  Let $-\im\Lie_\T$ (essentially Lie derivative with respect to $\T$, a smooth
nowhere zero real vector field) and $P$ be commuting differential operators,
respectively of orders 1 and $m\geq 1$, the latter formally normal, both acting
on sections of a vector bundle over a closed manifold. It is shown that if
$P+(-i\Lie_\T)^m$ is elliptic then the restriction of $-\im\Lie_\T$ to
$\Dom\subset \ker P\subset L^2$ yields a selfadjoint operator
$-\im\Lie_\T|_\Dom:\Dom\subset\ker P\to \ker P$ with compact resolvent ($\Dom$
is specified carefully). It is also shown that, in the presence of an
additional hypothesis on microlocal hypoellipticity of $P$, $-\im\Lie_\T|_\Dom$
is semi-bounded. These results are applied to CR manifolds on which $\T$ acts
as an infinitesimal CR transformation which are then shown to yield versions of
Kodaira's vanishing theorem.
