Illinois Journal of Mathematics

An extremal function for the multiplier algebra of the universal Pick space

Frank Wikström

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Let $H^2_m$ be the Hilbert function space on the unit ball in $\C{m}$ defined by the kernel $k(z,w) = (1-\langle z,w \rangle)^{-1}$. For any weak zero set of the multiplier algebra of $H^2_m$, we study a natural extremal function, $E$. We investigate the properties of $E$ and show, for example, that $E$ tends to $0$ at almost every boundary point. We also give several explicit examples of the extremal function and compare the behaviour of $E$ to the behaviour of $\delta^*$ and $g$, the corresponding extremal function for $H^\infty$ and the pluricomplex Green function, respectively.

Article information

Illinois J. Math., Volume 48, Number 3 (2004), 1053-1065.

First available in Project Euclid: 13 November 2009

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Zentralblatt MATH identifier

Primary: 32U35: Pluricomplex Green functions
Secondary: 32F45: Invariant metrics and pseudodistances 46E22: Hilbert spaces with reproducing kernels (= [proper] functional Hilbert spaces, including de Branges-Rovnyak and other structured spaces) [See also 47B32] 46J15: Banach algebras of differentiable or analytic functions, Hp-spaces [See also 30H10, 32A35, 32A37, 32A38, 42B30] 47B32: Operators in reproducing-kernel Hilbert spaces (including de Branges, de Branges-Rovnyak, and other structured spaces) [See also 46E22]


Wikström, Frank. An extremal function for the multiplier algebra of the universal Pick space. Illinois J. Math. 48 (2004), no. 3, 1053--1065. doi:10.1215/ijm/1258131070.

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