Isometries of the disc algebra

Abstract:

The linear isometries $u:A \to A$ of the disc algebra $A$ into itself are completely described. Such isometries $u$ must be one of two distinct types. The first type is $uf = \psi \cdot f(\phi )$, where $\psi \in A$ and $\phi \in {H^\infty }$ satisfy certain described conditions. The second type is $uf = E(\psi \cdot f(\phi ))$, where $\phi :Q \to T$ is any continuous function from a closed zero measure subset $Q$ of the unit circle $T$ onto itself, $\psi \in C(Q)$ is unimodular, and $E:Y \to A$ is a norm 1 extension operator, where $Y = \left \{ {\psi \cdot f(\phi ):f \in A} \right \} \subset C(Q)$. Isometries of $C(K)$ spaces into the disc algebra are also described.