The title pretty much says it all. For example, we sometimes think of the three momentum operators as components of a "vector operator" $\hat{ \vec{p}}=(\hat p_x,\hat p_y,\hat p_z)$. The terminology is often justified via commutation relations and can made rigorous using representation theory [1] [2].
But does there exist (or can we construct) a space in which the vector operator $\hat{\vec p}$ lives, as an actual vector (or generalized vector) such that the standard operators $\hat p_i$ are genuine components? This would be nice because then, for instance, we could derive the transformation laws for how the $\hat p_i$ change under a change of basis, rather than imposing them by hand.
A first thought would be something like End$(H)\times{}$End$(H)\times{}$End$(H)$. But this space has infinite dimension [assuming $H=L^2(\mathbb R^3)$], and we want a three-dimensional space. We also want the components to be operators, which a standard vector space can't accommodate. But maybe a module would work?
