Complex Structure Λ on K

A complex structure on a real Hilbert space $K$ is a bounded operator $\Lambda$ with $\Lambda^2=-I$.

In the paper, $\Lambda z = i z$ on $K=H$ viewed as real, and it relates symplectic and adjoint operations: a regular $T$ is symplectic iff $\Lambda T\Lambda^{-1}=T^{*-1}$.

Key properties:

• Lets one write $K=M\oplus M$ with $\Lambda(x\oplus y)=-y\oplus x$.
• Identifies $U(H)=O(K)\cap Sp(K)$.

Example: On $\mathbb R^{2n}$, $\Lambda(p,q)=(-q,p)$.