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

Abstract

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