Spectra of the translations and Wiener-Hopf operators on $L_\omega ^2({\mathbb R}^+)$
HTML articles powered by AMS MathViewer
- 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 \[ \mathcal {O}= \{ z \in \mathbb {C}: \operatorname {Im} z < \alpha _0\}.\]References
- A. Beurling and P. Malliavin, On Fourier transforms of measures with compact support, Acta Math. 107 (1962), 291–309. MR 147848, DOI 10.1007/BF02545792
- 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, DOI 10.1112/blms/22.3.281
- 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, DOI 10.1016/0022-1236(91)90020-6
- Klaus-Jochen Engel and Rainer Nagel, A short course on operator semigroups, Universitext, Springer, New York, 2006. MR 2229872
- G. I. Eskin, Boundary value problems for elliptic pseudodifferential equations, Translations of Mathematical Monographs, vol. 52, American Mathematical Society, Providence, R.I., 1981. Translated from the Russian by S. Smith. MR 623608
- Larry Gearhart, Spectral theory for contraction semigroups on Hilbert space, Trans. Amer. Math. Soc. 236 (1978), 385–394. MR 461206, DOI 10.1090/S0002-9947-1978-0461206-1
- 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
- I. Herbst, The spectrum of Hilbert space semigroups, J. Operator Theory 10 (1983), no. 1, 87–94. MR 715559
- Lars Hörmander, Estimates for translation invariant operators in $L^{p}$ spaces, Acta Math. 104 (1960), 93–140. MR 121655, DOI 10.1007/BF02547187
- Ronald Larsen, The multiplier problem, Lecture Notes in Mathematics, Vol. 105, Springer-Verlag, Berlin-New York, 1969. MR 0435737
- Violeta Petkova, Symbole d’un multiplicateur sur $L^2_\omega ({\Bbb R})$, Bull. Sci. Math. 128 (2004), no. 5, 391–415 (French, with English and French summaries). MR 2066346, DOI 10.1016/j.bulsci.2004.03.001
- Violeta Petkova, Wiener-Hopf operators on $L^2_\omega (\Bbb R^+)$, Arch. Math. (Basel) 84 (2005), no. 4, 311–324. MR 2135041, DOI 10.1007/s00013-004-1167-z
- Violeta Petkova, Wiener-Hopf operators on spaces of functions on $\Bbb R^+$ with values in a Hilbert space, Integral Equations Operator Theory 59 (2007), no. 3, 355–378. MR 2363014, DOI 10.1007/s00020-007-1530-0
- Violeta Petkova, Multipliers on a Hilbert space of functions on $\Bbb R$, Serdica Math. J. 35 (2009), no. 2, 207–216. MR 2567484
- Violeta Petkova, Spectral theorem for multipliers on $L^2_\omega (\Bbb R)$, Arch. Math. (Basel) 93 (2009), no. 4, 357–368. MR 2558528, DOI 10.1007/s00013-009-0043-2
- William C. Ridge, Approximate point spectrum of a weighted shift, Trans. Amer. Math. Soc. 147 (1970), 349–356. MR 254635, DOI 10.1090/S0002-9947-1970-0254635-5
- William C. Ridge, Spectrum of a composition operator, Proc. Amer. Math. Soc. 37 (1973), 121–127. MR 306457, DOI 10.1090/S0002-9939-1973-0306457-2
- F.-O. Speck, General Wiener-Hopf factorization methods, Research Notes in Mathematics, vol. 119, Pitman (Advanced Publishing Program), Boston, MA, 1985. With a foreword by E. Meister. MR 790315
- Lutz Weis, The stability of positive semigroups on $L_p$ spaces, Proc. Amer. Math. Soc. 123 (1995), no. 10, 3089–3094. MR 1273529, DOI 10.1090/S0002-9939-1995-1273529-2
- 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, DOI 10.1090/S0002-9939-98-04612-7
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