Matrix semigroups whose ring commutators have real spectra are realizable

TL;DR

This paper investigates matrix semigroups with the property that their ring commutators have real spectra, establishing conditions under which they are similar to real matrices and providing a structure theorem for certain compact groups.

Contribution

It proves that irreducible semigroups with real spectrum commutators are similar to real matrices and characterizes compact groups with this property.

Findings

Irreducible semigroups are similar to real matrices.

A structure theorem for compact groups with real spectrum commutators.

Semigroups with this property can be characterized by similarity to real matrices.

Abstract

We study matrix semigroups in which ring commutators have real spectra. We prove that irreducible semigroups with this property are simultaneously similar to semigroups of real-entried matrices. We also obtain a structure theorem for compact groups satisfying the property under investigation.