Norm of the discrete Ces\`aro operator minus identity

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.