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Proceedings of the American Mathematical Society
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
Published electronically: April 2, 2014
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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

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


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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: http://dx.doi.org/10.1090/S0002-9939-2014-11988-5
PII: S 0002-9939(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