Perturbations of real parts of eigenvalues of bounded linear operators in a Hilbert space

Let A be a bounded linear operator in a complex separable Hilbert space ℌ, and S be a selfadjoint operator in ℌ. Assuming that A − S belongs to the Schattenvon Neumann ideal \(\cal{S}_{p}\ (p>1)\) $${\mathcal{S}}_p(p>1)$$ , we derive a bound for \(\sum_k\vert\rm{R}\ \lambda_{k}(A)-\lambda_{k}(S)\vert^{p}\) $$\sum _k\mid \mathrm{R}{\lambda }_k(A)-{\lambda }_k(S){\mid }^p$$ , where λ_k(A) (k = 1, 2, …) are the eigenvalues of A. Our results are formulated in terms of the “extended” eigenvalue sets in the sense introduced by T. Kato. In addition, in the case p = 2 we refine the Weyl inequality between the real parts of the eigenvalues of A and the eigenvalues of its Hermitian component.