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# Weighted Mean Operators on $l_p$

Published:2000-12-01
Printed: Dec 2000
• David Borwein
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## Abstract

The weighted mean matrix $M_a$ is the triangular matrix $\{a_k/A_n\}$, where $a_n > 0$ and $A_n := a_1 + a_2 + \cdots + a_n$. It is proved that, subject to $n^c a_n$ being eventually monotonic for each constant $c$ and to the existence of $\alpha := \lim \frac{A_n}{na_n}$, $M_a \in B(l_p)$ for $1 < p < \infty$ if and only if $\alpha < p$.
 Keywords: weighted means, operators on $l_p$, norm estimates
 MSC Classifications: 47B37 - Operators on special spaces (weighted shifts, operators on sequence spaces, etc.) 47A30 - Norms (inequalities, more than one norm, etc.) 40G05 - Cesaro, Euler, Norlund and Hausdorff methods