Bounds on Variation of Spectral Subspaces under J-Self-adjoint Perturbations

Abstract.

Let A be a self-adjoint operator on a Hilbert space $\mathfrak{H}$. Assume that the spectrum of A consists of two disjoint components σ₀ and σ₁. Let V be a bounded operator on $\mathfrak{H}$, off-diagonal and J-self-adjoint with respect to the orthogonal decomposition $\mathfrak{H} = \mathfrak{H}_{0}\, \oplus\, \mathfrak{H}_{1}$ where $\mathfrak{H}_{0}$ and $\mathfrak{H}_{1}$ are the spectral subspaces of A associated with the spectral sets σ₀ and σ₁, respectively. We find (optimal) conditions on V guaranteeing that the perturbed operator L = A + V is similar to a self-adjoint operator. Moreover, we prove a number of (sharp) norm bounds on the variation of the spectral subspaces of A under the perturbation V. Some of the results obtained are reformulated in terms of the Krein space theory. As an example, the quantum harmonic oscillator under a ${\mathcal{PT}}$-symmetric perturbation is discussed.