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Let $H$ be a (the) real separable Hilbert space. The Hilbert--Schmidt theorem says that a compact self-adjoint operator $A$ has an eigenfunction expansion. Instead of operator, we can think of a symmetric bilinear form and write $$ A = \sum_{k\ge 1} \lambda_k \varphi_k\otimes \varphi_k \tag{1} $$

My question is:

Are there any multilinear analogues of (1)? Which $n$-linear symmetric forms can be represented in a form $$ A = \sum_{k\ge 1} \lambda_k \varphi_k^{\otimes n}\ ?$$

(There is no compactness notion for multilinear forms, but we can assume that they are e.g. Hilbert--Schmidt.)

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Not in general. For even $n=2m$, $m >1$, a $n$-linear symmetric form $A$ is a bilinear symmetric form (or operator) on $H^{\otimes_s m}\otimes H^{\otimes_s m}$ and it will have a spectral decomposition of the form $$A=\sum_k \lambda_k u_k \otimes u_k$$ with $u_k \in H^{\otimes_s m}$, which is not in general in the form you wrote down (ie the $u_k$'s will not in general be in the form $\varphi_k \otimes \varphi_k\otimes \cdots \otimes \varphi_k$ ($m$-times) for some $\varphi_k \in H$).

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  • $\begingroup$ Thank you, though it is not what my wishful thinking imagined, this may be helpful. $\endgroup$
    – zhoraster
    Commented Mar 4, 2011 at 4:03
  • $\begingroup$ However, $u_k$ here cannot be arbitrary, and, as far as I see, they must be symmetric forms as well. So your answer gives the desired statement for $n=2^r$, or I'm missing something? And for odd $n$ the question remains. You wrote "the $u_k$'s will not in general be in the form...", but can you give any example when they are not? $\endgroup$
    – zhoraster
    Commented Mar 4, 2011 at 4:44
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There is a thing called Majorana representation of the symmetric states, somehow related to your question.

For $\dim H = 2$ and $\psi$ living in a symmetric subspace of $H^{\otimes n}$, we have

$$\psi = \sum_{\hbox{perm}} \phi_{P(1)}\otimes\phi_{P(2)}\otimes\cdots\otimes \phi_{P(n)},$$ where $\phi_i \in H$. The representation is unambiguous (leave alone constant factors and the permutation of the indices). Explicit decomposition employs finding roots of a polynomial.

The bad thing is I do not know if there is a generalization for $\dim H > 2$ (a naive one does not work).

Schmidt decomposition is very relevant to quantum entanglement. Nevertheless, there is no straightforward generalization for $n>2$ (or in your words - for multilinear forms). See e.g.:

  • A. Acin et al, Generalized Schmidt decomposition and classification of three-quantum-bit states, arXiv:quant-ph/0003050
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