Have they been studied? In particular, what is the analogue of the Schmidt theorem for compact operators in Hilbert spaces?


[Helemskii A. Ya., Lectures and Exercises on Functional Analysis](http://books.google.ru/books?id=wjzZCLzx6hUC&printsec=frontcover#v=onepage&q&f=false), Ch. 3, $\S4$, 



> 
> **Theorem** (Schmidt). 
>  Let $H$ and $K$ be Hilbert spaces and $T:H\to K$ a compact operator. Then there exist 
> 
> 
> a) a finite or countable orthonormal system $e'\_1$, $e'\_2,\ldots$, in $H$,
> 
> 
> b) an orthonormal system $e''\_1$, $e''\_2,\ldots$, in $K$ of the same cardinality,
> 
> 
> c) a (finite or infinite) sequence $s\_1\ge s\_2\ge s\_3\ge\ldots$ of positive numbers with the index set of the same cardinality, tending to zero if it is infinite,
> 
> 
> such that
> 
> 
> $$ Tx=\sum\_n s\_n\langle x,e'\_n\rangle e''\_n. $$
>  (In other words, $T(e'\_n)=s\_n e''\_n$ for all $n$ and $T$ takes every vector orthogonal to all $e\_n'$ to zero.)
> 
> 
> 


An equivalent statement: a compact operator in Hilbert spaces is weakly unitary equivalent to a diagonal operator. 


The numbers $\{s\_n\}$ are called the $s$-numbers of the operator $T$. What would they be in the quaternioniс case? Is where a variant of the theorem in which they are complex?



Reference for compact operators in quaternionic Hilbert spaces
Quaternionic functional analysis is reviewed in <http://arxiv.org/abs/math/0609160> (Chi-Keung Ng, On quaternionic functional analysis). In fact "Hilbert spaces of any one of the three kinds - real, complex and quaternionic - can be seen as Hilbert spaces of the other kinds, equipped with extra structure" (John C. Baez, Division Algebras and Quantum Theory, <http://arxiv.org/abs/1101.5690> ).


