Proposition 1

Let $E_1$ and $E_2$ be complex vector spaces, and let $u_1$ and $u_2$ be semilinear involutions on $E_1$ and $E_2$ respectively. Let $f$ be a linear mapping of $E_1$ into $E_2$. Then $f\circ u_1=u_2\circ f$ if and only if $f(E^{u_1}_1)\subset E^{u_2}_2$.

If $f\circ u_1=u_2\circ f$, we obtain immediately $f(E^{u_1}_1)\subset E^{u_2}_2$. Conversely, suppose that this condition is satisfied. Let $x\in$ E. Write $x=x_1+ix_2$ with $(x_1, x_2)\in E^{u_1}_1\times E^{u_1}_1$ (lemma above). We then have $f(u_1(x)) =f(x_1)-if(x_2)$ and $u_2(f(x)) =u_2(f(x_1))-iu_2(f(x_2)) =$ $f(x_1)-if(x_2) =f(u_1(x))$.