Let U and V be finite-dimensional vector spaces over a field F. Let S be a linear subspace of Hom(U,V). We say that it is intransitive when S x ≠ V for all x ∈ U, and we say that it is primitively intransitive when, in addition, there is no proper linear subspace W of V such that pi^V/W S is an intransitive subspace of Hom(U,V/W), where pi^V/W : V twoheadrightarrow V/W stands for the canonical projection. Atkinson and Lloyd have proved that if |F| ≥ n and S is primitively intransitive then dim S ≤ fracn(n-1)2. The conjecture states that the result holds without the cardinality assumption on F.

Atkinson-Lloyd conjecture for primitively intransitive spaces of linear operators

Let $U$ and $V$ be finite-dimensional vector spaces over a field $\mathbb{F}$ . Let $\mathcal{S}$ be a linear subspace of $\mathrm{Hom}(U,V)$ . We say that it is intransitive when $\mathcal{S} x \neq V$ for all $x \in U$ , and we say that it is primitively intransitive when, in addition, there is no proper linear subspace $W$ of $V$ such that $\pi^{V/W} \mathcal{S}$ is an intransitive subspace of $\mathrm{Hom}(U,V/W)$ , where $\pi^{V/W} : V \twoheadrightarrow V/W$ stands for the canonical projection.

Atkinson and Lloyd have proved that if $|\mathbb{F}| \geq n$ and $\mathcal{S}$ is primitively intransitive then $\dim \mathcal{S} \leq \frac{n(n-1)}{2}$ . The conjecture states that the result holds without the cardinality assumption on $\mathbb{F}$ .

References

https://www.cambridge.org/core/journals/journal-of-the-australian-mathematical-society/article/primitive-spaces-of-matrices-of-bounded-rank/A2F2D9825F53DD6601C3A15DEFED391E

https://arxiv.org/abs/2508.06934

linear-algebra

Posted by Clément de Seguins Pazzis Posted 1 month ago

Progress summary

Refreshed 1 month ago

Claimed progress

The original bound is known for sufficiently large fields, while a recent paper appears to extend related results but does not clearly settle the conjecture for every field.

Atkinson and Lloyd conjectured that every primitively intransitive operator space satisfies the stated dimension bound without any restriction on the field. Their published theorem imposed the hypothesis $|\mathbb{F}|\geq n$ .

Known results

Atkinson and Lloyd: if $|\mathbb{F}|\geq n$ , then $\dim\mathcal{S}\leq n(n-1)/2$ .

Recent related result, reported September 2026

The paper Spaces of matrices with few eigenvalues (II) states an Atkinson-type theorem for primitively intransitive spaces and says its main results hold beyond the usual field-cardinality assumptions, apart from the cases $|\mathbb{F}|\leq 2$ . The retrieved text does not explicitly identify this as a proof of the exact Atkinson–Lloyd conjecture, so this is claimed progress rather than a verified resolution.

Current status (as of September 2026): The conjecture is proved under $|\mathbb{F}|\geq n$ , while its unrestricted formulation remains open; a recent paper may cover related cases for $|\mathbb{F}|>2$ but does not explicitly establish this exact statement.

Sources

No solutions have been posted yet.