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Bounds for the norm of lower triangular matrices on the Cesàro weighted sequence space
Journal of Inequalities and Applications volume 2017, Article number: 67 (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 nonnegative 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.
Introduction
Let \(p\ge1\) and ω denote the set of all realvalued sequences. The space \(l_{p}\) is the set of all real sequences \(x=(x_{n})\in\omega\) such that
If \(w=(w_{n})\in\omega\) is a nonnegative sequence, we define the weighted sequence space \(l_{p}(w)\) as follows:
with norm \(\Vert \cdot \Vert _{p,w}\), which is defined in the following way:
Let \(C=(c_{n,k})\) denote the Cesàro matrix. We recall that the elements \(c_{n,k}\) of the matrix C are given by
The sequence space defined by
is called the Cesàro weighted sequence space, and the norm \(\Vert \cdot \Vert _{p,w,c}\) of the space is defined by
The Cesàro sequence spaces were studied in [1], where \(w_{n}=1\) for all n. It is significant that in the special case \(w_{n}=1\), we have \(l_{p}(w)=l_{p}\) and \(c_{p}(w)=c_{p}\).
Let \((w_{n})\) be a nonnegative sequence and \(A=(a_{n,k})\) be a lower triangular matrix with nonnegative entries. In this paper, we shall consider the inequality of the form
and the inequality of the form
where \(x=(x_{n})\) is a nonnegative sequence. The constant U does not depend on x, and we seek the smallest possible value of U. We write \(\Vert A \Vert _{p,w,c}\) for the norm of A as an operator from \(l_{p}(w)\) into \(c_{p}(w)\), \(\Vert A \Vert _{p,c}\) for the norm of A as an operator from \(l_{p}\) into \(c_{p}\), \(\Vert A \Vert _{c,p,w}\) for the norm of A as an operator from \(c_{p}(w)\) into \(l_{p}(w)\), \(\Vert A \Vert _{c,p}\) for the norm of A as an operator from \(c_{p}\) into \(l_{p}\), \(\Vert A \Vert _{p,w}\) for the norm of A as an operator from \(l_{p}(w)\) into itself and \(\Vert A \Vert _{p}\) for the norm of A as an operator from \(l_{p}\) into itself.
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].
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 nonnegative 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 nonnegative 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 nonnegative 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 \(p^{*}=p/(p1)\).
Lemma 2.1
[3], Lemma 2.1
Let a and x be two nonnegative sequences, then for all n,
Lemma 2.2
[3], Lemma 2.2
Let \(N\geq1\), and let a and x be two nonnegative sequences. If \(x_{N}\geq x_{N+1}\geq\cdots\geq0\) and \(x_{n}=0\) for \(n< N\), then
for all n.
Lemma 2.3
[7], Lemma 1.4
Let \(p>1\) and \(w=(w_{n})\) be a decreasing sequence with nonnegative entries and \(\sum_{n=1}^{\infty}\frac{w_{n}}{n}\) be divergent. Let \(N\geq1\) and the matrix \(C_{N}=(c_{n,k}^{N})\) be with the following entries:
Then \(\Vert C_{N} \Vert _{p,w}\) is determined by nonnegative decreasing sequences and \(\Vert C_{N} \Vert _{p,w}=p^{*}\).
Note that \(C_{1}\) is the wellknown Cesàro matrix.
Lemma 2.4
[7], Lemma 1.5
If \(p>1\) and x and w are two nonnegative sequences and also w is decreasing, then
We set \(a_{0,0}=0\) and \(a_{n,0}=0\) for \(n\geq1\) and
We are now ready to present the main result of this section.
Theorem 2.5
Suppose that \(p>1\) and \(w=(w_{n})\) is a decreasing sequence with nonnegative entries. If \(A=(a_{n,k})\) is a lower triangular matrix with nonnegative entries, then we have the following statements.

(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 \(\Vert A \Vert _{p,w,c}\geq p^{*}m_{A}\).
Therefore if \(w=(w_{n})\) is a decreasing sequence with nonnegative entries and \((\frac{w_{n}}{w_{n+1}})\) is decreasing and \(\sum_{n=1}^{\infty}\frac{w_{n}}{n}=\infty\), then
In particular, if \(w_{n}=1\) for all n and if \(M_{A}<\infty\), then A is a bounded matrix operator from \(l_{p}\) into \(c_{p}\) and \(p^{*}m_{A}\leq \Vert A \Vert _{p,c}\leq p^{*}M_{A}\).
