Relative residual bounds for eigenvalues in gaps of the essential spectrum

TL;DR

This paper establishes bounds on how close eigenvalues of a compressed operator are to those of the original operator within spectral gaps, based on maximal angles between subspaces, extending previous matrix results to broader operators.

Contribution

It generalizes eigenvalue gap bounds from matrices to not necessarily semibounded self-adjoint operators using subspace angle analysis.

Findings

Eigenvalue distance depends on maximal subspace angles.

Generalization from matrices to broader operator classes.

Provides bounds in spectral gap analysis.

Abstract

The relative distance between eigenvalues of the compression of a not necessarily semibounded self-adjoint operator to a closed subspace and some of the eigenvalues of the original operator in a gap of the essential spectrum is considered. It is shown that this distance depends on the maximal angles between pairs of associated subspaces. This generalises results by Drma\v{c} in [Linear Algebra Appl. 244 (1996), 155--163] from matrices to not necessarily (semi)bounded operators.