Remote Access Proceedings of the American Mathematical Society
Green Open Access

Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)

 
 

 

Spectra of the translations and Wiener-Hopf operators on $ L_\omega^2({\mathbb{R}}^+)$


Author: Violeta Petkova
Journal: Proc. Amer. Math. Soc. 142 (2014), 2491-2505
MSC (2010): Primary 47B35; Secondary 47B37, 47A10, 47A25
DOI: https://doi.org/10.1090/S0002-9939-2014-11988-5
Published electronically: April 2, 2014
MathSciNet review: 3195770
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: We study bounded operators $ T$ on the weighted space $ L^2_{\omega }(\mathbb{R}^+)$ commuting either with the ``right shift operators'' $ (R _t)_{t \geq 0}$ or ``left shift operators'' $ (L_{-t})_{t \geq 0},$ and we establish the existence of a symbol $ \mu $ of $ T$. We characterize completely the spectrum $ \sigma (R_t)$ of the operator $ R_t$ proving that

$\displaystyle \sigma (R _t) = \{z \in \mathbb{C}: \vert z\vert \leq e^{\alpha _0 t}\},$

where $ \alpha _0$ is the growth bound of $ (R_t)_{t\geq 0}$. We obtain a similar result for the spectrum of $ L_{-t},\: t >0.$ Moreover, for a bounded operator $ T$ commuting with $ R _t, \: t \geq 0,$ we establish the inclusion $ \overline {\mu ({\mathcal O})}\subset \sigma (T)$, where

$\displaystyle \mathcal {O}= \{ z \in \mathbb{C}: \operatorname {Im} z < \alpha _0\}.$


References [Enhancements On Off] (What's this?)

  • [1] A. Beurling and P. Malliavin, On Fourier transforms of measures with compact support, Acta Math. 107 (1962), 291-309. MR 0147848 (26 #5361)
  • [2] A. Böttcher, B. Silbermann, and I. M. Spitkovskiĭ, Toeplitz operators with piecewise quasisectorial symbols, Bull. London Math. Soc. 22 (1990), no. 3, 281-286. MR 1041144 (90m:47042), https://doi.org/10.1112/blms/22.3.281
  • [3] Albrecht Böttcher and Ilya M. Spitkovsky, Toeplitz operators with PQC symbols on weighted Hardy spaces, J. Funct. Anal. 97 (1991), no. 1, 194-214. MR 1105659 (92k:47046), https://doi.org/10.1016/0022-1236(91)90020-6
  • [4] Klaus-Jochen Engel and Rainer Nagel, A short course on operator semigroups, Universitext, Springer, New York, 2006. MR 2229872 (2007e:47001)
  • [5] G. I. Eskin, Boundary value problems for elliptic pseudodifferential equations, Translations of Mathematical Monographs, vol. 52, American Mathematical Society, Providence, RI, 1981. Translated from the Russian by S. Smith. MR 623608 (82k:35105)
  • [6] Larry Gearhart, Spectral theory for contraction semigroups on Hilbert space, Trans. Amer. Math. Soc. 236 (1978), 385-394. MR 0461206 (57 #1191)
  • [7] I. C. Gohberg and M. G. Kreĭn, Fundamental aspects of defect numbers, root numbers and indexes of linear operators, Uspehi Mat. Nauk (N.S.) 12 (1957), no. 2(74), 43-118 (Russian). MR 0096978 (20 #3459)
  • [8] I. Herbst, The spectrum of Hilbert space semigroups, J. Operator Theory 10 (1983), no. 1, 87-94. MR 715559 (84m:47052)
  • [9] Lars Hörmander, Estimates for translation invariant operators in $ L^{p}$spaces, Acta Math. 104 (1960), 93-140. MR 0121655 (22 #12389)
  • [10] Ronald Larsen, The multiplier problem, Lecture Notes in Mathematics, Vol. 105, Springer-Verlag, Berlin, 1969. MR 0435737 (55 #8694)
  • [11] Violeta Petkova, Symbole d'un multiplicateur sur $ L^2_\omega ({\mathbb{R}})$, Bull. Sci. Math. 128 (2004), no. 5, 391-415 (French, with English and French summaries). MR 2066346 (2005h:47067), https://doi.org/10.1016/j.bulsci.2004.03.001
  • [12] Violeta Petkova, Wiener-Hopf operators on $ L^2_\omega (\mathbb{R}^+)$, Arch. Math. (Basel) 84 (2005), no. 4, 311-324. MR 2135041 (2005k:47061), https://doi.org/10.1007/s00013-004-1167-z
  • [13] Violeta Petkova, Wiener-Hopf operators on spaces of functions on $ \mathbb{R}^+$ with values in a Hilbert space, Integral Equations Operator Theory 59 (2007), no. 3, 355-378. MR 2363014 (2008g:47056), https://doi.org/10.1007/s00020-007-1530-0
  • [14] Violeta Petkova, Multipliers on a Hilbert space of functions on $ \mathbb{R}$, Serdica Math. J. 35 (2009), no. 2, 207-216. MR 2567484 (2010k:42012)
  • [15] Violeta Petkova, Spectral theorem for multipliers on $ L^2_\omega (\mathbb{R})$, Arch. Math. (Basel) 93 (2009), no. 4, 357-368. MR 2558528 (2010k:47013), https://doi.org/10.1007/s00013-009-0043-2
  • [16] William C. Ridge, Approximate point spectrum of a weighted shift, Trans. Amer. Math. Soc. 147 (1970), 349-356. MR 0254635 (40 #7843)
  • [17] William C. Ridge, Spectrum of a composition operator, Proc. Amer. Math. Soc. 37 (1973), 121-127. MR 0306457 (46 #5583)
  • [18] F.-O. Speck, General Wiener-Hopf factorization methods, with a foreword by E. Meister, Research Notes in Mathematics, vol. 119, Pitman (Advanced Publishing Program), Boston, MA, 1985. MR 790315 (87a:47045)
  • [19] Lutz Weis, The stability of positive semigroups on $ L_p$ spaces, Proc. Amer. Math. Soc. 123 (1995), no. 10, 3089-3094. MR 1273529 (95m:47074), https://doi.org/10.2307/2160665
  • [20] Lutz Weis, A short proof for the stability theorem for positive semigroups on $ L_p(\mu )$, Proc. Amer. Math. Soc. 126 (1998), no. 11, 3253-3256. MR 1469440 (99a:47064), https://doi.org/10.1090/S0002-9939-98-04612-7

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC (2010): 47B35, 47B37, 47A10, 47A25

Retrieve articles in all journals with MSC (2010): 47B35, 47B37, 47A10, 47A25


Additional Information

Violeta Petkova
Affiliation: LMAM, Université de Lorraine (Metz), UMR 7122, Ile du Saulcy, 57045 Metz Cedex 1, France
Address at time of publication: IECL, Université de Lorraine (Metz), Bât A, Ile du Saulcy, 57045 Metz Cedex 1, France
Email: violeta.petkova@univ-lorraine.fr

DOI: https://doi.org/10.1090/S0002-9939-2014-11988-5
Keywords: Translations, spectrum of Wiener-Hopf operator, semigroup of translations, weighted spaces, symbol
Received by editor(s): March 5, 2012
Received by editor(s) in revised form: July 10, 2012, July 30, 2012, and August 11, 2012
Published electronically: April 2, 2014
Communicated by: Michael Hitrik
Article copyright: © Copyright 2014 American Mathematical Society

American Mathematical Society