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Bohr phenomenon for operator-valued functions

Part of: General theory of linear operators Geometric function theory Series expansions

Published online by Cambridge University Press:  08 January 2021

Bappaditya Bhowmik  and
Nilanjan Das
Bappaditya Bhowmik
Affiliation:
Department of Mathematics, Indian Institute of Technology Kharagpur, Kharagpur721302, India ([email protected]; [email protected])
Nilanjan Das
Affiliation:
Department of Mathematics, Indian Institute of Technology Kharagpur, Kharagpur721302, India ([email protected]; [email protected])
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Abstract

We establish Bohr inequalities for operator-valued functions, which can be viewed as analogues of a couple of interesting results from scalar-valued settings. Some results of this paper are motivated by the classical flavour of Bohr inequality, while others are based on a generalized concept of the Bohr radius problem.

Keywords

Bohr radiusharmonic functionbiholomorphic function

MSC classification

Primary: 47A56: Functions whose values are linear operators (operator and matrix valued functions, etc., including analytic and meromorphic ones) 30B10: Power series (including lacunary series) 47A63: Operator inequalities 30C80: Maximum principle; Schwarz's lemma, Lindelöf principle, analogues and generalizations; subordination
Type
Research Article
Information
Proceedings of the Edinburgh Mathematical Society , Volume 64 , Issue 1 , February 2021 , pp. 72 - 86
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Copyright
Copyright © The Author(s) 2021. Published by Cambridge University Press on Behalf of The Edinburgh Mathematical Society

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References

Abu Muhanna, Y., Bohr's phenomenon in subordination and bounded harmonic classes, Complex Var. Elliptic Equ. 55(11) (2010), 1071–1078.CrossRefGoogle Scholar
Abu Muhanna, Y. and Ali, R. M., Bohr's phenomenon for analytic functions into the exterior of a compact convex body, J. Math. Anal. Appl. 379(2) (2011), 512–517.CrossRefGoogle Scholar
Aizenberg, L., Multidimensional analogues of Bohr's theorem on power series, Proc. Amer. Math. Soc. 128(4) (2000), 1147–1155.Google Scholar
Aizenberg, L., Aytuna, A. and Djakov, P., An abstract approach to Bohr's phenomenon, Proc. Amer. Math. Soc. 128(9) (2000), 2611–2619.CrossRefGoogle Scholar
Balasubramanian, R., Calado, B. and Queffélec, H., The Bohr inequality for ordinary Dirichlet series, Studia Math. 175(3) (2006), 285–304.Google Scholar
Bhowmik, B. and Das, N., Bohr phenomenon for subordinating families of certain univalent functions, J. Math. Anal. Appl. 462(2) (2018), 1087–1098.CrossRefGoogle Scholar
Blasco, O., The Bohr radius of a Banach space, in Vector measures, integration and related topics (eds G. P. Curbera, G. Mockenhaupt and W. J. Ricker), pp. 59–64, Operator Theory: Advances and Applications, Volume 201 (Birkhäuser Verlag, Basel, 2010).CrossRefGoogle Scholar
Boas, H. P. and Khavinson, D., Bohr's power series theorem in several variables, Proc. Amer. Math. Soc. 125(10) (1997), 2975–2979.CrossRefGoogle Scholar
Bohr, H., A theorem concerning power series, Proc. Lond. Math. Soc. 13(2) (1914), 1–5.Google Scholar
Bombieri, E. and Bourgain, J., A remark on Bohr's inequality, Int. Math. Res. Not. 2004(80) (2004), 4307–4330.CrossRefGoogle Scholar
Conway, J. B., A Course in Functional Analysis, 2nd ed., Graduate Texts in Mathematics, Volume 96 (Springer-Verlag, New York, 1990).Google Scholar
Defant, A., García, D. and Maestre, M., Bohr's power series theorem and local Banach space theory, J. Reine Angew. Math. 557 (2003), 173–197.Google Scholar
Defant, A., García, D., Maestre, M. and Pérez-García, D., Bohr's strip for vector valued Dirichlet series, Math. Ann. 342(3) (2008), 533–555.CrossRefGoogle Scholar
Dixon, P. G., Banach algebras satisfying the non-unital von Neumann inequality, Bull. London Math. Soc. 27(4) (1995), 359–362.CrossRefGoogle Scholar
Goldberg, M. and Tadmor, E., On the numerical radius and its applications, Linear Algebra Appl. 42 (1982), 263–284.Google Scholar
Graham, I. and Kohr, G., Geometric function theory in one and higher dimensions, Monographs and Textbooks in Pure and Applied Mathematics, Volume 255 (Marcel Dekker, Inc., New York, 2003).CrossRefGoogle Scholar
Hensgen, W., On the dual space of $H^{p}(X)$, $1<p<\infty$, J. Funct. Anal. 92(2) (1990), 348–371.CrossRefGoogle Scholar
Kayumov, I. R. and Ponnusamy, S., Bohr's inequalities for the analytic functions with lacunary series and harmonic functions, J. Math. Anal. Appl. 465(2) (2018), 857–871.CrossRefGoogle Scholar
Liu, T. and Wang, J., An absolute estimate of the homogeneous expansions of holomorphic mappings, Pacific J. Math. 231(1) (2007), 155–166.CrossRefGoogle Scholar
Newburgh, J. D., The variation of spectra, Duke Math. J. 18 (1951), 165–176.CrossRefGoogle Scholar
Paulsen, V. I., Completely bounded maps and operator algebras, Cambridge Studies in Advanced Mathematics, Volume 78 (Cambridge University Press, Cambridge, 2002).Google Scholar
Paulsen, V. I. and Singh, D., Extensions of Bohr's inequality, Bull. London Math. Soc. 38(6) (2006), 991–999.CrossRefGoogle Scholar
Paulsen, V. I., Popescu, G. and Singh, D., On Bohr's inequality, Proc. London Math. Soc. 85(2) (2002), 493–512.CrossRefGoogle Scholar
Popescu, G., Multivariable Bohr inequalities, Trans. Amer. Math. Soc. 359(11) (2007), 5283–5317.CrossRefGoogle Scholar
Popescu, G., Bohr inequalities for free holomorphic functions on polyballs, Adv. Math. 347 (2019), 1002–1053.Google Scholar
Rogosinski, W., On the coefficients of subordinate functions, Proc. Lond. Math. Soc. 48 (1943), 48–82.Google Scholar