Let $F^{i \times j}$ be the vector space of $i \times j$ matrices. Let $m$ and $n$ be integers, $n < m$. Let $S$ be a subspace of $F^{m \times m}$. I would need to find $m \times n$ matrices $X$ and $Y$, such that the "superoperator" $T: F^{m \times m} \to F^{n \times n}$, $T(A) = X^\dagger A Y$, is trace-preserving on the subspace $S$.
In other words, I have a bunch of square matrices $M_i$ (the span of which is $S$), and I would like to find two non-square matrices $X$ and $Y$ and multiply all $M_i \mapsto X^\dagger M_i Y$ to get smaller square matrices, but so that the traces are preserved in this process for all $M_i$. What happens to other properties of the $M_i$'s under this mapping, I don't care. What happens to the traces of other matrices, outside the span of the $M_i$'s, I don't care. The question is, how do I find such $X$ and $Y$.
An immediate observation to make is that no such $X$ and $Y$ exist if $n$ is too small, or one is otherwise having a bad day. On the other hand, in some cases when $S$ is small enough and $n$ not too small, non-trivial cases of this have solutions. I don't expect to find a neat, exact closed form solution (this would be a part of a numerical optimization procedure anyway), but I'm more looking for ideas or hints as to where to start looking. For instance, are there any useful characterizations for being trace-preserving on a subspace? Any useful keywords that I might have missed? I've spent some time reading stuff about quantum channels, but haven't found anything too useful.