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

Gil’ Michael

Opuscula Mathematica

2024

Vol. 44, no. 2

241--248

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.

Spectrum localization of a perturbed operator in a strip and applications

Gil Michael

Opuscula Mathematica

2021

Vol. 41, no. 3

395–-412

Let A and A be bounded operators in a Hilbert space. We consider the following problem: let the spectrum of A lie in some strip. In what strip the spectrum of A lies if A and Ä are “close”? Applications of the obtained results to integral operators and matrices are also discussed. In addition, we apply our perturbation results to approximate the spectral strip of a Hilbert-Schmidt operator by the spectral strips of finite matrices.