Non-holomorphic functional calculus for commuting operators with real spectrum
Annali della Scuola Normale Superiore di Pisa - Classe di Scienze, Série 5, Tome 1 (2002) no. 4, pp. 925-955.

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 n , depending on the growth of exp(ia·t), t n , when |t|. 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.

Classification : 47A60, 47A13, 32A25, 32A65, 46F05
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Andersson, Mats; Berndtsson, Bo. Non-holomorphic functional calculus for commuting operators with real spectrum. Annali della Scuola Normale Superiore di Pisa - Classe di Scienze, Série 5, Tome 1 (2002) no. 4, pp. 925-955. http://archive.numdam.org/item/ASNSP_2002_5_1_4_925_0/

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