Bounds on the Spectrum and Reducing Subspaces of a J-Self-adjoint Operator

Given a self-adjoint involution J on a Hilbert space H, we consider a J-self-adjoint operator L = A + V on 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 PT-symmetric 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.