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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

Multivariable Bohr inequalities


Author: Gelu Popescu
Journal: Trans. Amer. Math. Soc. 359 (2007), 5283-5317
MSC (2000): Primary 47A20, 47A56; Secondary 47A13, 47A63
Published electronically: May 8, 2007
MathSciNet review: 2327031
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Abstract: Operator-valued multivariable Bohr type inequalities are obtained for:

(i)
a class of noncommutative holomorphic functions on the open unit ball of $ B(\mathcal{H})^n$, generalizing the analytic functions on the open unit disc;
(ii)
the noncommutative disc algebra $ \mathcal{A}_n$ and the noncommutative analytic Toeplitz algebra $ F_n^\infty$;
(iii)
a class of noncommutative selfadjoint harmonic functions on the open unit ball of $ B(\mathcal{H})^n$, generalizing the real-valued harmonic functions on the open unit disc;
(iv)
the Cuntz-Toeplitz algebra $ C^*(S_1,\ldots, S_n)$, the reduced (resp. full) group $ C^*$-algebra $ C_{red}^*(\mathbb{F}_n)$ (resp.  $ C^*(\mathbb{F}_n)$) of the free group with $ n$ generators;
(v)
a class of analytic functions on the open unit ball of $ \mathbb{C}^n$.

The classical Bohr inequality is shown to be a consequence of Fejér's inequality for the coefficients of positive trigonometric polynomials and Haager- up-de la Harpe inequality for nilpotent operators. Moreover, we provide an inequality which, for analytic polynomials on the open unit disc, is sharper than Bohr's inequality.


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Additional Information

Gelu Popescu
Affiliation: Department of Mathematics, The University of Texas at San Antonio, San Antonio, Texas 78249
Email: gelu.popescu@utsa.edu

DOI: http://dx.doi.org/10.1090/S0002-9947-07-04170-0
PII: S 0002-9947(07)04170-0
Keywords: Multivariable operator theory, Bohr's inequality, holomorphic function, harmonic function, von Neumann inequality, Poisson transform, noncommutative disc algebra, noncommutative analytic Toeplitz algebra, Fej\' er's inequality
Received by editor(s): December 27, 2004
Received by editor(s) in revised form: August 17, 2005
Published electronically: May 8, 2007
Additional Notes: This research was supported in part by an NSF grant.
Article copyright: © Copyright 2007 American Mathematical Society