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 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 = z ∈ ℂ: Im z α 0.
@article{bwmeta1.element.doi-10_2478_s11533-012-0139-y,
author = {Violeta Petkova},
title = {Multipliers and Wiener-Hopf operators on weighted L p spaces},
journal = {Open Mathematics},
volume = {11},
year = {2013},
pages = {561-573},
zbl = {1258.47048},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.doi-10_2478_s11533-012-0139-y}
}
Violeta Petkova. Multipliers and Wiener-Hopf operators on weighted L p spaces. Open Mathematics, Tome 11 (2013) pp. 561-573. http://gdmltest.u-ga.fr/item/bwmeta1.element.doi-10_2478_s11533-012-0139-y/
[1] Engel K.-J., Nagel R., A Short Course on Operator Semigroups, Universitext, Springer, New York, 2006
[2] Fašangová E., Miana P.J., Spectral mapping inclusions for the Phillips functional calculus in Banach spaces and algebras, Studia Math., 2005, 167(3), 219–226 http://dx.doi.org/10.4064/sm167-3-3[Crossref] | Zbl 1080.47017
[3] Gearhart L., Spectral theory for contraction semigroups on Hilbert space, Trans. Amer. Math. Soc., 1978, 236, 385–394 http://dx.doi.org/10.1090/S0002-9947-1978-0461206-1[Crossref] | Zbl 0326.47038
[4] Latushkin Yu., Montgomery-Smith S., Evolutionary semigroups and Lyapunov theorems in Banach spaces, J. Funct. Anal., 1995, 127(1), 173–197 http://dx.doi.org/10.1006/jfan.1995.1007[Crossref]
[5] Petkova V., Wiener-Hopf operators on L ω2(ℝ+), Arch. Math. (Basel), 2005, 84(4), 311–324 http://dx.doi.org/10.1007/s00013-004-1167-z[Crossref]
[6] Petkova V., Multipliers on Banach spaces of functions on a locally compact abelian group, J. Lond. Math. Soc., 2007, 75(2), 369–390 http://dx.doi.org/10.1112/jlms/jdm002[Crossref] | Zbl 1138.43005
[7] Petkova V., Multipliers on a Hilbert space of functions on R, Serdica Math. J., 2009, 35(2), 207–216 | Zbl 1224.42019
[8] Petkova V., Spectral theorem for multipliers on L ω2(ℝ), Arch. Math. (Basel), 2009, 93(4), 357–368 http://dx.doi.org/10.1007/s00013-009-0043-2[Crossref][WoS] | Zbl 1217.47010
[9] Petkova V., Spectra of the translations and Wiener-Hopf operators on L ω2(ℝ+), Proc. Amer. Math. Soc. (in press) [WoS]
[10] Ridge W.C., Approximate point spectrum of a weighted shift, Trans. Amer. Math. Soc., 1970, 147(2), 349–356 http://dx.doi.org/10.1090/S0002-9947-1970-0254635-5[Crossref] | Zbl 0192.47803
[11] Rudin W., Fourier Analysis on Groups, Interscience Tracts in Pure and Applied Mathematics, 12, Interscience, New York-London, 1962
[12] Weis L., The stability of positive semigroups on L p-spaces, Proc. Amer. Math. Soc., 1995, 123(10), 3089–3094 | Zbl 0851.47028
[13] Weis L., A short proof for the stability theorem for positive semigroups on L p(µ), Proc. Amer. Math. Soc., 1998, 126(11), 3253–3256 http://dx.doi.org/10.1090/S0002-9939-98-04612-7[Crossref] | Zbl 0904.47028