Similarity of some dual operators in characteristic two

AbstractLet k be a field of characteristic two, with involution x↦x¯. Let (V,·) be a finite dimensional Hermitian space over k and Λ:V→V a linear operator whose dual is Λ+1. We prove that ϕΛϕ-1=Λ+1, where ϕ is an isometry and ϕ2=1. If E is a given subspace of V, then ϕ can also be chosen to stabilize E, but the equality ϕΛϕ-1=Λ+1 is only true modulo a combination of certain bracket operators. As a corollary, we solve the following congruence problem. Given a square matrix A over k, there is a non-singular matrix S satisfying AT=S¯TAS and SS¯=1