The unique extremal QC mapping and uniqueness of Hahn-Banach extensions

Abstract

Let $\chi$ be an essentialy bounded complex valued measurable function defined on the unit dise $\Delta$, and let $\Lambda_\chi$ be the corrensponding linear functional on the space $\Cal B$ of analytic $L^1$-integrable functions.An outline of proof of main steps of the following is given: If $|\chi|$ is a constant function in $\Delta$, then the uniqueness of Hahn-Banach extension of $\Lambda_\chi$ from $B$ to $L^1$, when $\|\Lambda_\chi\|=\|\chi\|_ınfty$, implies that $\chi$ is the unique complex dilatation. We give a short review of some related results.