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://www.numdam.org/item/ASNSP_2002_5_1_4_925_0/

[1] D. W. Albrecht, Explicit formulae for Taylor's functional calculus, Multivariable operator theory (Seattle, WA, 1993), 1-5, Contemp. Math. 185, AMS. Providence, RI. 1995. | MR | Zbl

[2] D. W. Albrecht, Integral formulae for special cases of Taylor's functional calculus, Studia Math. 105 (1993), 51-68. | MR | Zbl

[3] M. Andersson, Taylor's functional calculus with Cauchy-Fantappie-Leray formulas, Int. Math. Res. Not. 6 (1997), 247-258. | MR | Zbl

[4] M. Andersson, Correction to Taylor's functional calculus with Cauchy-Fantappie-Leray formulas, Int. Math. Res. Not. 2 (1998), 123-124. | MR | Zbl

[5] M. Andersson - B. Berndtsson, Almost holomorphic extensions of ultradifferentiable functions, to appear in J. d'Analyse. | MR | Zbl

[6] A. Beurling, On quasianalyticity and general distributions, Lecture notes, Stanford, 1961.

[7] B. Droste, Holomorphic approximation of ultradifferentiable functions, Math. Ann. 257 (1981), 293-316. | MR | Zbl

[8] B. Droste, Extension of analytic functional calculus mappings and duality by ¯-closed forms with growth, Math. Ann. 261 (1982), 185-200. | MR | Zbl

[9] E. M. Dynkin, An operator calculus based on the Cauchy-Green formula 30 (1972), 33-39. | MR | Zbl

[10] J. Eschmeier - M. Putinar, “Spectral Decompositions and Analytic Sheaves”, Clarendon Press, Oxford, 1996. | MR | Zbl

[11] L. Hörmander, “The Analysis of Linear Partial Differential Operators I”, Sec. Ed. Springer-Verlag, 1990. | Zbl

[12] V. Kordula - Müller, Vasilescu-Martinelli formula for operators in Banach spaces, Studia Math. 113 (1995), 127-139. | MR | Zbl

[13] T. H. Nguyen, Calcul fonctionnel dependant de la croissance des coefficients spectraux, Ann. Inst. Fourier (Grenoble) 27 (1977), 169-199. | Numdam | MR | Zbl

[14] M. Putinar, The superposition property for Taylor's functional calculus, J. Operator Theory 7 (1982), 149-155. | MR | Zbl

[15] W. Rudin, “Functional Analysis”, McGraw-Hill, 1973. | MR | Zbl

[16] S. Sandberg, On non-holomorphic functional calculus for commuting operators, To appear in Math. Scand. | MR | Zbl

[17] J. L. Taylor, A joint spectrum for several commuting operators, J. Funct. Anal. 6 (1970), 172-191. | MR | Zbl

[18] J. L. Taylor, The analytic-functional calculus for several commuting operators, Acta Math. 125 (1970), 1-38. | MR | Zbl

[19] F. H. Vasilescu, “Analytic functional calculus and spectral decompositions”, Mathematics and its Applications (East European Series), D. Reidel Publ. Co., Dordrecht-Boston, Mass., 1982. | MR | Zbl

[20] L. Waelbroeck, Calcul symbolique lié à la croissance de la résolvant, Rend. Sem. Mat. Fis. Milano, 34 (1964). | MR | Zbl

[21] J. Wermer, The existence of invariant subspaces, Duke Math. J. 19 (1952), 615-622. | MR | Zbl