Helly-type theorem for eigenvectors

TL;DR

This paper establishes a Helly-type theorem for eigenvectors, showing that a certain number of operators sharing an eigenvector implies all do, and explores related invariant subspace problems.

Contribution

It proves a sharp bound for common eigenvectors in families of linear operators and investigates partial results for common invariant subspaces.

Findings

If any ⌊3d/2⌋ operators share an eigenvector, then all do.

The bound ⌊3d/2⌋ is optimal and cannot be lowered.

Partial results on common invariant subspaces for O(d) operators.

Abstract

We prove that if any $\lfloor3d/2 \rfloor$ or fewer elements of a finite family of linear operators $\mathbb K^d\to \mathbb K^d$ ($\mathbb K$ is an arbitrary field) have a common eigenvector then all operators in the family have a common eigenvector. Moreover, $\lfloor 3d/2\rfloor$ cannot be replaced by a smaller number. Also, we study the following problem, achieving partial results: prove that if any $l=O(d)$ or fewer elements of a finite family of linear operators $\mathbb K^d\to \mathbb K^d$ have a common non-trivial invariant subspace then all operators in the family have a common non-trivial invariant subspace.