# Proceedings of the American Mathematical Society

Published by the American Mathematical Society, the Proceedings of the American Mathematical Society (PROC) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

ISSN 1088-6826 (online) ISSN 0002-9939 (print)

The 2020 MCQ for Proceedings of the American Mathematical Society is 0.85.

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## Bounds for the operator norms of some Nörlund matricesHTML articles powered by AMS MathViewer

by P. D. Johnson Jr., R. N. Mohapatra and David Ross
Proc. Amer. Math. Soc. 124 (1996), 543-547 Request permission

## Abstract:

Suppose $(p_n)_{n \geq 0}$ is a non-increasing sequence of non-negative numbers with $p_0 = 1$, $P_n = \sum _{j=0}^n p_j$, $n = 0, 1 \dots$, and $A = A(p_n) = (a_{nk})$ is the lower triangular matrix defined by $a_{nk} = p_{n-k} / P_n$, $0 \leq k \leq n$, and $a_{nk} = 0$, $n < k$. We show that the operator norm of $A$ as a linear operator on $\ell _p$ is no greater than $p / (p-1)$, for $1 < p < \infty$; this generalizes, yet again, Hardy’s inequality for sequences, and simplifies and improves, in this special case, more generally applicable results of D. Borwein, Cass, and Kratz. When the $p_n$ tend to a positive limit, the operator norm of $A$ on $\ell _p$ is exactly $p/(p-1)$. We also give some cases when the operator norm of $A$ on $\ell _p$ is less than $p/(p-1)$.
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• P. D. Johnson Jr.
• Affiliation: Department of Discrete and Statistical Sciences 120 Math Annex Auburn University, Alabama 36849-5307
• Email: johnspd@mail.auburn.edu
• R. N. Mohapatra
• Affiliation: Department of Mathematics University of Central Florida Orlando, Florida 32816-6690
• David Ross
• Affiliation: Department of Mathematics Embry Riddle Aeronautical University Daytona Beach, Florida 32114
• Received by editor(s): February 4, 1994
• Received by editor(s) in revised form: September 7, 1994
• Communicated by: J. Marshall Ash
• © Copyright 1996 American Mathematical Society
• Journal: Proc. Amer. Math. Soc. 124 (1996), 543-547
• MSC (1991): Primary 40G05
• DOI: https://doi.org/10.1090/S0002-9939-96-03081-X
• MathSciNet review: 1301506