J. Math. Sci. Univ. Tokyo
Vol. 4 (1997), No. 1, Page 33--52.

Avanissian, V.
Let $\Cal H_{\Bbb R}$ be a real Hilbert space and let $\Cal H_{\Bbb C}$ be the complexification of $\Cal H_{\Bbb R}$. The first part of this paper treats the problem of the existence of the minimal norm $\tilde\ell$ on $\Cal H_{\Bbb C}$ such that % \align & \tilde\ell(z)\le\|z\|_{\Cal H_{\Bbb C}}\m\n \hbox{for}\m z\in\Cal H_{\Bbb C} \ & \tilde\ell(x)=\|x\|_{\Cal H_{\Bbb R}}\m\n \hbox{for}\m x\in\Cal H_{\Bbb R}. \endalign % We prove the following theorem : a)\m The minimal norme $\tilde\ell$ exists in $\Cal H_{\Bbb C}$. b)\m Let $D\subset\Bbb C^N$ be a bounded, convex, balanced domain. There exists a maximal bounded convex, balanced domain $\tilde D\subset\Bbb C^N$ such that % $$\tilde D\supset D,\m\n \tilde D\cap\Bbb R^N=D\cap\Bbb R^N.$$ % c)\m Let $\Cal H_{\Bbb C}=\Bbb C^N$, then the minimal norm $\tilde\ell$ is the supporting function of the unit closed Lie ball in $\Bbb C^N$. (a) and b) extend a result of K. T. Hahn and Peter Plug) where $\Cal H_{\Bbb R}=\Bbb R^N$ and $D$ is the unit euclidean ball in $\Cal C^N$. The second part of the paper gives a geometrical interpretation of the minimal norm $\tilde\ell$ in $\Cal H_{\Bbb C}$. If $\Cal N$ is a norm in $\Bbb C^N$, log $\Cal N(z)$ is plurisubharmonic function. The final part of the paper studies the plurisubharmonic functions $V$ in $\Bbb C^N$ such that $\forall k\in\Bbb C$, $V(kz)=|k|V(z)$, $V(z)\le\|z\|$ for $z\in\Bbb C^N$, $V(x)=\|x\|$ for $x\in\Bbb R^N$, $\|z\|$ is euclidean norm in $\Bbb C^N$.