## Illinois Journal of Mathematics

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

Frank Wikström

#### Abstract

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

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

Dates
First available in Project Euclid: 13 November 2009

https://projecteuclid.org/euclid.ijm/1258131070

Digital Object Identifier
doi:10.1215/ijm/1258131070

Mathematical Reviews number (MathSciNet)
MR2114269

Zentralblatt MATH identifier
1071.32026

#### Citation

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. https://projecteuclid.org/euclid.ijm/1258131070