Abstract. A non-negative pluriharmonic polynomial ${\msbm R}e p(z)$ on the unit ball of ${\msbm C}^n$ is used as a weight against the rotationally invariant measure on the unit sphere. The resulting Hardy space carries the canonical n-tuple $S$ of multiplication by the coordinate functions. By means of compressions of $S$ to co-analytically invariant subspaces, and known estimates of the numerical radius of a nilpotent matrix we obtain bounds for the coefficients of $p$, in terms of the arithmetic mean and degree of $p$, and dimension $n$.

AMS Subject Classification (1991): 42B05, 47A12, 31C10

Received June 4, 2005, and in revised form September 21, 2005. (Registered under 5913/2009.)