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, Ch. 3, $\S4$,

Theorem(Schmidt). Let $H$ and $K$ be Hilbert spaces and $T:H\to K$ a compact operator. Then there exista) 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?

someimpact, but, lacking more specifics, it's not clear what one gains by having had a quaternionic structure. $\endgroup$