and \(A_I:=(A-A^*)/(2i)\). Assuming that \(A_I\) is a Hilbert-Schmidt operator, we investigate perturbations of the imaginary parts of the eigenvalues of \(A\). Our results are formulated in terms of the "extended" eigenvalue sets in the sense introduced by T. Kato. Besides, we refine the classical Weyl inequality \(\sum_{k=1}^\infty (\operatorname{Im} \lambda_k(A))^2 \leq N_2^2(A_I)\), where \(\lambda_k(A)\) \((k=1,2, \ldots )\) are the eigenvalues of \(A\) and \(N_2(\cdot)\) is the Hilbert-Schmidt norm. In addition, we discuss applications of our results to the Jacobi operators.