Sharpening the Norm Bound in the Subspace Perturbation Theory

Let be a (possibly unbounded) self-adjoint operator on a separable Hilbert space Assume that is an isolated component of the spectrum of , that is, where Suppose that is a bounded self-adjoint operator on such that and let , Denote by the spectral projection of associated with the spectral set and let be the spectral projection of corresponding to the closed -neighborhood of Introducing the sequence we prove that the following bound holds: where the estimating function , , is given by with . The bound obtained is essentially stronger than the previously known estimates for Furthermore, this bound ensures that and, thus, that the spectral subspaces and are in the acute-angle case whenever , where Our proof of the above results is based on using the triangle inequality for the maximal angle between subspaces and on employing the a priori generic estimate for the variation of a spectral subspace. As an example, the boundedly perturbed quantum harmonic oscillator is discussed.