Norm of the discrete Ces\`aro operator minus identity

TL;DR

This paper determines the exact operator norm of the discrete Cesàro operator minus identity on b1l^p spaces for all p, confirming a conjecture and answering a longstanding question in functional analysis.

Contribution

It provides the complete norm calculation for the operator C - I on b1l^p spaces, verifying a recent conjecture and resolving a question posed in 1996.

Findings

Norm of C - I on b1l^p is 1/(p-1) for 1 < p  2

Norm of C - I on b1l^p is p/(p-1) for p > 2

Discrete and continuous cases have identical operator norms

Abstract

The norm of $C-I$ on $\ell^p$, where $C$ is the Ces\`aro operator, is shown to be $1/(p-1)$ when $1<p\le2$. This verifies a recent conjecture of G. J. O. Jameson. The norm of $C-I$ on $\ell^p$ is also determined when $2< p<\infty$. The two parts together answer a question raised by G. Bennett in 1996. Operator norms in the continuous case, Hardy's averaging operator minus identity, are already known. Norms in the discrete and continuous cases coincide.