Sums of two square-zero selfadjoint or skew-selfadjoint endomorphisms

Abstract

Let $V$ be a finite-dimensional vector space over a field $\mathbb{F}$, equipped with a symmetric or alternating non-degenerate bilinear form $b$. When the characteristic of $\mathbb{F}$ is not $2$, we characterize the endomorphisms $u$ of $V$ that split into $u=a_1+a_2$ for some pair $(a_1,a_2)$ of $b$-selfadjoint (respectively, $b$-skew-selfadjoint) endomorphisms of $V$ such that $(a_1)^2=(a_2)^2=0$. In the characteristic $2$ case, we obtain a similar classification for the endomorphisms of $V$ that split into the sum of two square-zero $b$-alternating endomorphisms of $V$ when $b$ is alternating (an endomorphism $v$ is called $b$-alternating whenever $b(x,v(x))=0$ for all $x \in V$). Finally, if the field $\mathbb{F}$ is equipped with a non-identity involution, we characterize the pairs $(h,u)$ in which $h$ is a Hermitian form on a finite-dimensional space over $\mathbb{F}$, and $u$ is the sum of two square-zero $h$-selfadjoint endomorphisms.