Definition: Let V be complex finite dimensional inner product space Complex Conjugate: $J$ is called complex conjugate on V iff $(i) J$ is antilinear $(ii) J^2 =I$
Definition: Anti-unitary Complex Conjugate: $K$ is called complex conjugate on V iff $(i)$ $K$ is antilinear $(ii) J^2 =I (iii) <Ku, Kv> = <v, u>$
Question: Let $T$ be linear transformation on V such that there exist a complex conjugate $J$ on V such that $JTJ = T$. Can we find an Anti-unitary complex conjuagte $K$ on V such that $KTK = T$?
Extra Comments
This led me to question: https://math.stackexchange.com/questions/1678399/v-complex-ips-and-lt-with-real-matrix-entries
And complec conjugate may not imply unitary complex conjugate is proved in https://math.stackexchange.com/questions/1678401/a-conjugate-lt-in-an-inner-product-space/