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.)



An analogue of Hilbert-Schmidt theorem for multilinear forms
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 $\varphi\_k \in H$). 


