Hyperinvariant subspaces of locally nilpotent linear transformations

TL;DR

This paper characterizes the structure of hyperinvariant subspaces for infinite-dimensional, locally nilpotent linear transformations, extending finite-dimensional results to a broader class of infinite spaces.

Contribution

It extends the description of hyperinvariant subspaces to infinite-dimensional spaces with locally nilpotent operators, generalizing prior finite-dimensional results.

Findings

Describes the lattice of hyperinvariant subspaces for the given class of transformations.

Extends finite-dimensional results to infinite-dimensional vector spaces.

Provides a comprehensive framework for understanding hyperinvariant subspaces in this context.

Abstract

A subspace $X$ of a vector space over a field $K$ is hyperinvariant with respect to an endomorphism $f$ of $V$ if it is invariant for all endomorphisms of $V$ that commute with $f$. We assume that $f$ is locally nilpotent, that is, every $ x \in V $ is annihilated by some power of $f$, and that $V$ is an infinite direct sum of $f$-cyclic subspaces. In this note we describe the lattice of hyperinvariant subspaces of $V$. We extend results of Fillmore, Herrero and Longstaff (Linear Algebra Appl. 17 (1977), 125--132) to infinite dimensional spaces.