Every time I run into a commutator of two observables such as $[\hat{X},\hat{Y}]$, with $\hat{X}$ and $\hat{Y}$ being two operators from different state spaces: $\hat{X}\in\xi_{x}\land\hat{Y}\in\xi_{y}$; it is said that their commutator equals zero because each operator acts on its own subspace.
Could someone give the proof for that statement? Thank you.
If our Hilbert space can be separated to two (or more!) orthogonal subspaces $|\xi_x\rangle$ and $|\xi_y\rangle$, then every state in it can be written as a a direct product of states from this two orthogonal subspaces $$|\xi_x, \xi_y \rangle = |\xi_x\rangle \otimes |\xi_y\rangle$$ In this case, an operator $\hat{X}$ that acts only on states in the subspace spanned by $|\xi_x\rangle$ will act like the identity operator on all states in the orthogonal subspace spanned by $|\xi_y\rangle$, and likewise for an operator $\hat{Y}$ that acts only on states in the subspace spanned by $|\xi_y\rangle$.
From here, it is clear to see that for every state $$\hat{X} \hat{Y} |\xi_x, \xi_y\rangle = \hat{Y} \hat{X} |\xi_x, \xi_y\rangle$$ As this is true for every state, they commute.
