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.
DOI: 10.14232/actasm-018-570-5
AMS Subject Classification
(1991): 47B15
Keyword(s):
spectral theorem,
real normal operator,
Williamson's normal form,
infinite mode quantum systems
Received August 3, 2018 and in final form November 22, 2018. (Registered under 70/2018.)
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