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Under what conditions is the common eigenspace of two commuting hermitian operators isomorphic to the direct product of their individual eigenspaces?
As I'm not being able to precisely phrase my doubt, consider this example: Hilbert space of a two dimensional particle is the direct product of eigenspaces of cartesian X and Y operators. But as far as I know, the common eigenspace of Lz and L2 operators isn't the direct product of their eigenspaces. What is the difference between these two cases?
As I'm not being able to precisely phrase my doubt, consider this example: Hilbert space of a two dimensional particle is the direct product of eigenspaces of cartesian X and Y operators. But as far as I know, the common eigenspace of Lz and L2 operators isn't the direct product of their eigenspaces. What is the difference between these two cases?