Algebraic aspects of spectral theory

TL;DR

This paper explores algebraic facets of spectral theory, focusing on applications to matrix algebras with polynomial or analytic function entries, without relying on analytical methods.

Contribution

It introduces algebraic perspectives in spectral theory and applies these to matrix algebras with polynomial and analytic function entries.

Findings

Spectral properties of matrices with polynomial entries analyzed algebraically.

Applications to matrix algebras with analytic function entries.

No analytical methods needed for the spectral considerations.

Abstract

We describe some aspects of spectral theory that involve algebraic considerations but need no analysis. Some of the important applications of the results are to the algebra of $n\times n$ matrices with entries that are polynomials or more general analytic functions.