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Proceedings of the American Mathematical Society

Published by the American Mathematical Society, the Proceedings of the American Mathematical Society (PROC) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

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

The 2020 MCQ for Proceedings of the American Mathematical Society is 0.85.

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Spectra of the translations and Wiener-Hopf operators on $L_\omega ^2({\mathbb R}^+)$
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by Violeta Petkova PDF
Proc. Amer. Math. Soc. 142 (2014), 2491-2505 Request permission

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
  • 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
  • © Copyright 2014 American Mathematical Society
  • 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
  • MathSciNet review: 3195770