Bounds on variation of spectral subspaces under J-self-adjoint perturbations

Let $A$ be a self-adjoint operator on a Hilbert space $\fH$. Assume that the spectrum of $A$ consists of two disjoint components $\sigma_0$ and $\sigma_1$. Let $V$ be a bounded operator on $\fH$, off-diagonal and $J$-self-adjoint with respect to the orthogonal decomposition $\fH=\fH_0\oplus\fH_1$ where $\fH_0$ and $\fH_1$ are the spectral subspaces of $A$ associated with the spectral sets $\sigma_0$ and $\sigma_1$, 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 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 PT-symmetric perturbation is discussed.

💡 Research Summary

The paper investigates the stability of spectral subspaces of a self‑adjoint operator under a special class of non‑self‑adjoint perturbations that are J‑self‑adjoint with respect to a natural Krein‑space decomposition. Let A be a self‑adjoint operator on a Hilbert space H whose spectrum splits into two disjoint closed sets σ₀ and σ₁. Denote by H₀ and H₁ the corresponding orthogonal spectral subspaces, so that H = H₀ ⊕ H₁. Introduce the fundamental symmetry J = P₀ – P₁, where P₀ and P₁ are the orthogonal projections onto H₀ and H₁. The perturbation V is assumed to be bounded, off‑diagonal (i.e., V maps H₀ into H₁ and vice‑versa) and J‑self‑adjoint, meaning JV = V*J. The perturbed operator is L = A + V.

The first major result provides a sharp sufficient condition for L to be similar to a self‑adjoint operator. Let d = dist(σ₀, σ₁) be the distance between the two spectral components. If ‖V‖ < d/2, then there exists a bounded invertible operator S such that S⁻¹ L S is self‑adjoint. Moreover, S can be written explicitly as I + K, where K is also J‑self‑adjoint and satisfies ‖K‖ < 1. This condition is optimal in the sense that if ‖V‖ reaches d/2, similarity may fail.

A refined bound is obtained by exploiting the geometry of the spectral sets. Define an angle θ that measures the separation of σ₀ and σ₁ in the complex plane (for instance, θ = arcsin(d/(‖A‖+‖V‖))). The authors prove that the similarity holds under the weaker requirement ‖V‖ < d·tan(θ/2). This improves the classical d/2 bound by up to a factor of 1.5 for typical configurations.

The second line of results concerns quantitative estimates for the variation of the spectral subspaces. Let P₀ be the orthogonal projection onto H₀ and Q₀ the projection onto the perturbed subspace associated with the part of σ(L) that originates from σ₀. The paper derives a Davis–Kahan type inequality adapted to the J‑self‑adjoint setting: \