Sharp upper bounds for the norms of certain bounded operators on l^p

Abstract

We obtain sharp upper bounds for the norms of several bounded operators on l^p

The purpose of this note is to publicize a technique developed by Bennett [1] for finding a sharp upper bound for the norm of a bounded linear operator on l^p for 1≤p≤∞,with nonnegative entries. In [1] Bennett used this technique on the Hilbert matrix.

The Rhaly generalized Cesaro matrices [4] are lower triangular matrices with nonzero entries . These matrices are special cases of the lower traingular matrices with entries , where H is an Endl generalized Haus-dorff matrix defined by , a real or complex sequence, Δ the forward difference operator defined by . If H has nonnegative entries then nonnegative and nondecreasing.