Operators with smooth functional calculi

We introduce a class of (tuples of commuting) unbounded operators on a Banach space, admitting smooth functional calculi, which contains all operators of Helffer-Sjostrand 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.