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 $\mathrm{\Delta }$, and let ${\mathrm{\Lambda }}_{\chi }$ be the corrensponding linear functional on the space $\mathcal{B}$ 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 $\mathrm{\Delta }$, then the uniqueness of Hahn-Banach extension of ${\mathrm{\Lambda }}_{\chi }$ from $B$ to $L^1$, when $\Vert {\mathrm{\Lambda }}_{\chi }\Vert =\Vert \chi {\Vert }_{ı}nfty$, implies that $\chi$ is the unique complex dilatation. We give a short review of some related results.