Bounds on the spectrum and reducing subspaces of a $J$-self-adjoint operator Sergio AlbeverioAlexander MotovilovChristiane Tretter 47A1047B5047A5547A62subspace perturbation problemKrein space$J$-self-adjoint operator$PT$-symmetry$PT$-symmetric operatoroperator Riccati equationoperator angleDavis-Kahan theorems 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. Indiana University Mathematics Journal 2010 text pdf 10.1512/iumj.2010.59.4225 10.1512/iumj.2010.59.4225 en Indiana Univ. Math. J. 59 (2010) 1737 - 1776 state-of-the-art mathematics http://iumj.org/access/