Proof
(i) Let \((x_{n})\) be a nonnegative sequence. By using Lemma 2.1, we get
By applying Lemma 2.4, we deduce that
(ii) We have \(m_{A}=\sup_{N\geq1}\beta_{N}\), where
Let \(N\geq1\), so that \(\beta_{N}\geq0\). If \(y=(y_{n})\) is a decreasing sequence with nonnegative entries and \(\Vert y \Vert _{p,w}=1\), we set \(x_{1}=x_{2}=\cdots=x_{N1}=0\) and
for all \(n\geq1\). So \(\Vert x \Vert _{p,w}= \Vert y \Vert _{p,w}=1\), and from Lemma 2.2 it follows that
By Lemma 2.3, we conclude that \(\Vert A \Vert _{p,w,c}\geq p^{*}\beta_{N}\), so
□
In what follows we assume that \(w=(w_{n})\) is a decreasing sequence with nonnegative entries and \((\frac{w_{n}}{w_{n+1}})\) is decreasing and \(\sum_{n=1}^{\infty}\frac{w_{n}}{n}=\infty\).
At first we bring a corollary of Theorem 2.5 for a lower triangular matrix \(A=(a_{n,k})\). The rows of \(C_{1}A\) are increasing, where \(C_{1}\) is the Cesàro matrix and
Corollary 2.6
Suppose that \(p>1\) and \(A=(a_{n,k})\) is a nonnegative lower triangular matrix that \(\sum_{i=k1}^{n}a_{i,k1}\le \sum_{i=k}^{n}a_{i,k}\) for \(1< k \leq n\). Then
In particular, \(\Vert I \Vert _{p,w,c}=p^{*}\), where I is the identity matrix.
Proof
Since the finite sequence \((\sum_{i=k}^{n}a_{i,k} )_{k=1}^{n}\) is increasing for each n, we have
for \(1\leq k \le n\). Hence
Moreover,
and
Hence, according to Theorem 2.5, we obtain the desired result. □
Example 2.7
Let \(A=(a_{n,k})\) be defined by
Since the finite sequence \((\sum_{i=k}^{n}a_{i,k} )_{k=1}^{n}\) is increasing for each n and \(\sup_{n\geq1} a_{n,n}=2\), by Corollary 2.6, we have \(\Vert A \Vert _{p,w,c}=2p^{*}\).
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
Suppose that \(p>1\) and \(A=(a_{n,k})\) is a lower triangular matrix with \(\sum_{i=k1}^{n}a_{i,k1}\ge\sum_{i=k}^{n}a_{i,k}\) for \(1< k \leq n\). Then
In particular, for summability matrices the lefthand side of the above inequality reduces to \(p^{*}\).
Moreover, if the righthand side of the above inequality is finite, then A is a bounded matrix operator from \(l_{p}(w)\) into \(c_{p}(w)\).
Proof
Since the finite sequence \((\sum_{i=k}^{n}a_{i,k} )_{k=1}^{n}\) is decreasing for each n, we have
and \((\sum_{i=1}^{n}a_{i,1}\sum_{i=0}^{n}a_{i,0} )^{+} =\sum_{i=1}^{n}a_{i,1}\). Hence \(M_{A}=\sup_{n\geq1}\sum_{i=1}^{n}a_{i,1}\). Moreover,
for \(1< k \leq n\), so
Therefore, by Theorem 2.5, we prove the desired result. □
The two examples of Corollary 2.8 are given as follows.
Example 2.9
Suppose that \(\alpha\ge2\) and the matrix \(A=(a_{n,k})\) is defined by
Since \(\sum_{i=k}^{n}a_{i,k}=\sum_{i=k}^{n} \frac{1}{i^{\alpha}}\) and \(\sum_{i=k1}^{n}a_{i,k1}\ge\sum_{i=k}^{n}a_{i,k}\) for \(1< k \leq n\), we have \(0\le \Vert A \Vert _{p,w,c}\leq p^{*}\zeta(\alpha)\), where \(\zeta(\alpha )=\sum_{n=1}^{\infty}\frac{1}{n^{\alpha}}\).
Example 2.10
Suppose that the matrix \(A=(a_{n,k})\) is defined by
Since \(\sum_{i=k}^{n}a_{i,k}=\sum_{i=k}^{n}\frac{1}{i(i+1)}\), by Corollary 2.8, we have \(0\le \Vert A \Vert _{p,w,c}\leq p^{*}\).
We apply the above corollary to the following two special cases.
Let \((a_{n})\) be a nonnegative sequence with \(a_{1}>0\), and \(A_{n}=a_{1}+\cdots+a_{n}\). The Nörlund matrix \(N_{a}=(a_{n,k})\) is defined as follows:
Also the weighted mean matrix \(M_{a}=(a_{n,k})\) is defined by
Corollary 2.11
Suppose that \(p>1\) and \(N_{a}=(a_{n,k})\) is the Nörlund matrix and \((a_{n})\) is an increasing sequence. Then
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
Suppose that \(p>1\) and \(M_{a}=(a_{n,k})\) is the weighted mean matrix and \((a_{n})\) is a decreasing sequence. Then
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
Suppose that \(p>1\) and \(A=(a_{n,k})\) is a nonnegative lower triangular matrix. If A is a bounded matrix operator from \(l_{p}(w)\) into itself, then A is a bounded matrix operator from \(l_{p}(w)\) into \(c_{p}(w)\) and
Proof
We have
Hence, by Lemma 2.3, we conclude that \(\Vert A \Vert _{p,w,c}= \Vert C_{1}A \Vert _{p,w}\le p^{*} \Vert A \Vert _{p,w}\). □
We apply the above theorem to the following two Nörlund and weighted mean matrices.
