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

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 $ \Sigma_{k=1}^\infty (Im \lambda_k (A))^2 ≤ N_2^2 (A_I) $, where $ λk(A) (k = 1, 2, . . .) $ are the eigenvalues of $ A $ and $ N_2(·) $ is the Hilbert-Schmidt norm. In addition, we discuss applications of our results to the Jacobi operators.