Existence of an invariant form under a linear map

英文摘要：

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