Complex powers for a class of infinite order hypoelliptic operators

We prove the complex powers of a class of infinite order hypoelliptic pseudodifferential operators can always be represented as hypoelliptic pseudodifferential operators modulo ultrasmoothing operators. We apply this result to the study of semigroups generated by square roots of non-negative hypoelliptic infinite order operators. For this purpose, we derive precise estimates of the corresponding heat kernel.