Multipliers and Wiener-Hopf operators on weighted L p spaces

Violeta Petkova

Open Mathematics (2013)

  • Volume: 11, Issue: 3, page 561-573
  • ISSN: 2391-5455

Abstract

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We study multipliers M (bounded operators commuting with translations) on weighted spaces L ω p (ℝ), and establish the existence of a symbol µM for M, and some spectral results for translations S t and multipliers. We also study operators T on the weighted space L ω p (ℝ+) commuting either with the right translations S t , t ∈ ℝ+, or left translations P +S −t , t ∈ ℝ+, and establish the existence of a symbol µ of T. We characterize completely the spectrum σ(S t ) of the operator S t proving that σ ( S t ) = { z ∈ ℂ : | z | ⩽ e t α 0 } , where α 0 is the growth bound of (S t )t≥0. A similar result is obtained for the spectrum of (P +S −t ), t ≥ 0. Moreover, for an operator T commuting with S t , t ≥ 0, we establish the inclusion [...] , where 𝒪 = z ∈ ℂ: Im z α 0.

How to cite

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Violeta Petkova. "Multipliers and Wiener-Hopf operators on weighted L p spaces." Open Mathematics 11.3 (2013): 561-573. <http://eudml.org/doc/268958>.

@article{VioletaPetkova2013,
abstract = {We study multipliers M (bounded operators commuting with translations) on weighted spaces L ω p (ℝ), and establish the existence of a symbol µM for M, and some spectral results for translations S t and multipliers. We also study operators T on the weighted space L ω p (ℝ+) commuting either with the right translations S t , t ∈ ℝ+, or left translations P +S −t , t ∈ ℝ+, and establish the existence of a symbol µ of T. We characterize completely the spectrum σ(S t ) of the operator S t proving that $\sigma (S_t ) = \lbrace z \in \mathbb \{C\}:|z| \leqslant e^\{t\alpha _0 \} \rbrace ,$ where α 0 is the growth bound of (S t )t≥0. A similar result is obtained for the spectrum of (P +S −t ), t ≥ 0. Moreover, for an operator T commuting with S t , t ≥ 0, we establish the inclusion [...] , where $\mathcal \{O\}$ = z ∈ ℂ: Im z α 0.},
author = {Violeta Petkova},
journal = {Open Mathematics},
keywords = {Mutilpliers; Semi-groups of translations; Spectrum of translation; Weiner-Hopf operators; multipliers; semigroups of translations; spectrum of a translation; Wiener-Hopf operators},
language = {eng},
number = {3},
pages = {561-573},
title = {Multipliers and Wiener-Hopf operators on weighted L p spaces},
url = {http://eudml.org/doc/268958},
volume = {11},
year = {2013},
}

TY - JOUR
AU - Violeta Petkova
TI - Multipliers and Wiener-Hopf operators on weighted L p spaces
JO - Open Mathematics
PY - 2013
VL - 11
IS - 3
SP - 561
EP - 573
AB - We study multipliers M (bounded operators commuting with translations) on weighted spaces L ω p (ℝ), and establish the existence of a symbol µM for M, and some spectral results for translations S t and multipliers. We also study operators T on the weighted space L ω p (ℝ+) commuting either with the right translations S t , t ∈ ℝ+, or left translations P +S −t , t ∈ ℝ+, and establish the existence of a symbol µ of T. We characterize completely the spectrum σ(S t ) of the operator S t proving that $\sigma (S_t ) = \lbrace z \in \mathbb {C}:|z| \leqslant e^{t\alpha _0 } \rbrace ,$ where α 0 is the growth bound of (S t )t≥0. A similar result is obtained for the spectrum of (P +S −t ), t ≥ 0. Moreover, for an operator T commuting with S t , t ≥ 0, we establish the inclusion [...] , where $\mathcal {O}$ = z ∈ ℂ: Im z α 0.
LA - eng
KW - Mutilpliers; Semi-groups of translations; Spectrum of translation; Weiner-Hopf operators; multipliers; semigroups of translations; spectrum of a translation; Wiener-Hopf operators
UR - http://eudml.org/doc/268958
ER -

References

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  9. [9] Petkova V., Spectra of the translations and Wiener-Hopf operators on L ω2(ℝ+), Proc. Amer. Math. Soc. (in press) [WoS] 
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  11. [11] Rudin W., Fourier Analysis on Groups, Interscience Tracts in Pure and Applied Mathematics, 12, Interscience, New York-London, 1962 
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