TY  - JOUR
ID  - Gil2024
LB  - Gil2024
AU  - Gil', Michael
TI  - An inequality for imaginary parts of eigenvalues of non-compact operators with Hilbert-Schmidt Hermitian components
ST  - An inequality for imaginary parts of eigenvalues of non-compact operators with Hilbert-Schmidt Hermitian components
JO  - Opuscula Math.
JA  - Opuscula Math.
JF  - Opuscula Mathematica
PY  - 2024
VL  - 44
IS  - 2
SP  - 241
EP  - 248
DO  - https://doi.org/10.7494/OpMath.2024.44.2.241
UR  - https://doi.org/10.7494/OpMath.2024.44.2.241
AB  - 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.
KW  - Hilbert space
KW  - linear operators
KW  - eigenvalues
SN  - 1232-9274
ER  -