Bounds for the operator norms of some Nörlund matrices

Authors:
P. D. Johnson Jr., R. N. Mohapatra and David Ross

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

Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

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 $p/(p-1)$.

- David Borwein,
*Nörlund operators on*$\ell _p$, Canad. Math. Bull.**36**(1993), 8–14. - D. Borwein and F. P. Cass,
*Nörlund matrices as bounded operators on*$\ell _p$, Arch. Math.**42**(1984), 464–469. - D. Borwein and A. Jakimovski,
*Matrix operators on*$\ell _P$, Rocky Mountain J. Math.**9**(1979), 463–477. - F. P. Cass and W. Kratz,
*Nörlund and weighted mean matrices as bounded operators on*$\ell _p$, Rocky Mountain J. Math.**29**(1990), 59–74. - G. S. Davies and G. M. Petersen,
*On an inequality of Hardy’s (II)*, Quart. J. Math. Oxford Ser. (2)**15**(1964), 35–40. - Tomlinson Fort,
*Infinite series*, Oxford University Press, London, 1930. - G. H. Hardy,
*An inequality for Hausdorff means*, J. London Math. Soc.**18**(1943), 46–50. - G. H. Hardy, J. E. Littlewood, and G. Polya,
*Inequalities*, Cambridge University Press, London, 1934. - J. Németh,
*Generalizations of the Hardy-Littlewood inequality*, Acta Sci. Math. (Szeged)**32**(1971), 295–299.

Retrieve articles in *Proceedings of the American Mathematical Society*
with MSC (1991):
40G05

Retrieve articles in all journals with MSC (1991): 40G05

Additional Information

**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

Article copyright:
© Copyright 1996
American Mathematical Society