Norm of the discrete Cesàaro operator minus identity

Abstract

The norm of C−I on l, where C is the Cesàro operator, is shown to be 1/(p − 1) when 1 < p ≤ 2. This verifies a recent conjecture of G. J. O. Jameson. The norm of C − I on l is also determined when 2 < p < ∞. 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. The Cesàro operator, C, maps a sequence (xn) to (yn), where yn = 1 n n