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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
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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 \[ \sigma (R _t) = \{z \in \mathbb {C}: |z| \leq e^{\alpha _0 t}\},\] 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 \[ \mathcal {O}= \{ z \in \mathbb {C}: \operatorname {Im} z < \alpha _0\}.\]


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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

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