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

TL;DR

This paper determines the exact norm of the backward shift operator on the Hardy space H^4 as the fourth root of the golden ratio and characterizes all extremal functions achieving this norm.

Contribution

It provides a precise calculation of the backward shift norm on H^4 and characterizes the extremal functions explicitly.

Findings

The norm of the backward shift on H^4 is 0 ext{th root of }rac{1+\u2215 5}{2} (the golden ratio).

All extremal functions are of a specific form involving inner functions and a constant.

The extremal functions are characterized explicitly in terms of inner functions with a particular value at zero.

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 $I(0) = \sqrt{\frac{\varphi}{2}}$.