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 AI := (A − A∗)/(2i). Assuming that AI 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 [formula], where λk(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.

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Opracowanie rekordu ze środków MEiN, umowa nr SONP/SP/546092/2022 w ramach programu "Społeczna odpowiedzialność nauki" - moduł: Popularyzacja nauki i promocja sportu (2022-2023)