Bounds on the spectrum and reducing subspaces of a J-self-adjoint operator

Abstract

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.

(http://www.iumj.indiana.edu/IUMJ/FULLTEXT/2010/59/4225)

References in corpus (4)

• Bounds on variation of spectral subspaces under J-self-adjoint perturbations
• On the existence of solutions to the operator Riccati equation and the tanΘtheorem
• $PT$ symmetric non-selfadjoint operators, diagonalizable and non-diagonalizable, with real discrete spectrum
• Point Interactions: PT-Hermiticity and Reality of the Spectrum

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• Spectral properties of unbounded J-self-adjoint matrices
• On invariant graph subspaces of a J-self-adjoint operator in the Feshbach case
• Solvability of the operator Riccati equation in the Feshbach case
• Many-body excitations in trapped Bose gas: A non-Hermitian view