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

Abstract

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.