Complex Structure Λ on K

A complex structure on a real Hilbert space $K$ is a bounded real-linear operator $\Lambda:K\to K$ satisfying $\Lambda^2=-I$ . It defines complex scalar multiplication by

(a+ib)z=az+b\Lambda z.

When $\Lambda$ is orthogonal, this complex structure is compatible with the real Hilbert-space inner product.

Remarks

In the paper, $\Lambda z=iz$ on $K=H$ viewed as a real Hilbert space. It relates symplectic and adjoint operations: a regular $T$ is symplectic if and only if $\Lambda T\Lambda^{-1}=T^{*-1}$ .

The compatible unitary group is $U(H)=O(K)\cap Sp(K)$ .

Examples

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