Abstract
In this paper, we obtain the Köthe–Toeplitz duals of the domain of an arbitrary invertible summability matrix E in the space \(\ell_{p}\). As a consequence, we apply our results to the Fibonacci and Euler sequence spaces and show that some recent works by Altay, Başar, and Mursaleen (Inf. Sci. 176:1450–1462, 2006) are all the special cases of our results.
Similar content being viewed by others
1 Introduction and preliminaries
Let ω, \(\ell_{\infty}\), \(\ell_{p}\), and c be the sets of all sequences, bounded sequences, p-absolutely summable sequences, and convergent sequences, respectively. The multiplier space of the sequence spaces X and Y is defined by
and the α-, β-, and γ-duals of the space X, which are denoted by \(X^{\alpha}\), \(X^{\beta}\), and \(X^{\gamma}\), are
Here
and
For an infinite matrix A, the domain of A in the space X, which is a sequence space, is defined by
Recently in [9], the author defined and studied the domain of an arbitrary invertible summability matrix \(E=(E_{n,k})_{n,k\geq0}\) in the space \(\ell_{p}\), i.e., \(E_{p}:=(\ell_{p})_{E}\). One can easily show that the sequence space \(E_{p}\) is a normed space with \({ \Vert x \Vert _{{E_{p}}}}: = { \Vert {{E}x} \Vert _{{\ell_{p}}}}\), and the inclusion \(E_{q}\subseteq E_{p}\) holds while \(1\le q\le p\). Moreover, applying Hölder’s inequality, we have
which implies the inclusion \(\ell_{p}\subseteq E_{p}\) for \(1\le p<\infty\) provided that
Eventually, one can easily check that if the map \(E: E_{p}\to\ell_{p}\) is onto, then the space \(E_{p}\) is linearly isomorphic to \(\ell_{p}\) and in such a case the columns of the matrix \(E^{-1}\) form a Schauder basis for \(E_{p}\), where \(1\le p<\infty\).
It is known that, for the infinite summability matrix E, there may be left or right inverses, or even if both exist, they may not be unique. In this paper we deal with the case in which the left and right inverses are equal, and we denote it by \(E^{-1}\). Further, to give full knowledge on the definitions and calculations with infinite matrices, we refer the readers to the textbook [3].
In this paper, we are going to find out the α-, β-, and γ-duals of the space \(E_{p}\) for \(p\in[1,\infty]\). We assume throughout that \(\mathfrak{F}\) is the collection of all finite subsets of \(\mathbb{N}\) and \(\frac{1}{p} + \frac{1}{q} = 1\). Further, we denote by \((X:Y)\) the class of all infinite matrices which transform X into Y.
2 Main results
In this section, we assume that the transformation \(E: E_{p}\to\ell_{p}\) is surjective and state theorems determining the α-, β-, and γ-duals of \(E_{p}\), where \(p\in[1,\infty]\). We consider only the case \(1 < p < \infty\) in the proof of Theorems 2.1–2.3 below, because the cases \(p = 1\) and \(p =\infty \) can be proved similarly.
Theorem 2.1
Define the sets \(G_{q}\) and \(G_{\infty}\) as follows:
and
Then \((E_{1})^{\alpha}=G_{\infty}\) and \((E_{p})^{\alpha}=G_{q}\), where \(1< p\le \infty\).
Proof
First, consider the following equations:
in which the rows of the matrix A are the product of the rows of the matrix \(E^{-1}\) with the sequence \(b=(b_{n})\) and y is the E-transform of the sequence x. Therefore, we realize by (2.1) that \(bx=(b_{n}x_{n})\in\ell_{1}\) while \(x\in E_{p} \) if and only if \(Ay\in\ell_{1}\) whenever \(y\in\ell_{p}\). That is, \(b=(b_{n})\in (E_{p})^{\alpha}\) if and only if \(A\in (\ell_{p}:\ell_{1} )\). So, by 76 of [8], we obtain that
This implies that \((E_{p})^{\alpha}=G_{q}\). □
Theorem 2.2
Define the sets \(H_{1}\), \(H_{2}\), \(H_{3}\), and \(D_{q}\) by
and
Then \((E_{1})^{\beta}=H_{1}\cap H_{2}\), \((E_{\infty})^{\beta}=H_{1}\cap H_{3}\), and \((E_{p})^{\beta}=H_{1}\cap D_{q}\), where \(1< p<\infty\).
