Curvature and the backward shift operators

Abstract

Let φ α {\varphi _\alpha } be a Möbius transformation of the unit disk D {\mathbf {D}} , | α | > 1 \left | \alpha \right | > 1 . We characterize all the operators T T in B 1 ( D ) {B_1}\left ( {\mathbf {D}} \right ) which are unitarily equivalent to φ α ( T ) {\varphi _\alpha }\left ( T \right ) for all α \alpha with | α | > 1 \left | \alpha \right | > 1 , using curvature techniques.