Non-Holomorphic Functional Calculus for Commuting Operators with Real Spectrum

We consider n-tuples of commuting operators a = a(1) ,..., a(n) on a Banach space with real spectra. The holomorphic functional calculus for a is extended to algebras of ultra-differentiable functions on R-n, depending on the growth of parallel to exp(ia . t)parallel to, t is an element of R-n, when |t| -> infinity. In the non-quasi-analytic case we use the usual Fourier transform, whereas for the quasi-analytic case we introduce a variant of the FBI transform, adapted to ultradifferentiable classes.