  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 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.
