Classification of linear operators satisfying $(Au,v)=(u,A^rv)$ or $(Au,A^rv)=(u,v)$ on a vector space with indefinite scalar product

Abstract

We classify all linear operators A ∶ V → V satisfying (Au, v) = (u, A r v) and all linear operators satisfying (Au, A r v) = (u, v) with r = 2, 3, . . . on a complex, real, or quaternion vector space with scalar product given by a nonsingular symmetric, skew-symmetric, Hermitian, or skew-Hermitian form.