Proof
Consider the equation
in which y is the E-transform of x and \(S= (s_{n,k} )\) is defined by
for all \(n,k\in\mathbb{N}\). Accordingly, we derive from (2.2) that \(bx=(b_{n}x_{n})\in cs\) whenever \(x=(x_{n})\in E_{p}\) if and only if \(S y \in c\) while \(y=(y_{n})\in\ell_{p}\). This implies that \(b=(b_{n})\in (E_{p} )^{\beta}\) if and only if \(S\in(\ell _{p}:c)\). Hence, we deduce from 16 of [8] that
which shows that \((E_{p})^{\beta}=H_{1}\cap D_{q}\). □
Theorem 2.3
Define the set \(D_{1}\) by
Then \((E_{1})^{\gamma}=H_{2}\), \((E_{\infty})^{\gamma}=D_{1}\), and \((E_{p})^{\gamma}=D_{q}\), where \(1< p<\infty\).
Proof
Using 1, 5, and 6 of [8], the proof can be easily adopted from one of Theorems 2.1 and 2.2 above, and so we omit the details. □
3 Special cases
In the following we present several special cases of Theorems 2.1–2.3. First, consider the Fibonacci sequence spaces defined by
and
which are the matrix domain of the Fibonacci matrix in \(\ell_{p}\) [5], where the Fibonacci matrix \(F=(F_{n,k})_{n,k\ge0}\) is defined by
Here \(\{f_{k}\}_{k=0}^{\infty}\) is a sequence of Fibonacci numbers defined by \({f_{n}} = {f_{n - 1}} + {f_{n - 2}}\) for all \(n \ge1\), where \(f_{-1}=0\) and \(f_{0}=1\). The inverse of the Fibonacci matrix, \(F^{-1}=(c_{n,k})\), is
Applying Theorems 2.1, 2.2, and 2.3, we have the following results.
Corollary 3.1
The α-, β-, and γ-duals of Fibonacci sequence spaces \(F_{p}\) (\(1\le p\le\infty\)) are as follows:
-
1.
\({({F_{1}})^{\alpha}} = \{ { ( {{b_{n}}} ) \in\omega: \sup_{n,k \in{\Bbb {N}}} \sum_{n = k}^{k + 1} { \vert {{{( - 1)}^{n - k}}\frac{{{f_{k}}{f_{k + 1}}}}{{f_{n}^{2}}}{b_{n}}} \vert } < \infty} \}\),
-
2.
\((F_{p})^{\alpha}= \{ { ( {{b_{n}}} )\in\omega:\sup_{K \in\mathfrak{F} } \sum_{k} {{{ \vert {\sum_{n \in K\cap\{k,k+1\} } (-1)^{n-k}\frac{f_{k}f_{k+1}}{f_{n}^{2}}b_{n} } \vert }^{q}} < \infty} } \}\),
-
3.
\((F_{\infty})^{\alpha}= \{ { ( {{b_{n}}} )\in\omega :\sup_{K \in\mathfrak{F} } \sum_{k} {{{ \vert {\sum_{n \in K\cap\{ k,k+1\}} (-1)^{n-k}\frac{f_{k}f_{k+1}}{f_{n}^{2}}b_{n} } \vert }} < \infty } } \}\),
-
4.
\({ ( {{F_{1}}} )^{\beta}}={ ( {{F_{1}}} )^{\gamma}} = \{ { ( {{b_{n}}} ) \in\omega:\sup_{n,k \in \mathbb {N}} \vert {\sum_{n = k}^{k + 1} {{{( - 1)}^{n - k}}\frac {{{f_{k}}{f_{k + 1}}}}{{f_{n}^{2}}}{b_{n}}} } \vert < \infty} \}\),
-
5.
\({ ( {{F_{p}}} )^{\beta}}={ ( {{F_{p}}} )^{\gamma}} = \{ { ( {{b_{n}}} ) \in\omega:\sup_{n \in\mathbb {N}} \sum_{k = 0}^{n} {{{ \vert {\sum_{j = k}^{k + 1} {{{( - 1)}^{j - k}}\frac{{{f_{k}}{f_{k + 1}}}}{{f_{j}^{2}}}{b_{j}}} } \vert }^{q}}} < \infty} \}\),
-
6.
