Global Hypoellipticity and Global Solvability for a Class of Operators on Compact Manifolds

We consider operators such as pseudo-differential operators on a manifoldM1. LetM2be another manifold, and considerPalso as an operator onM1×M2. We prove that, whenPis injective in ′ and symmetric, the following four properties ofPare equivalent: global hypoellipticity onM1×M2, global solvability onM1×M2, global hypoellipticity onM1, and global sovability onM1. We apply this for a specific class of pseudo-differential operators onS1×M2