# (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

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topAndersson, 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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