Characteristic subspaces and hyperinvariant frames

TL;DR

This paper investigates the structure of characteristic subspaces that are not hyperinvariant in finite-dimensional vector spaces over the field with two elements, focusing on hyperinvariant frames and their properties.

Contribution

It introduces the concept of hyperinvariant frames for characteristic subspaces and characterizes the invariant subspaces within these frames, providing a method to construct characteristic non-hyperinvariant subspaces.

Findings

All invariant subspaces in the interval [W_H, W^h] are characteristic.

The paper provides a construction method for characteristic non-hyperinvariant subspaces.

Abstract

Let $f$ be an endomorphism of a finite dimensional vector space $V$ over a field $K$. An $f$-invariant subspace of $V$ is called hyperinvariant (respectively characteristic) if it is invariant under all endomorphisms (respectively automorphisms) that commute with $f$. We assume $|K| = 2$, since all characteristic subspaces are hyperinvariant if $|K| > 2$. The hyperinvariant hull $W^h$ of a subspace $ W$ of $ V$ is defined to be the smallest hyperinvariant subspace of $V$ that contains $ W$, the hyperinvariant kernel $W_H$ of $ W$ is the largest hyperinvariant subspace of $V$ that is contained in $W$, and the pair $( W_H, W^h) $ is the hyperinvariant frame of $W$. In this paper we study hyperinvariant frames of characteristic non-hyperinvariant subspaces $W$. We show that all invariant subspaces in the interval $[ W_H, W^h ]$ are characteristic. We use this result for the construction of characteristic non-hyperinvariant subspaces.