\({ ( {{F_{\infty}}} )^{\beta}} = \{ { ( {{b_{n}}} ) \in\omega:\sum_{k} { \vert {\sum_{j = k}^{k + 1} {{{( - 1)}^{j - k}}\frac{{{f_{k}}{f_{k + 1}}}}{{f_{j}^{2}}}{b_{j}}} } \vert }\textit{ converges uniformly in }n} \}\),
-
7.
\({ ( {{F_{\infty}}} )^{\gamma}} = \{ { ( {{b_{n}}} ) \in\omega\,:\sup_{n \in{\Bbb {N}}} \sum_{k = 0}^{n} { \vert {\sum_{j = k}^{k + 1} {{{( - 1)}^{j - k}}\frac{{{f_{k}}{f_{k + 1}}}}{{f_{j}^{2}}}{b_{j}}} } \vert } < \infty} \}\).
Next consider the Euler sequence spaces of order θ, defined as
and
which are the matrix domain of the Euler matrix in \(\ell_{p}\) [1], where the Euler matrix \(E(\theta)= (e_{n,k} )\) is defined by
Since the inverse of \(E(\theta)\) is \(E(\frac{1}{\theta})\), we observe that Theorems 4.4, 4.5, and 4.6 of [1] are all the special cases of Theorems 2.1, 2.2, and 2.3, respectively, in which the matrix E is replaced by \(E(\theta)\).
We refer the readers to [1, 2, 4, 6], and [7] for some results which are all the special cases of Theorems 2.1, 2.2, and 2.3.
4 Conclusions
In this study, we obtain the α-, β-, and γ-duals of the domain of an arbitrary invertible summability matrix E in \(\ell_{p}\) and show that the recent works by Altay, Başar, and Mursaleen are all the special cases of our results.
References
Altay, B., Başar, F., Mursaleen, M.: On the Euler sequence spaces which include the spaces \(\ell _{p}\) and \(\ell_{\infty}\) I. Inf. Sci. 176, 1450–1462 (2006)
Başar, F., Altay, B.: On the space of sequences of p-bounded variation and related matrix mappings. Ukr. Math. J. 55(1), 136–147 (2003)
Boos, J.: Classical and Modern Methods in Summability. Oxford University Press, New York (2000)
Kara, E.E.: Some topological and geometrical properties of new Banach sequence spaces. J. Inequal. Appl. 2013, 38 (2013)
Kara, E.E., Başarir, M.: An application of Fibonacci numbers into infinite Toeplitz matrices. Caspian J. Math. Sci. 1(1), 43–47 (2012)
Kara, E.E., İlkhan, M.: On some Banach sequence spaces derived by a new band matrix. Br. J. Math. Comput. Sci. 9(2), 141–159 (2015)
Mursaleen, M., Noman, A.K.: On some new sequence spaces of non-absolute type related to the spaces \(\ell_{p}\) and \(\ell_{\infty}\) II. Math. Commun. 16, 383–398 (2011)
Stieglitz, M., Tietz, H.: Matrix transformationen von Folgenräumen. Eine Ergebnisübersicht. Math. Z. 154(1), 1–16 (1977)
Talebi, G.: On the Taylor sequence spaces and upper boundedness of Hausdorff matrices and Nörlund matrices. Indag. Math. 28(3), 629–636 (2017)
Acknowledgements
The author would like to thank the reviewers for their careful reading and making some useful comments which improved the presentation of the paper.
Funding
The research for this manuscript is not funded by anybody else.
Author information
Authors and Affiliations
Contributions
The author approved the final manuscript.
Corresponding author
Ethics declarations
Competing interests
The author declares that he has no competing interests.
Additional information
The paper is dedicated to the doyen of the martyrs, the chief of the youth of paradise, Imam Hossein ibn Ali (peace be upon him) in the memory of the 1379th occasion of his Arbaeen.
Publisher’s Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Rights and permissions
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.
About this article
Cite this article
Talebi, G. On multipliers of matrix domains. J Inequal Appl 2018, 296 (2018). https://doi.org/10.1186/s13660-018-1887-4
Received:
Accepted:
Published:
DOI: https://doi.org/10.1186/s13660-018-1887-4