v, J_0w) = \omega(v,w)$.

$g(v,w) = \omega(v, J_0w)$ and $\omega(v,w)=g(J_0v,w)$ for any $v,w 
\in V$.

In particular, $\omega(v, J_0w)$ is positive definite, $J_0 \in 
\jj_\omega$. This shows that $\jj$ is non-empty.

[*Step 2. For any $J \in \jj_\omega$ the operator $J + J_0$ is invertible*]{}. Really, otherwise we have $Jv = - J_0v$ for some non-zero $v\in V$ and then $0< \omega(v, Jv) = -\omega(v, J_0v) <0$, a contradiction.

For $J \in \jj$ with $J + J_0$ invertible we set $\sfL(J) \deff -(J - J_0) (J+
J_0)\inv$. The equivalent definitions are $$\matrix\format \r&\,\c\, & \l \\
\sfL (J) &=&  -(J - J_0) (J+J_0)\inv = 
-\bigl((J - J_0)J_0\bigr) \cdot \bigl( (J+J_0) J_0 \bigr)\inv \cr
\vph&=&(\bfone + JJ_0 )(\bfone - JJ_0  )\inv 
      = (\bfone - JJ_0)\inv (\bfone + JJ_0).
\endmatrix
\eqno\rom(1.2.4)$$

[*Step 3. If $J + J_0$ is invertible, then $W= \sfL(J)$ is $J_0$-antilinear, $W J_0= - J_0 W$*]{}. Really, $$\matrix \format \c &\,\c\, &\r &\,\c \,&\r &\,\c\, &\c
\vbox to 11pt{} \\
W J_0 &=& -(J - J_0) (J+J_0)\inv J_0 &=& 
(\bfone - JJ_0)\inv (\bfone + JJ_0) J_0 &=&
\cr 
&&
(\bfone -JJ_0)\inv (J_0- J)&=&
- (\bfone -JJ_0)\inv J(\bfone + JJ_0) &=& 
\cr
&&
(\bfone -JJ_0)\inv J\inv (\bfone +JJ_0) &=& 
(J 