On the similarity of dual operators
Let \(k\) be a field with an involution \(\sigma\in\Aut(k)\) and let \(\langle\;,\;\rangle: V\times W\rightarrow k\) be a non-degenerate sesquilinear form, where \(V\) and \(W\) a linear spaces over \(k\) of the same finite dimension. Let \(\Lambda\in\operatorname{End}(V)\) and \(\Lambda^*\in \operatorname{End}(W)\) be dual operators. The aim of the paper is to prove the following result: If \(\Lambda\) and \(\Lambda^*\) are similar operators, then there exists a Hermitian operator \(\varphi:V\rightarrow W\) such that \(\Lambda^*=\varphi\Lambda\varphi^{-1}.\) This result was proved by \textit{D. Z. Djokovič} [Linear Algebra Appl. 6, 155--158 (1973; Zbl 0251.15013)], but that proof was incomplete.
scientific article; zbMATH DE number 5285776