Non-holomorphic functional calculus for commuting operators with real spectrum

Mats Andersson; Bo Berndtsson

Annali della Scuola Normale Superiore di Pisa - Classe di Scienze (2002)

  • Volume: 1, Issue: 4, page 925-955
  • ISSN: 0391-173X

Abstract

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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 ℝ n , depending on the growth of ∥ exp ( i a · t ) ∥ , t ∈ ℝ 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.

How to cite

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Andersson, Mats, and Berndtsson, Bo. "Non-holomorphic functional calculus for commuting operators with real spectrum." Annali della Scuola Normale Superiore di Pisa - Classe di Scienze 1.4 (2002): 925-955. <http://eudml.org/doc/84492>.

@article{Andersson2002,
abstract = {We consider $n$-tuples of commuting operators $a=a_1,\ldots ,a_n$ on a Banach space with real spectra. The holomorphic functional calculus for $a$ is extended to algebras of ultra-differentiable functions on $\mathbb \{R\}^n$, depending on the growth of $\Vert \exp (ia\cdot t)\Vert $, $t\in \mathbb \{R\}^n$, when $|t|\rightarrow \infty $. 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.},
author = {Andersson, Mats, Berndtsson, Bo},
journal = {Annali della Scuola Normale Superiore di Pisa - Classe di Scienze},
keywords = {functional calculus; joint spectrum; algebras of ultra-differentiable functions; Fourier transform; FBI transform},
language = {eng},
number = {4},
pages = {925-955},
publisher = {Scuola normale superiore},
title = {Non-holomorphic functional calculus for commuting operators with real spectrum},
url = {http://eudml.org/doc/84492},
volume = {1},
year = {2002},
}

TY - JOUR
AU - Andersson, Mats
AU - Berndtsson, Bo
TI - Non-holomorphic functional calculus for commuting operators with real spectrum
JO - Annali della Scuola Normale Superiore di Pisa - Classe di Scienze
PY - 2002
PB - Scuola normale superiore
VL - 1
IS - 4
SP - 925
EP - 955
AB - We consider $n$-tuples of commuting operators $a=a_1,\ldots ,a_n$ on a Banach space with real spectra. The holomorphic functional calculus for $a$ is extended to algebras of ultra-differentiable functions on $\mathbb {R}^n$, depending on the growth of $\Vert \exp (ia\cdot t)\Vert $, $t\in \mathbb {R}^n$, when $|t|\rightarrow \infty $. 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.
LA - eng
KW - functional calculus; joint spectrum; algebras of ultra-differentiable functions; Fourier transform; FBI transform
UR - http://eudml.org/doc/84492
ER -

References

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