Equivalence of commutativity and complete commutativity for unbounded operators

Determine whether, for unbounded normal operators on a Hilbert space, usual commutativity (S_μ S_ν = S_ν S_μ) implies complete commutativity, i.e., commutativity of the associated spectral projectors E_μ(λ) for all λ, in order to characterize the conditions for simultaneous measurability of quantities represented by such operators.

Background

In the paper, von Neumann formalizes simultaneous measurability via "complete commutativity," defined as the mutual commutation of the spectral projectors associated with the operators. He notes that for bounded operators, usual operator commutativity suffices to ensure complete commutativity.

For unbounded operators, domain issues complicate the matter, and the equivalence was explicitly left unresolved at the time, which is significant for rigorously grounding the criterion for simultaneous measurement in quantum mechanics.