(Ultra)differentiable functional calculus and current extension of the resolvent mapping

Mats Andersson[1]

  • [1] Chalmers University of Technology and the University of Göteborg, Department of Mathematics, 412 96 Göteborg (Suède)

Annales de l’institut Fourier (2003)

  • Volume: 53, Issue: 3, page 903-926
  • ISSN: 0373-0956

Abstract

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Let a = ( a 1 , ... , a n ) be a tuple of commuting operators on a Banach space X . 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 a . If a is a tuple that admits a smooth functional calculus we can define an operation translation by a on X -valued smooth functions (and forms). As an application we get a new simple proof of the so-called ( β ) ℰ 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.

How to cite

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Andersson, Mats. "(Ultra)differentiable functional calculus and current extension of the resolvent mapping." Annales de l’institut Fourier 53.3 (2003): 903-926. <http://eudml.org/doc/116058>.

@article{Andersson2003,
abstract = {Let $a=(a_1,\ldots ,a_n)$ be a tuple of commuting operators on a Banach space $X$. 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 $a$. If $a$ is a tuple that admits a smooth functional calculus we can define an operation translation by $a$ on $X$-valued smooth functions (and forms). As an application we get a new simple proof of the so-called $(\beta )_\{\{\mathcal \{E\}\}\}$ 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.},
affiliation = {Chalmers University of Technology and the University of Göteborg, Department of Mathematics, 412 96 Göteborg (Suède)},
author = {Andersson, Mats},
journal = {Annales de l’institut Fourier},
keywords = {commuting operators; generalized scalar operator; functional calculus; Bishop’s property $(\beta )$; Taylor spectrum; ultradifferentiable function; resolvent mapping; current; Bishop's property beta},
language = {eng},
number = {3},
pages = {903-926},
publisher = {Association des Annales de l'Institut Fourier},
title = {(Ultra)differentiable functional calculus and current extension of the resolvent mapping},
url = {http://eudml.org/doc/116058},
volume = {53},
year = {2003},
}

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
PB - Association des Annales de l'Institut Fourier
VL - 53
IS - 3
SP - 903
EP - 926
AB - Let $a=(a_1,\ldots ,a_n)$ be a tuple of commuting operators on a Banach space $X$. 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 $a$. If $a$ is a tuple that admits a smooth functional calculus we can define an operation translation by $a$ on $X$-valued smooth functions (and forms). As an application we get a new simple proof of the so-called $(\beta )_{{\mathcal {E}}}$ 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.
LA - eng
KW - commuting operators; generalized scalar operator; functional calculus; Bishop’s property $(\beta )$; Taylor spectrum; ultradifferentiable function; resolvent mapping; current; Bishop's property beta
UR - http://eudml.org/doc/116058
ER -

References

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  11. B. Malgrange, Ideals of Differentiable Functions, (1966), Oxford University Press Zbl0177.17902MR212575
  12. M. Putinar, Uniqueness of Taylor's functional calculus, Proc. Amer. Math. Soc 89 (1983), 647-650 Zbl0573.47034MR718990
  13. S. Sandberg, On non-holomorphic functional calculus for commuting operators Zbl1066.32008MR1997876
  14. R. Scarfiello, Sur le changement de variables dans les distributions et leurs transformées de Fourier, Nuovo Cimento 12 (1954), 471-482 Zbl0058.10301MR70966
  15. L. Schwartz, Théorie des Distributions, (1966), Hermann Zbl0078.11003MR209834
  16. J.L. Taylor, A joint spectrum for several commuting operators, J. Funct. Anal 6 (1970), 172-191 Zbl0233.47024MR268706
  17. J.L. Taylor, The analytic-functional calculus for several commuting operators, Acta Math 125 (1970), 1-38 Zbl0233.47025MR271741

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