Abstract

We consider $n$-tuples of commuting operators $a=a_1,\ldots,a_n$ on a Banach space with real spectra. The holomorphic functional calculus for $a$ is extended to algebras of ultra-differentiable functions on $\mathbb{R}^n$, depending on the growth of $\Vert \exp \Vert $, $t\in \mathbb{R}^n$, when $|t|\rightarrow \infty $. 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