Bohr’s inequality for non-commutative Hardy spaces
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- by Sneh Lata and Dinesh Singh PDF
- Proc. Amer. Math. Soc. 150 (2022), 201-211 Request permission
Abstract:
In this paper we extend the classical Bohr’s inequality to the setting of the non-commutative Hardy space $H^1$ associated with a semifinite von Neumann algebra. As a consequence, we obtain Bohr’s inequality for operators in the von Neumann-Schatten class $\mathcal C_1$ and square matrices of any finite order. Interestingly, we establish that the optimal bound for $r$ in the above mentioned Bohr’s inequality concerning von Neumann-Schatten class is 1/3 whereas it is 1/2 in the case of $2\times 2$ matrices and reduces to $\sqrt {2}-1$ for the case of $3\times 3$ matrices. We also obtain a generalization of our above-mentioned Bohr’s inequality for finite matrices where we show that the optimal bound for $r$, unlike above, remains 1/3 for every fixed order $n\times n,\ n\ge 2$.References
- Yusuf Abu Muhanna and Rosihan M. Ali, Bohr’s phenomenon for analytic functions and the hyperbolic metric, Math. Nachr. 286 (2013), no. 11-12, 1059–1065. MR 3092270, DOI 10.1002/mana.201200197
- Lev Aizenberg and Nikolai Tarkhanov, A Bohr phenomenon for elliptic equations, Proc. London Math. Soc. (3) 82 (2001), no. 2, 385–401. MR 1806876, DOI 10.1112/S0024611501012813
- R. Balasubramanian, B. Calado, and H. Queffélec, The Bohr inequality for ordinary Dirichlet series, Studia Math. 175 (2006), no. 3, 285–304. MR 2261747, DOI 10.4064/sm175-3-7
- David P. Blecher and Louis E. Labuschagne, Von Neumann algebraic $H^p$ theory, Function spaces, Contemp. Math., vol. 435, Amer. Math. Soc., Providence, RI, 2007, pp. 89–114. MR 2359421, DOI 10.1090/conm/435/08369
- David P. Blecher and Louis E. Labuschagne, A Beurling theorem for noncommutative $L^p$, J. Operator Theory 59 (2008), no. 1, 29–51. MR 2404463
- David P. Blecher and Louis E. Labuschagne, Outers for noncommutative $H^p$ revisited, Studia Math. 217 (2013), no. 3, 265–287. MR 3119759, DOI 10.4064/sm217-3-4
- Harald Bohr, A Theorem Concerning Power Series, Proc. London Math. Soc. (2) 13 (1914), 1–5. MR 1577494, DOI 10.1112/plms/s2-13.1.1
- P. G. Dixon, Banach algebras satisfying the non-unital von Neumann inequality, Bull. London Math. Soc. 27 (1995), no. 4, 359–362. MR 1335287, DOI 10.1112/blms/27.4.359
- Stavros Evdoridis, Saminathan Ponnusamy, and Antti Rasila, Improved Bohr’s inequality for locally univalent harmonic mappings, Indag. Math. (N.S.) 30 (2019), no. 1, 201–213. MR 3906130, DOI 10.1016/j.indag.2018.09.008
- Theodore W. Gamelin, Uniform algebras, Prentice-Hall, Inc., Englewood Cliffs, N.J., 1969. MR 0410387
- Guo Xing Ji, Maximality of semi-finite subdiagonal algebras, J. Shaanxi Normal Univ. Nat. Sci. Ed. 28 (2000), no. 1, 15–17 (Chinese, with English and Chinese summaries). MR 1758662
- L. E. Labuschagne, A noncommutative Szegö theorem for subdiagonal subalgebras of von Neumann algebras, Proc. Amer. Math. Soc. 133 (2005), no. 12, 3643–3646. MR 2163602, DOI 10.1090/S0002-9939-05-08064-0
- Vern I. Paulsen, Gelu Popescu, and Dinesh Singh, On Bohr’s inequality, Proc. London Math. Soc. (3) 85 (2002), no. 2, 493–512. MR 1912059, DOI 10.1112/S0024611502013692
- Vern I. Paulsen and Dinesh Singh, Bohr’s inequality for uniform algebras, Proc. Amer. Math. Soc. 132 (2004), no. 12, 3577–3579. MR 2084079, DOI 10.1090/S0002-9939-04-07553-7
- Vern I. Paulsen and Dinesh Singh, Extensions of Bohr’s inequality, Bull. London Math. Soc. 38 (2006), no. 6, 991–999. MR 2285252, DOI 10.1112/S0024609306019084
- Gilles Pisier and Quanhua Xu, Non-commutative $L^p$-spaces, Handbook of the geometry of Banach spaces, Vol. 2, North-Holland, Amsterdam, 2003, pp. 1459–1517. MR 1999201, DOI 10.1016/S1874-5849(03)80041-4
- Gelu Popescu, Bohr inequalities on noncommutative polydomains, Integral Equations Operator Theory 91 (2019), no. 1, Paper No. 7, 55. MR 3908841, DOI 10.1007/s00020-019-2505-7
- S. Sidon, Über einen Satz von Herrn Bohr, Math. Z. 26 (1927), no. 1, 731–732 (German). MR 1544888, DOI 10.1007/BF01475487
- M. Tomić, Sur un théorème de H. Bohr, Math. Scand. 11 (1962), 103–106 (French). MR 176040, DOI 10.7146/math.scand.a-10653
Additional Information
- Sneh Lata
- Affiliation: Department of Mathematics, School of Natural Sciences, Shiv Nadar University, NH-91, Tehsil Dadri, Gautam Budh Nagar 201314, Uttar Pradesh, India
- MR Author ID: 878501
- Email: sneh.lata@snu.edu.in
- Dinesh Singh
- Affiliation: SGT University, Gurugram 122505, Haryana, India
- MR Author ID: 188765
- Email: dineshsingh1@gmail.com
- Received by editor(s): October 26, 2020
- Received by editor(s) in revised form: March 30, 2021
- Published electronically: October 12, 2021
- Additional Notes: The first author’s research was supported in part by a grant from SERB (DST) under MATRICS Scheme, India
- Communicated by: Javad Mashreghi
- © Copyright 2021 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 150 (2022), 201-211
- MSC (2020): Primary 46L10, 46L51, 46L52, 47B10; Secondary 46J10, 46J15
- DOI: https://doi.org/10.1090/proc/15609
- MathSciNet review: 4335870