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

Abstract

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.

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• On Domains of PT Symmetric Operators Related to -y''(x) + (-1)^n x^{2n}y(x)
• The Rayleigh-Schrödinger perturbation series of quasi-degenerate systems
• PT Symmetric Schrödinger Operators: Reality of the Perturbed Eigenvalues
• On $J$-self-adjoint extensions of the Phillips symmetric operator
• Spectral enclosures for a class of block operator matrices
• Spectral properties of unbounded J-self-adjoint matrices
• On invariant graph subspaces of a J-self-adjoint operator in the Feshbach case
• Bounds on the non-real spectrum of differential operators with indefinite weights
• Solvability of the operator Riccati equation in the Feshbach case
• Many-body excitations in trapped Bose gas: A non-Hermitian view
• Applications of fixed point theorems in the theory of invariant subspaces