Hyperinvariant, characteristic and marked subspaces

TL;DR

This paper investigates three special types of invariant subspaces in finite-dimensional vector spaces under an endomorphism, establishing their relationships and conditions under which they coincide, especially over fields with more than two elements.

Contribution

It characterizes hyperinvariant, characteristic, and marked subspaces, proving their equivalence under certain conditions and clarifying their interrelations.

Findings

Hyperinvariant subspaces are exactly those that are characteristic and marked.

Over fields with more than two elements, all characteristic subspaces are hyperinvariant.

The paper provides conditions under which these subspace types coincide.

Abstract

Let $V$ be a finite dimensional vector space over a field $K$ and $f$ a $K$-endomorphism of $V$. In this paper we study three types of $f$-invariant subspaces, namely hyperinvariant subspaces, which are invariant under all endomorphisms of $V$ that commute with $f$, characteristic subspaces, which remain fixed under all automorphisms of $V$ that commute with $f$, and marked subspaces, which have a Jordan basis (with respect to $f_{|X}$) that can be extended to a Jordan basis of $V$. We show that a subspace is hyperinvariant if and only if it is characteristic and marked. If $K$ has more than two elements then each characteristic subspace is hyperinvariant.