Could you please help me to understand the polar decomposition theorem for $Sp(H, J_Q)$ and $O(H,J_R)$ where $H$ is infinite dimensional separable Hilbert space and $J_R$ and $J_Q$ stands for conjugation and anticonjugation operators such that: $$O(H,J_R)=\{T \in GL(H):<Tx,J_R(Ty)>=<x,J_Ry>\}$$ $$Sp(H,J_Q)={T \in GL(H):<Tx,J_Q(Ty)>=<x,J_Qy>}.$$
(Note that $x$ and $y$ are arbitrary vectors in $H$).