Under what conditions is the common eigenspace of two commuting hermitian operators isomorphic to the direct product of their individual eigenspaces?(adsbygoogle = window.adsbygoogle || []).push({});

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 L_{z}and L^{2}operators isn't the direct product of their eigenspaces. What is the difference between these two cases?

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# Commuting operators and Direct product spaces

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