Hessenberg decomposition of matrix fields and bounded operator fields

TL;DR

This paper generalizes Hessenberg decomposition from matrices to continuous matrix fields over topological spaces, providing new structural insights and extending classical results to infinite-dimensional operator fields.

Contribution

It introduces a generalization of Hessenberg decomposition for matrix fields over topological spaces and extends classical vector bundle results to these settings.

Findings

Derived new structure results on self-adjoint matrix fields

Established eigenvalue separation results

Generalized classical vector bundle theorems to finite-dimensional normal spaces

Abstract

Hessenberg decomposition is the basic tool used in computational linear algebra to approximate the eigenvalues of a matrix. In this article, we generalize Hessenberg decomposition to continuous matrix fields over topological spaces. This works in great generality: the space is only required to be normal and to have finite covering dimension. As applications, we derive some new structure results on self-adjoint matrix fields, we establish some eigenvalue separation results, and we generalize to all finite-dimensional normal spaces a classical result on trivial summands of vector bundles. Finally, we develop a variant of Hessenberg decomposition for fields of bounded operators on a separable, infinite-dimensional Hilbert space.