  We introduce a class of (tuples of commuting) unbounded operators on a Banach
space, admitting smooth functional calculi, that contains all operators of
Helffer-Sj\"ostrand type and is closed under the action of smooth proper
mappings. Moreover, the class is closed under tensor product of commuting
operators. In general an operator in this class has no resolvent in the usual
sense so the spectrum must be defined in terms of the functional calculus. We
also consider invariant subspaces and spectral decompositions.
