I am not sure if this is completely ill defined, and not much can be said about. Any pointer/suggestion would be helpful.
Suppose $T \in \mathbb{R}^{m \times n}$ with $m < n$ ("fat matrix") and $K_1 \in \mathbb{R}^{n\times n}$ and $K_2 \in \mathbb{R}^{m\times m}$.
Can one say what properties $K_1$ and $K_2$ should satisfy for $$ TK_1 =K_2T $$