Conjugation of a Complex Hilbert Space

In the setting of Complex Hilbert Spaces and Bounded Linear Maps: Standing Notation, let $H$ be a complex Hilbert space.

1. (Conjugation)¶ A conjugation of $H$ is a map $J:H\to H$ such that, for all $\xi,\eta\in H$ and $c\in\mathbb{C}$,

2. (Conjugated maps)¶ For a conjugation $J$ of $H$ and a map $A:H\to H$, $JAJ$ denotes the map $H\to H$, $\xi\mapsto J(A(J\xi))$; for a set $\mathcal{S}$ of maps $H\to H$, $J\mathcal{S}J=\{JAJ:\ A\in\mathcal{S}\}$.

3. (Fixed vectors)¶ For a conjugation $J$ of $H$, the set of fixed vectors of $J$ is $H^{J}=\{\xi\in H:\ J\xi=\xi\}$.