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

Abstract

Let V be a finite-dimensional vector space over a field F, equipped with a symmetric or alternating non-degenerate bilinear form b. When the characteristic of F is not 2, we characterize the endomorphisms u of V that split into u=a1+a2 for some pair (a1,a2) of b-selfadjoint (respectively, b-skew-selfadjoint) endomorphisms of V such that (a1)2=(a2)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∈V).Finally, if the field 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 F, and u is the sum of two square-zero h-selfadjoint endomorphisms.