Title: Bounds on the spectrum and reducing subspaces of a $J$-self-adjoint operator

Authors: Sergio Albeverio, Alexander K. Motovilov and Christiane Tretter

Issue: Volume 59 (2010), Issue 5, 1737-1776

Abstract: Given a self-adjoint involution $J$ on a Hilbert space $\mathfrak{H}$, we consider a $J$-self-adjoint operator $L = A + V$ on $\mathfrak{H}$ where $A$ is a possibly unbounded self-adjoint operator commuting with $J$ and $V$ a bounded $J$-self-adjoint operator anti-commuting with $J$. We establish optimal estimates on the position of the spectrum of $L$ with respect to the spectrum of $A$ and we obtain norm bounds on the operator angles between maximal uniformly definite reducing subspaces of the unperturbed operator $A$ and the perturbed operator $L$. All the bounds are given in terms of the norm of $V$ and the distances between pairs of disjoint spectral sets associated with the operator $L$ and/or the operator $A$. As an example, the quantum harmonic oscillator under a $\mathcal{P}\mathcal{T}$-symmetric perturbation is discussed. The sharp norm bounds obtained for the operator angles generalize the celebrated Davis-Kahan trigonometric theorems to the case of $J$-self-adjoint perturbations.