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This question was posed on MathStackExchange but did not get an answer (even with a bounty).

In all books that I have checked the spectral theorem (every self-adjoint unbounded operator on a Hilbert space is unitary equivalent to a multiplication operator on some $L_2(\mu)$) is only stated for complex Hilbert spaces (and the use of the Cayley transformation for the reduction to the bounded case requires indeed complex scalars). However, since the spectrum of a a self-adjoint operator is real the theorem should (or could) be true in real Hilbert spaces. I could imagine an argument by complexification but there are a number of things to do, in particucar, I think that the unitary operator between the complexified Hilbert space and the complex $L_2(\mu)$ does not automatically map the original real Hilbert space to the real-valued functions). Hence the question:

Is the the spectral theorem in real Hilbert spaces true? (In this case I would also like to have a reference.)

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    $\begingroup$ If you use the version of the spectral theorem where the operator is written as an integral of its spectral measure then there is no longer any problems with comparing the initial Hilbert space and the space of real valued functions. $\endgroup$
    – Simon Henry
    Commented Jan 17, 2014 at 10:30
  • $\begingroup$ Try the brilliant proof by Leinfelder in his article "A geometric proof of the spectral theorem for unbounded self-adjoint operators" in Math. Ann. vol. 242 which does not use the complexity of the Hilbert space. $\endgroup$
    – 7891user
    Commented Jan 17, 2014 at 10:39
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    $\begingroup$ Although the complexification is not the unique way, it is a general and natural tool, so I'd suggest to consider it fixing all details. $\endgroup$
    – Pietro Majer
    Commented Jan 17, 2014 at 10:52
  • $\begingroup$ I am not so sure whether things work out well: link.springer.com/article/10.1007%2FBF01364454#page-1. $\endgroup$
    – Marc Palm
    Commented Jan 17, 2014 at 12:02
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    $\begingroup$ Yes because, as you mentioned the spectrum is real and if $U$ is a subset of $\mathbb{R}$ then $E_{x,y}(U)$ is defined as $\langle x ,P y \rangle$ where $P$ is the spectral projection of $T$ corresponding to $U$, and as far as i know you don't need complex number to define them. $\endgroup$
    – Simon Henry
    Commented Jan 17, 2014 at 13:16

4 Answers 4

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Simon Henry's comment is close to an answer. As I asked for a reference :

Remark 20.18 in R. Meise and D. Vogt, Introduction to Functional Analysis.

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It seems worthwhile to mention that this is quite comprehensively treated in the recent preprint "The Functional Calculus Approach to the Spectral Theorem" by Markus Haase, where the entire theory is developed for the real and the complex case simultaneously.

Section 7 contains the spectral theorem for finitely many unbounded commuting self-adjoint operators on a real Hilbert space (Theorem 7.6).

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Suggestion: As you point out, Leinfelder does in fact use Cayley. However, he shows that if you have a self-adjoint operator, then you can represent the space as a Hilbert sum of closed subspaces, each of which is invariant under $A$ (or any operator which commutes with it), and on which it is bounded. If you apply this to the complexification of $A$ as in your problem, then you get a corresponding splitting. Once you have that, then it is a simple matter to get the result you are looking for.

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I posted a solution at StackExchange that uses only real techniques, but I don't know of any reference that uses the technique.

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  • $\begingroup$ A link would be helpful. $\endgroup$
    – Jochen Wengenroth
    Commented Sep 21, 2014 at 15:11
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    $\begingroup$ You posted the link to the question, and my answer is there; it was the one accepted. The techniques are borrowed from the Complex case to use harmonic functions instead, which is what is effectively used in the Complex case anyway. $\endgroup$
    – TrialAndError
    Commented Sep 21, 2014 at 16:03

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