What makes an operator a "genuine operator"? My professor defined these operators, $P_{T}=\rho_1+\rho_2$, and, $x_{cm}=\frac{x_1+x_2}{2}$, in class and said something along the lines of, “These are genuine momentum-position operators.” Now, I recognize that these are classically, the total momentum and center of mass, but what makes them “genuine momentum-position operators”?
For that matter, how would one define an operator which genuinely belongs to a particular class of operators? For example, in the addition of angular momentum, we define an operator, $$\vec{J}=\vec{J_1}+\vec{J_2}.$$ How can we say that this is truly an angular momentum operator? How do we know?
 A: If I understand your question, an operator is said to belong to a particular class of operators if it satisfies a particular commutation relation characteristic to all operators of a class. For example, position-momentum operators satisfy, $$[\hat{x},\hat{p}]=i\hbar.$$
Consider the position and momentum operators, your professor defined. We have $$[x_{cm},P_T]=[\frac{x_1}{2}+\frac{x_2}{2},p_1+p_2]=[\frac{x_1}{2},p_1]+[\frac{x_2}{2},p_2]=\frac{1}{2}i\hbar+\frac{1}{2}i\hbar=i\hbar.$$
Similarly, the commutation relation obeyed by angular momentum operators is, $$[L_i,L_j]=i\hbar\epsilon_{ijk}L_k,$$ where $\epsilon_{ijk}$ is the Levi-Civita symbol. 
Using this, we can verify that $J=J_1+J_2=J_{1x}+J_{2x}+J_{1y}+J_{2y}+J_{1z}+J_{2z}$ is truly an angular momentum operator. Consider the commutation, $$[J_x,J_y]=[J_{1x}+J_{2x},J_{1y}+J_{2y}].$$ This evaluates to, $$[J_{1x},J_{1y}]+[J_{2x},J_{2y}]=i\hbar J_{1z}+i\hbar J_{2z}=i\hbar(J_{1z}+J_{2z})=i\hbar J_{z},$$ which we expect if $J$ is a true angular momentum operator.
Similar arguments can be made for other operators (i.e. the raising/lowering operators, etc.). Hope that helps.
