A Lower Bound on the Cesaro Operator

If the sequence a = (a„>^ E /2, i.e., ||a||2 = 2~=0 | a„ |2 < oo, define Sa as the sequence of averages 1 " 2 ak n + 1 , It follows easily from the Marcinkiewicz Interpolation Theorem that S is a bounded operator from I2 to I2; this can also be proved directly using the Cauchy-Buniakowski-Schwarz inequality [1]. S is known as the Cesàro operator. S is, of course, not bounded below, but the following property does hold, confirming a conjecture of Allen Shields and Sheldon Axler. Theorem. If a0> ax > ■ ■ ■ ̂ 0, a=(an)%, then \\Sa\\2 > *r2\\a\\2/6, with equality if and only if ax = a2 = ■ ■ ■ = 0. Proof. If we expand the squares and group terms, we find that