Let be a tuple of commuting operators on a Banach space . We discuss various conditions equivalent to that the holomorphic (Taylor) functional calculus has an extension to the real-analytic functions or various ultradifferentiable classes. In particular, we discuss the possible existence of a functional calculus for smooth functions. We relate the existence of a possible extension to existence of a certain (ultra)current extension of the resolvent mapping over the (Taylor) spectrum of . If is a tuple that admits a smooth functional calculus we can define an operation translation by on -valued smooth functions (and forms). As an application we get a new simple proof of the so-called property.The main tool that we introduce in this paper, and which we think has an independent interest, is Fourier transforms of forms and currents. We prove some basic properties including the inversion formula and compute the Fourier transforms of some special currents.
Soit un -uplet d’opérateurs commutant entre eux sur un espace de Banach . Nous discutons diverses conditions équivalentes pour que le calcul fonctionnel holomorphe (de Taylor) s’étende aux fonctions analytiques réelles ou à diverses classes ultra-différentiables. En particulier, nous abordons la possibilité d’un calcul fonctionnel pour les fonctions lisses, qui est liée à l’existence d’un prolongement de l’application résolvante en tant que (ultra-)courant sur le spectre (de Taylor) de . Si est un -uplet admettant un calcul fonctionnel lisse on peut définir une opération de translation par sur les fonctions lisses (et les formes) à valeurs dans . Nous obtenons comme application une nouvelle démonstration simple de la propriété .L’outil principal que nous introduisons dans cet article, et dont nous pensons qu’il a un interêt propre, est la transformée de Fourier des formes et des courants. Nous en démontrons quelques propriétés fondamentales telles que la formule d’inversion et calculons la transformée de Fourier de certains courants particuliers.
Keywords: commuting operators, generalized scalar operator, functional calculus, Bishop’s property $(\beta )$, Taylor spectrum, ultradifferentiable function, resolvent mapping, current
Mot clés : opérateurs commutants, opérateur scalaire généralisé, calcul fonctionnel, propriété $(\beta )$ de Bishop, spectre de Taylor, fonction ultradifférentiable, application résolvante, courant
Andersson, Mats 1
@article{AIF_2003__53_3_903_0, author = {Andersson, Mats}, title = {(Ultra)differentiable functional calculus and current extension of the resolvent mapping}, journal = {Annales de l'Institut Fourier}, pages = {903--926}, publisher = {Association des Annales de l{\textquoteright}institut Fourier}, volume = {53}, number = {3}, year = {2003}, doi = {10.5802/aif.1965}, zbl = {1052.47009}, mrnumber = {2008446}, language = {en}, url = {https://aif.centre-mersenne.org/articles/10.5802/aif.1965/} }
TY - JOUR AU - Andersson, Mats TI - (Ultra)differentiable functional calculus and current extension of the resolvent mapping JO - Annales de l'Institut Fourier PY - 2003 SP - 903 EP - 926 VL - 53 IS - 3 PB - Association des Annales de l’institut Fourier UR - https://aif.centre-mersenne.org/articles/10.5802/aif.1965/ DO - 10.5802/aif.1965 LA - en ID - AIF_2003__53_3_903_0 ER -
%0 Journal Article %A Andersson, Mats %T (Ultra)differentiable functional calculus and current extension of the resolvent mapping %J Annales de l'Institut Fourier %D 2003 %P 903-926 %V 53 %N 3 %I Association des Annales de l’institut Fourier %U https://aif.centre-mersenne.org/articles/10.5802/aif.1965/ %R 10.5802/aif.1965 %G en %F AIF_2003__53_3_903_0
Andersson, Mats. (Ultra)differentiable functional calculus and current extension of the resolvent mapping. Annales de l'Institut Fourier, Volume 53 (2003) no. 3, pp. 903-926. doi : 10.5802/aif.1965. https://aif.centre-mersenne.org/articles/10.5802/aif.1965/
[1] Taylor's functional calculus for commuting operators with Cauchy-Fantappie-Leray formulas, International Math. Research Notices, Volume 6 (1997), pp. 247-258 | DOI | Zbl
[2] Taylor's functional calculus and the resolvent mapping (2000) (Preprint Göteborg)
[3] Integral representation with weights I (to appear in Math. Annalen) | Zbl
[4] Non-holomorphic functional calculus for commuting operators with real spectrum (to appear in Annali Scuola Num. Sup. Pisa) | Numdam | Zbl
[5] On quasianalyticity and general distributions, Lecture notes, Stanford, 1961
[6] Theory of generalized spectral operators, Gordon and Breach, 1968 | MR | Zbl
[7] Extension of analytic functional calculus mappings and duality by -closed forms with growth, Math. Ann, Volume 261 (1982), pp. 185-200 | DOI | MR | Zbl
[8] An operator calculus based on the Cauchy-Green formula. (Russian) Investigations on linear operators and the theory of functions, III, Zap. Nauv. cn. Sem. Leningrad. Otdel. Mat. Inst. Steklov. (LOMI), Volume 30 (1972), pp. 33-39 | MR
[9] Spectral Decompositions and Analytic Sheaves, Clarendon Press, Oxford, 1996 | MR | Zbl
[10] BRST models for equivariant cohomology and representatives for the equivariant Thom class, Comm. Math. Phys, Volume 153 (1993), pp. 447-463 | DOI | MR | Zbl
[11] Ideals of Differentiable Functions, Oxford University Press, 1966 | MR | Zbl
[12] Uniqueness of Taylor's functional calculus, Proc. Amer. Math. Soc, Volume 89 (1983), pp. 647-650 | MR | Zbl
[13] On non-holomorphic functional calculus for commuting operators (to appear in Math. Scand) | MR | Zbl
[14] Sur le changement de variables dans les distributions et leurs transformées de Fourier, Nuovo Cimento, Volume 12 (1954), pp. 471-482 | DOI | MR | Zbl
[15] Théorie des Distributions, Hermann, 1966 | MR | Zbl
[16] A joint spectrum for several commuting operators, J. Funct. Anal, Volume 6 (1970), pp. 172-191 | DOI | MR | Zbl
[17] The analytic-functional calculus for several commuting operators, Acta Math, Volume 125 (1970), pp. 1-38 | DOI | MR | Zbl
Cited by Sources: