Spectral-Variation Bounds in Hyperbolic Geometry

TL;DR

This paper introduces a hyperbolic geometry-based spectral variation bound for non-normal matrices, improving classical bounds and providing a new perspective on eigenvalue localization with applications to infinite-dimensional operators.

Contribution

It develops a novel hyperbolic metric approach for spectral variation, surpassing classical bounds and extending to algebraic operators on Hilbert space.

Findings

Bound is sharper for small matrix differences

Provides a new characterization of eigenvalue localization

Applicable to algebraic operators on Hilbert space

Abstract

We derive new estimates for distances between optimal matchings of eigenvalues of non-normal matrices in terms of the norm of their difference. We introduce and estimate a hyperbolic metric analogue of the classical spectral-variation distance. The result yields a qualitatively new and simple characterization of the localization of eigenvalues. Our bound improves on the best classical spectral-variation bounds due to Krause if the distance of matrices is sufficiently small and is sharp for asymptotically large matrices. Our approach is based on the theory of model operators, which provides us with strong resolvent estimates. The latter naturally lead to a Chebychev-type interpolation problem with finite Blaschke products, which can be solved explicitly and gives stronger bounds than the classical Chebychev interpolation with polynomials. As compared to the classical approach our method does not rely on Hadamard's inequality and immediately generalize to algebraic operators on Hilbert space.