Multivariable Bohr inequalities

Popescu, Gelu

doi:10.1090/S0002-9947-07-04170-0

Multivariable Bohr inequalities
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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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