The lattice of characteristic subspaces of an endomorphism with Jordan-Chevalley decomposition

TL;DR

This paper explores the structure of characteristic subspaces of an endomorphism with Jordan-Chevalley decomposition, extending known results about invariant and hyperinvariant subspaces and generalizing Shoda's Theorem.

Contribution

It extends the understanding of characteristic subspaces in endomorphisms with Jordan-Chevalley decomposition and generalizes Shoda's Theorem on characteristic non-hyperinvariant subspaces.

Findings

Lattices of characteristic subspaces can be derived from the nilpotent part of the Jordan-Chevalley decomposition.

Generalization of Shoda's Theorem characterizes the existence of characteristic non-hyperinvariant subspaces.

The results unify the structure of invariant, hyperinvariant, and characteristic subspaces for such endomorphisms.

Abstract

Given an endomorphism A over a finite dimensional vector space having Jordan-Chevalley decomposition, the lattices of invariant and hyperinvariant subspaces of A can be obtained from the nilpotent part of this decomposition. We extend this result for lattices of characteristic subspaces. We also obtain a generalization of Shoda's Theorem about the characterization of the existence of characteristic non hyperinvariant subspaces.