We study differential operators D which commute with a fixed normal elliptic operator E on a compact manifold M . We use eigenfunction expansions relative to E to obtain simple conditions giving global hypoellipticity. These conditions are equivalent to D having parametrices in certain spaces of functions or distributions. An example is given by M = compact Lie group and and E = Casimir operator, with D any invariant differential operator. The connections with global subelliptic estimates are investigated.
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Greenfield et al. (1973) studied this question.
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