Isomorphisms between lattices of hyperinvariant subspaces

TL;DR

This paper investigates conditions under which the lattices of hyperinvariant subspaces of two nilpotent endomorphisms are isomorphic, reducing the problem to the nilpotent case via Jordan-Chevalley decomposition.

Contribution

It characterizes isomorphisms between lattices of hyperinvariant subspaces for nilpotent endomorphisms, extending understanding in linear algebra and invariant subspace theory.

Findings

Provides criteria for lattice isomorphisms between nilpotent endomorphisms

Reduces the problem to the nilpotent case using Jordan-Chevalley decomposition

Enhances classification of hyperinvariant subspace lattices

Abstract

Given two nilpotent endomorphisms, we determine when their lattices of hyperinvariant subspaces are isomorphic. The study of the lattice of hyperinvariant subspaces can be reduced to the nilpotent case when the endomorphism has a Jordan-Chevalley decomposition; for example, it occurs if the underlying field is the field of complex numbers.