Extreme points of the unit ball of the operator Hardy space $H^\infty(E\to E_*)$

S. R. Treil'

Zapiski Nauchnykh Seminarov POMI, Investigations on linear operators and function theory. Part XV, Tome 149 (1986), pp. 160-164

Résumé

The following description of extreme points of the unit ball of the operator Hardy space $H^\infty(E\to E_*)$ is obtained. Theorem. Let $\theta\in H^\infty(E\to E_*)$ and $\|\theta\|_\infty\leqslant1$. Then $\theta$ is an extreme point of the unit ball of $H^\infty(E\to E_*)$ if and only if at least one of two following conditions holds: a)$\inf\{\|(I-\theta^*\theta)^{1/2}(e+\sum_{k\geqslant1}z^ke_k)\|:e_k\in E\}=0$, $\forall e\in E$; b)$\inf\{\|(I-\theta\theta^*)^{1/2}(e+\sum_{k\geqslant1}\bar z^ke_k)\|:e_k\in E\}=0$, $\forall e\in E$.