Existence of an invariant form under a linear map

Abstract

Let F be a field of characteristic different from 2 and V be a vector space over F. Let J: α → α J be a fixed involutory automorphism on F. In this paper we answer the following question: given an invertible linear map T: V → V, when does the vector space V admit a T-invariant nondegenerate J-hermitian, resp. J-skew-hermitian, form?