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On T × G T \times G , where T T is a compact real-analytic manifold and G G is a compact Lie group, we consider differential operators P P which are invariant by left translations on G G and are elliptic in T T . Under a mild technical condition, we prove that global hypoellipticity of P P implies its global analytic-hypoellipticity (actually Gevrey of any order s ≥ 1 s \geq 1 ). We also study the connection between the latter property and the notion of global analytic (resp. Gevrey) solvability, but in a much more general setup.

We introduce a new class of smooth pseudodifferential operators on the torus whose calculus allows us to show that global hypoellipticity with a finite loss of derivatives of certain systems of pseudodifferential operators is stable under perturbations by lower order systems of pseudodifferential operators whose order depends on the loss of derivatives. We also present some applications. K E Y W O R D Sglobal hypoellipticity, loss of derivatives, perturbations by lower order terms, smooth pseudodifferential operators M S C ( 2 0 2 0 ) 35H10, 35H20, 35S05