Weighted almost convergence and related infinite matrices
- Syed Abdul Mohiuddine^{1}Email authorView ORCID ID profile and
- Abdullah Alotaibi^{1}
https://doi.org/10.1186/s13660-017-1600-z
© The Author(s) 2018
Received: 8 November 2017
Accepted: 27 December 2017
Published: 10 January 2018
Abstract
The purpose of this paper is to introduce the notion of weighted almost convergence of a sequence and prove that this sequence endowed with the sup-norm \({\Vert \cdot \Vert } _{\infty}\) is a BK-space. We also define the notions of weighted almost conservative and regular matrices and obtain necessary and sufficient conditions for these matrix classes. Moreover, we define a weighted almost A-summable sequence and prove the related interesting result.
Keywords
MSC
1 Introduction and preliminaries
A sequence space X is called a BK-space if it is a Banach space with continuous coordinates \(p_{j}:X\to\mathbb {C}\), the set of complex fields, and \(p_{j}(s)=s_{j}\) for all \(s=(s_{j})\in X\) and every \(j\in \mathbb {N}\). A BK-space \(X\supset\psi\), the set of all finite sequences that terminate in zeros, is said to have AK if every sequence \(s=(s_{j})\in X\) has a unique representation \(s=\sum_{j=0}^{\infty }s_{j}e_{j}\).
We now recall the following result.
Lemma 1.1
([22])
Let X and Y be BK-spaces. (i) Then \((X,Y)\subset B(X,Y)\), that is, every \(A\in(X,Y)\) defines an operator \({\mathcal{L}}_{A}\in B(X,Y)\) by \({\mathcal{L}}_{A}(x)=Ax\) for all \(x\in X\), where \(B(X,Y)\) denotes the set of all bounded linear operators from X into Y. (ii) Then \(A\in(X,\ell_{\infty})\) if and only if \(\Vert A \Vert _{(X,\ell_{\infty})}=\sup_{n} \Vert A_{n} \Vert _{X}<\infty\). Moreover, if \(A\in(X,\ell _{\infty})\), then \(\Vert {\mathcal{L}}_{A} \Vert = \Vert A \Vert _{(X,\ell_{\infty})}\).
2 Weighted almost convergence
Definition 2.1
Example 2.2
Consider a sequence \(s=(s_{k})\) defined by \(s_{k}=1\) if k is odd and 0 for even k. Also, let \(t_{k}=1\) for all k. Then we see that \(s=(s_{k})\) is \(f(\bar{N})\)-convergent to \(1/2\) but not convergent.
Definition 2.3
The matrix A (or a matrix map A) is said to be weighted almost conservative if \(As\in f({\bar {N}})\) for all \(s=(s_{k})\in c\). One denotes this by \(A\in(c,f({\bar {N}}))\). If \(A\in(c,f({\bar{N}}))\) with \(f({\bar{N}})\mbox{-}\lim As=\lim s\), then we say that A is weighted almost regular matrix; one denotes such matrices by \(A\in(c,f({\bar{N}}))_{R}\).
Theorem 2.4
The space \(f(\bar{N})\) of weighted almost convergence endowed with the norm \({\Vert \cdot \Vert } _{\infty}\) is a BK-space.
Proof
Since \(c\subset f(\bar{N})\subset l_{\infty}\), there exist positive real numbers α and β with \(\alpha<\beta\) such that \(\alpha \Vert s \Vert _{\infty}\leq \Vert s \Vert _{f(\bar{N})}\leq\beta \Vert s \Vert _{\infty}\). That is to say, two norms \({\Vert \cdot \Vert } _{\infty}\) and \({\Vert \cdot \Vert } _{f(\bar{N})}\) are equivalent. It is well known that the spaces c and \(l_{\infty}\) endowed with the norm \({\Vert \cdot \Vert } _{\infty}\) are BK-spaces, and hence the space \(f(\bar{N})\) endowed with the norm \({\Vert \cdot \Vert } _{\infty}\) is also a BK-space. □
We prove the following characterization of weighted almost conservative matrices.
Theorem 2.5
Proof
In the following theorem, we obtain the characterization of weighted almost regular matrices.
Theorem 2.6
Proof
Necessity. Let \(A\in(c,f({\bar{N}}))_{R}\). We see that condition (13) holds by using the fact that A is also weighted almost conservative. Take \(e_{k},e\in c\). Then A-transforms of the sequences \(e_{k}\) and e are weighted almost convergent to 0 and 1, respectively, since \(e_{k}\to0\) and \(e\to1\). Hence \(e_{k}\in c\) gives condition (14) and \(e\in c\) proves the validity of (15).
Sufficiency. Let conditions (13)-(15) hold. It is easy to see that A is weighted almost conservative. So, for each \((s_{k})\in c\), \(\lim_{m\to\infty}\Phi_{mr}(s)=\Phi(s)\) uniformly in r. Thus we obtain from (11) and our hypotheses (13)-(15) that \(\Phi(s)=\xi=\lim s_{k}\). This yields A is weighted almost regular. □
We now obtain necessary and sufficient conditions for the matrix A which transform the absolutely convergent series into the space of weighted almost convergence.
Theorem 2.7
Proof
Theorem 2.8
If the matrix A in \((l_{1},f({\bar {N}}))\), then \(\Vert {\mathcal{L}}_{A} \Vert = \Vert A \Vert \).
Proof
Definition 2.9
In the applications of summability theory to function theory, it is important to know the region in which \(S=(S_{k}(z))\), the sequence of partial sums of the geometric series is A-summable to \(\frac {1}{1-z}\) for a given matrix A. In the following theorem, we find the region in which S is weighted almost A-summable to \(\frac{1}{1-z}\).
Theorem 2.10
Proof
Declarations
Acknowledgements
This work was supported by the Deanship of Scientific Research (DSR), King Abdulaziz University, Jeddah, under grant No. (130-694-D1435). The authors, therefore, gratefully acknowledge the DSR technical and financial support.
Authors’ contributions
The authors contributed equally and significantly in writing this paper. All authors read and approved the final manuscript.
Competing interests
The authors declare that they have no competing interests.
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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