Additive maps of rank r tensors and symmetric tensors

Abstract

Let $\mathbb{F}$ and $\mathbb{K}$ be fields and let n be a positive integer. Let $\mathcal{U}$ and $\mathcal{V}$ be linear spaces over $\mathbb{F}$ such that $n=\dim \mathcal{U}⩽\dim \mathcal{V}$ and let $\mathcal{W}$ and $\mathcal{Z}$ be linear spaces over $\mathbb{K}$. Let $\mathcal{U}⨂\mathcal{V}$ be the tensor product of $\mathcal{U}$ and $\mathcal{V}$ and let ${\mathcal{S}}^2(\mathcal{U})$ be the subspace of symmetric tensors of $\mathcal{U}⨂\mathcal{U}$. In this paper, we show that a map $\psi :\mathcal{P}\to \mathcal{Q}$ satisfies $$\psi (u+v)=\psi (u)+\psi (v)$$

for all rank r tensors $u,v\in \mathcal{P}$ if and only if $\psi$ is additive. Here r is a fixed integer such that $\frac{n}{2}⩽r⩽n$ and $(\mathcal{P},\mathcal{Q})=(\mathcal{U}⨂\mathcal{V},\mathcal{W}⨂\mathcal{Z})$ with $n⩾2$, or $(\mathcal{P},\mathcal{Q})=({\mathcal{S}}^2(\mathcal{U}),{\mathcal{S}}^2(\mathcal{W}))$ with $n⩾3$ and $\mathrm{char}\mathbb{F}\ne 2$. Examples showing the indispensability of assumption in our results are included.

Keywords:

• Additive maps
• tensor products
• symmetric tensors
• ranks

AMS Subject Classifications:

• 15A03
• 15A04
• 15A69
• 15A86

Acknowledgements

The authors are indebted to the referee for a careful reading of the original manuscript and for useful comments and suggestions.

Disclosure statement

No potential conflict of interest was reported by the authors.