Real Normal Operators and Williamson's Normal Form

A simple proof is provided to show that any bounded normal operator on a real Hilbert space is orthogonally equivalent to its transpose(adjoint). A structure theorem for invertible skew-symmetric operators, which is analogous to the finite dimensional situation is also proved using elementary techniques. The second result is used to establish the main theorem of this article, which is a generalization of Williamson’s normal form for bounded positive operators on infinite dimensional separable Hilbert spaces. This has applications in the study of infinite mode Gaussian states.