Hypoellipticity and local solvability in Gevrey classes

Let P be a linear partial differential operator with coefficients in the Gevrey class $G^s$. We prove first that if P is s–hypoelliptic then its transposed operator $^tP$ is s–locally solvable, thus extending to the Gevrey classes the well–known analogous result in the $C^\infty$ class. We prove also that if P is s–hypoelliptic then its null space is finite dimensional and its range is closed; this implies an index theorem for s–hypoelliptic operators. Generalizations of these results to other classes of functions are also considered.