Corollary 2.14
[7], Corollary 1.3
Suppose that \(p>1\) and \(N_{a}=(a_{n,k})\) is the Nörlund matrix and \((a_{n})\) is a decreasing sequence with \(a_{n}\downarrow\alpha\) and \(\alpha>0\). Then
Corollary 2.15
Suppose that \(p>1\) and \(N_{a}=(a_{n,k})\) is the Nörlund matrix and \((a_{n})\) is a decreasing sequence with \(a_{n}\downarrow\alpha\) and \(\alpha>0\). Then
Proof
By applying Theorem 2.13 and Corollary 2.14, we have the desired result. □
Corollary 2.16
[7], Corollary 1.4
Suppose that \(p>1\) and \(M_{a}=(a_{n,k})\) is the weighted mean matrix and \((a_{n})\) is an increasing sequence with \(a_{n}\uparrow\alpha\) and \(\alpha<\infty\). Then
Corollary 2.17
Suppose that \(p>1\) and \(M_{a}=(a_{n,k})\) is the weighted mean matrix and \((a_{n})\) is an increasing sequence with \(a_{n}\uparrow\alpha\) and \(\alpha<\infty\). Then
Proof
By using Theorem 2.13 and Corollary 2.16, the proof is clear. □
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 nonnegative entries, and let \(C_{1}\) be the Cesàro matrix. Then we have \(\Vert C_{1} \Vert _{p,w}\leq p^{*}\).
Theorem 3.2
Suppose that \(p>1\) and \(w=(w_{n})\) is a sequence with nonnegative entries and \(A=(a_{n,k})\) is a lower triangular matrix with nonnegative entries. We have
Moreover, if the righthand side of the above inequality is finite, then A is a bounded matrix operator from \(c_{p}(w)\) into \(l_{p}(w)\). In particular, if \(w_{n}=1\) for all n, then we have
Proof
Suppose that \(x\in c_{p}(w)\)
Hence
and
On the other hand, Proposition 3.1 concludes that \(\Vert x \Vert _{p,w,c}= \Vert C_{1}x \Vert _{p,w}\leq p^{*} \Vert x \Vert _{p,w}\), so
Therefore \(\frac{1}{p^{*}} \Vert A \Vert _{p,w}\leq \Vert A \Vert _{c,p,w}\), and the proof is complete. □
Corollary 3.3
If \(p>1\), then the generalized Cesàro matrix \(C_{N}\) is bounded from \(c_{p}(w)\) into \(l_{p}(w)\) and
Proof
Since
by using Lemma 2.3 and Theorem 3.2, the proof is obvious. □
We apply the above theorem to the following two special cases.
Corollary 3.4
Suppose that \(p>1\) and \(N_{a}=(a_{n,k})\) is the Nörlund matrix and \((a_{n})\) is a decreasing sequence with \(a_{n}\downarrow\alpha\) and \(\alpha>0\). Then
Proof
By Theorem 3.2 and Corollary 2.14, the proof is clear. □
Example 3.5
Let \(\alpha>0\) and \(a_{n}=\alpha+\frac{1}{n^{\alpha+1}}\) for all n. The sequence \((a_{n})\) is decreasing and \(a_{n}\downarrow\alpha\) and also \(a_{1}\sup_{n\ge1}\frac{n}{A_{n}}=1+\frac{1}{\alpha}\). So
Specially \(\Vert N_{a} \Vert _{c,p,w}\rightarrow1\), when \(\alpha\rightarrow\infty\).
Corollary 3.6
Suppose that \(p>1\) and \(M_{a}=(a_{n,k})\) is the weighted mean matrix and \((a_{n})\) is an increasing sequence with \(a_{n}\uparrow\alpha\) and \(\alpha<\infty\). Then
Proof
By using Theorem 3.2 and Corollary 2.16, the proof is obvious. □
Example 3.7
Let \(a_{n}=1\frac{1}{(n+1)^{2}}\) for all n. The sequence \((a_{n})\) is increasing and \(a_{n}\uparrow1\) and also
So
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].
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Acknowledgements
The second author would like to record his pleasure to Dr. Gholareza Talebi [Department of Mathematics, Faculty of Sciences, ValiEAsr University of Rafsanjan] for his valuable conversations during the preparation of the present paper.
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Foroutannia, D., Roopaei, H. Bounds for the norm of lower triangular matrices on the Cesàro weighted sequence space. J Inequal Appl 2017, 67 (2017). https://doi.org/10.1186/s1366001713396
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DOI: https://doi.org/10.1186/s1366001713396
Keywords
 norm
 lower triangular matrix
 Nörlund matrix
 weighted mean matrix
 weighted sequence space