The norm of the backward shift on $H^4$ is $\sqrt[4]{\varphi}$

Abstract

We prove that the backward shift operator on $H^4$ has norm equal to $\sqrt[4]{\varphi}$, with $\varphi = \frac{1 + \sqrt{5}}{2}$. Furthermore, we characterize all extremal functions; they are precisely the functions of the form \[ f(z) = \mu \left( I(z) - \sqrt{\frac{1}{2\varphi}}\right), \] where $\mu \in \mathbb{C}$ and $I$ is an inner function with $I(0) = \sqrt{\frac{\varphi}{2}}$.