Let's consider bounded operators for simplicity. Are there known (necesssary, sufficient) conditions on (self adjoint) operators $S,T$ on a Hilbert space $H$ to give that there exists Hilbert spaces $H_1, H_2$, a unitary map $U : H \to H_1 \otimes H_2$, and operators $A$ on $H_1$ and $B$ on $H_2$ such that $$U S U^{-1} = A \otimes I;\ \ UTU^{-1} = I \otimes B?$$ This question arises from considering quantum mechanics, if two observables have this form they are really acting on distinct physical subsystems. As such, they are particularly easy to work with. The question is if there is a way to recognize in some generality when this is the case.
We note immediately a necessary condition is that $S, T$ commute. This is of course not sufficient, as $S$ commutes with itself.