Bounds for the norm of lower triangular matrices on the Cesàro weighted sequence space
- Davoud Foroutannia^{1}Email author and
- Hadi Roopaei^{1}
https://doi.org/10.1186/s13660-017-1339-6
© The Author(s) 2017
Received: 22 December 2016
Accepted: 22 March 2017
Published: 4 April 2017
Abstract
This paper is concerned with the problem of finding bounds for the norm of lower triangular matrix operators from \(l_{p}(w)\) into \(c_{p} (w)\), where \(c_{p}(w)\) is the Cesàro weighted sequence space and \((w_{n})\) is a non-negative sequence. Also this problem is considered for lower triangular matrix operators from \(c_{p}(w)\) into \(l_{p} (w)\), and the norms of certain matrix operators such as Cesàro, Nörlund and weighted mean are computed.
Keywords
1 Introduction
The problem of finding the norm of a lower triangular matrix on the sequence spaces \(l_{p}\) and \(l_{p}(w)\) has been studied before in [2–8]. In the study, we will expand this problem for matrix operators from \(l_{p}(w)\) into \(c_{p} (w)\) and matrix operators from \(c_{p}(w)\) into \(l_{p} (w)\), and we consider certain matrix operators such as Cesàro, Nörlund and weighted mean. The study is an extension of some results obtained by [3, 7].
2 The norm of matrix operators from \(l_{p}(w)\) into \(c_{p}(w)\)
In this section, we tend to compute the bounds for the norm of lower triangular matrix operators from \(l_{p}(w)\) into \(c_{p} (w)\). In particular, we apply our results for lower triangular matrix operators from \(l_{p}\) into \(c_{p}\), when \(w_{n}=1\) for all n.
Throughout this paper, let \(A=(a_{n,k})\) be a matrix with non-negative real entries i.e., \(a_{n,k}\ge0\), for all n, k. This implies that \(\Vert Ax \Vert _{p,w,c}\le \Vert A \vert x \vert \Vert _{p,w,c}\), and hence the non-negative sequences are sufficient to determine the norm of A. We say that \(A=(a_{n,k})\) is lower triangular if \(a_{n,k}=0\) for \(n< k\). A non-negative lower triangular matrix is called a summability matrix if \(\sum_{k=1}^{n}a_{n,k}=1\) for all n.
We first state some lemmas from [3, 7], which are needed for our main result. Set \(\xi^{+}=\max(\xi,0)\) and \(\xi^{-}=\min(\xi,0)\) and \(p^{*}=p/(p-1)\).
Lemma 2.1
[3], Lemma 2.1
Lemma 2.2
[3], Lemma 2.2
Lemma 2.3
[7], Lemma 1.4
Note that \(C_{1}\) is the well-known Cesàro matrix.
Lemma 2.4
[7], Lemma 1.5
We are now ready to present the main result of this section.
Theorem 2.5
- (i)
\(\Vert A \Vert _{p,w,c}\leq p^{*}M_{A}\). Moreover, if \(M_{A}<\infty\), then A is a bounded matrix operator from \(l_{p}(w)\) into \(c_{p}(w)\).
- (ii)
If \(\sum_{n=1}^{\infty}\frac{w_{n}}{n}\) is divergent and \((\frac{w_{n}}{w_{n+1}})\) is decreasing, then \(\Vert A \Vert _{p,w,c}\geq p^{*}m_{A}\).
Proof
In what follows we assume that \(w=(w_{n})\) is a decreasing sequence with non-negative entries and \((\frac{w_{n}}{w_{n+1}})\) is decreasing and \(\sum_{n=1}^{\infty}\frac{w_{n}}{n}=\infty\).
Corollary 2.6
Proof
Example 2.7
Now, in the second case, we state some corollaries of Theorem 2.5 for a lower triangular matrix A, where the rows of \(C_{1}A\) are decreasing.
Corollary 2.8
Moreover, if the right-hand side of the above inequality is finite, then A is a bounded matrix operator from \(l_{p}(w)\) into \(c_{p}(w)\).
Proof
The two examples of Corollary 2.8 are given as follows.
Example 2.9
Example 2.10
We apply the above corollary to the following two special cases.
Corollary 2.11
Proof
Since \(N_{a}\) is a summability matrix and \(\sum_{i=1}^{n}a_{i,1}=\sum_{i=1}^{n}\frac{a_{i}}{A_{i}}\), by applying Corollary 2.8, we have the desired result. □
Corollary 2.12
Proof
Since \(M_{a}\) is a summability matrix and \(\sum_{i=1}^{n}a_{i,1}=\sum_{i=1}^{n}\frac{a_{1}}{A_{i}}\), by Corollary 2.8, the proof is obvious. □
Finally, in the third case, if the rows of \(C_{1}A\) are neither increasing nor decreasing, we present the following theorem.
Theorem 2.13
Proof
We apply the above theorem to the following two Nörlund and weighted mean matrices.
Corollary 2.14
[7], Corollary 1.3
Corollary 2.15
Corollary 2.16
[7], Corollary 1.4
Corollary 2.17
3 The norm of matrix operators from \(c_{p}(w)\) into \(l_{p}(w)\)
In this section, we compute the bounds for the norm of lower triangular matrix operators from \(c_{p}(w)\) into \(l_{p}(w)\). In particular, when \(w_{n}=1\) for all n, the bounds for the norm of lower triangular matrix operators from \(c_{p}\) into \(l_{p}\) are deduced. Moreover, we apply our results for Cesàro, Nörlund and weighted mean matrices.
We begin with a proposition which is needed to prove the main theorem of this section.
Proposition 3.1
([6], Proposition 5.1). Let \(p>1\) and \(w=(w_{n})\) be a decreasing sequence with non-negative entries, and let \(C_{1}\) be the Cesàro matrix. Then we have \(\Vert C_{1} \Vert _{p,w}\leq p^{*}\).
Theorem 3.2
Proof
Corollary 3.3
Proof
We apply the above theorem to the following two special cases.
Corollary 3.4
Example 3.5
Corollary 3.6
Example 3.7
4 Conclusions
In the present study, we considered the problem of finding bounds for the norm of lower triangular matrix operators from \(l_{p}(w)\) into \(c_{p} (w)\) and from \(c_{p}(w)\) into \(l_{p} (w)\). Moreover, we computed the norms of certain matrix operators such as Cesàro, Nörlund and weighted mean, and we extended some results of [3, 7].
Declarations
Acknowledgements
The second author would like to record his pleasure to Dr. Gholareza Talebi [Department of Mathematics, Faculty of Sciences, Vali-E-Asr University of Rafsanjan] for his valuable conversations during the preparation of the present paper.
Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
Authors’ Affiliations
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