Real Normal Operators and Williamson's Normal Form

TL;DR

This paper proves that bounded normal operators on real Hilbert spaces are orthogonally equivalent to their transpose and generalizes Williamson's normal form for positive operators, with applications in infinite mode Gaussian states.

Contribution

It provides a simple proof of the orthogonal equivalence of normal operators to their transpose and extends Williamson's normal form to infinite-dimensional settings.

Findings

Normal operators are orthogonally equivalent to their transpose.

A structure theorem for invertible skew-symmetric operators is established.

Williamson's normal form is generalized to infinite-dimensional positive operators.

Abstract

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.