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

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

2009-09-07

arxiv.org/abs/0909.1211v1

Indiana University Mathematics Journal 59:5 (2010), 1737-1776

Mathematics

Spectral Theory

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

Scientific paper

10.1512/iumj.2010.59.4225

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.

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