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Given a self-adjoint involution J on a Hilbert space H, we consider aJ-self-adjoint operator L=A+V on H where A is a possibly unbounded self-adjointoperator commuting with J and V a bounded J-self-adjoint operatoranti-commuting with J. We establish optimal estimates on the position of thespectrum of L with respect to the spectrum of A and we obtain norm bounds onthe operator angles between maximal uniformly definite reducing subspaces ofthe unperturbed operator A and the perturbed operator L. All the bounds aregiven in terms of the norm of V and the distances between pairs of disjointspectral sets associated with the operator L and/or the operator A. As anexample, the quantum harmonic oscillator under a PT-symmetric perturbation isdiscussed. The sharp norm bounds obtained for the operator angles generalizethe celebrated Davis-Kahan trigonometric theorems to the case of J-self-adjointperturbations.

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