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On Bohr's Inequality
Author(s) -
Paulsen Vern I.,
Popescu Gelu,
Singh Dinesh
Publication year - 2002
Publication title -
proceedings of the london mathematical society
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.899
H-Index - 65
eISSN - 1460-244X
pISSN - 0024-6115
DOI - 10.1112/s0024611502013692
Subject(s) - mathematics , bohr model , bounded function , commutative property , von neumann architecture , von neumann algebra , hilbert space , pure mathematics , norm (philosophy) , algebra over a field , discrete mathematics , mathematical analysis , physics , quantum mechanics , political science , law
Bohr's inequality says that if f ( z ) = ∑ n = 0 ∞a n z n is a bounded analytic function on the closed unit disc, then ∑ n = 0 ∞ ∣ a n ∣ r n ⩽ ∥ f ∥ ∞for 0 leq r ⩽ 1/3 and that 1/3 is sharp. In this paper we give an operator‐theoretic proof of Bohr's inequality that is based on von Neumann's inequality. Since our proof is operator‐theoretic, our methods extend to several complex variables and to non‐commutative situations. We obtain Bohr type inequalities for the algebras of bounded analytic functions and the multiplier algebras of reproducing kernel Hilbert spaces on various higher‐dimensional domains, for the non‐commutative disc algebra A n , and for the reduced (respectively full) group C*‐algebra of the free group on n generators. We also include an application to Banach algebras. We prove that every Banach algebra has an equivalent norm in which it satisfies a non‐unital version of von Neumann's inequality. 2000 Mathematical Subject Classification : 47A20, 47A56.