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 ^P∞_k=1(Im λ_k(A))² ≤ N₂²(A_I), where λ_k(A) (k = 1,2,...) are the eigenvalues of A and N₂(·) is the Hilbert–Schmidt norm. In addition, we discuss applications of our results to the Jacobi operators.