Compact operators on some Fibonacci difference sequence spaces
- Abdullah Alotaibi^{1},
- Mohammad Mursaleen^{1, 2}Email author,
- Badriah AS Alamri^{1} and
- Syed Abdul Mohiuddine^{1}
DOI: 10.1186/s13660-015-0713-5
© Alotaibi et al. 2015
Received: 19 March 2015
Accepted: 22 May 2015
Published: 18 June 2015
Abstract
In this paper, we characterize the matrix classes \((\ell _{1},\ell _{p}(\widehat{F}))\) (\(1\leq p<\infty \)), where \(\ell _{p}(\widehat{F})\) is some Fibonacci difference sequence spaces. We also obtain estimates for the norms of the bounded linear operators \(L_{A}\) defined by these matrix transformations and find conditions to obtain the corresponding subclasses of compact matrix operators by using the Hausdorff measure of noncompactness.
Keywords
sequence spaces Fibonacci numbers compact operators Hausdorff measure of noncompactnessMSC
46A45 11B39 46B501 Introduction and preliminaries
Let \(\mathbb{N} =\{0,1,2,\ldots\}\) and \(\mathbb{R} \) be the set of all real numbers. We shall write \(\lim_{k}\), \(\sup_{k}\), \(\inf_{k}\), and \(\sum_{k}\) instead of \(\lim_{k\rightarrow \infty }\), \(\sup_{k\in \mathbb{N} }\), \(\inf_{k\in \mathbb{N} }\), and \(\sum_{k=0}^{\infty }\), respectively. Let ω be the vector space of all real sequences \(x=(x_{k})_{k\in \mathbb{N} }\). By the term sequence space, we shall mean any linear subspace of ω. Let φ, \(\ell _{\infty }\), c, and \(c_{0}\) denote the sets of all finite, bounded, convergent and null sequences, respectively. We write \(\ell _{p}=\{x\in \omega :\sum_{k}\vert x_{k}\vert ^{p}<\infty \}\) for \(1\leq p<\infty \). Also, we shall use the conventions that \(e=(1,1,\ldots)\) and \(e^{(n)}\) is the sequence whose only non-zero term is 1 in the nth place for each \(n\in \mathbb{N} \). For any sequence \(x=(x_{k})\), let \(x^{[n]}=\sum_{k=0}^{n}x_{k}e^{(k)}\) be its n-section. Moreover, we write bs and cs for the sets of sequences with bounded and convergent partial sums, respectively.
A B-space is a complete normed space. A topological sequence space in which all coordinate functionals \(\pi_{k}\), \(\pi _{k}(x)=x_{k}\), are continuous is called a K-space. A BK-space is defined as a K-space which is also a B-space, that is, a BK-space is a Banach space with continuous coordinates. A BK-space \(X\supset\varphi\) is said to have AK if every sequence \(x=(x_{k})\in X\) has a unique representation \(x=\sum_{k}x_{k}e^{(k)}\). For example, the space \(\ell_{p}\) (\(1\leq p<\infty\)) is BK-space with \(\Vert x\Vert _{p}= ( \sum_{k}\vert x_{k}\vert ^{p} ) ^{1/p}\) and \(c_{0}\), c, and \(\ell_{\infty}\) are BK-spaces with \(\Vert x\Vert _{\infty}=\sup_{k}\vert x_{k}\vert \). Further, the BK-spaces \(c_{0}\) and \(\ell_{p}\) have AK, where \(1\leq p<\infty\) (cf. [1]).
A sequence \((b_{n})\) in a normed space X is called a Schauder basis for X if for every \(x\in X\) there is a unique sequence \((\alpha_{n})\) of scalars such that \(x=\sum_{n}\alpha_{n}b_{n}\), i.e., \(\lim_{m}\Vert x-\sum_{n=0}^{m}\alpha_{n}b_{n}\Vert =0\).
For arbitrary subsets X and Y of ω, we write \(( X,Y ) \) for the class of all infinite matrices that map X into Y. Thus, \(A\in ( X,Y ) \) if and only if \(A_{n}\in X^{\beta}\) for all \(n\in \mathbb{N}\) and \(Ax\in Y\) for all \(x\in X\).
The following results are very important in our study [2, 3].
Lemma 1.1
- (a)
Then we have \((X,Y)\subset \mathcal{B}(X,Y)\), that is, every matrix \(A\in (X,Y)\) defines an operator \(L_{A}\in \mathcal{B}(X,Y)\) by \(L_{A}(x)=Ax\) for all \(x\in X\).
- (b)
If X has AK, then \(\mathcal{B}(X,Y)\subset (X,Y)\), that is, for every operator \(L\in \mathcal{B}(X,Y)\) there exists a matrix \(A\in (X,Y)\) such that \(L(x)=Ax\) for all \(x\in X\).
Fibonacci numbers have many interesting properties and applications in arts, sciences and architecture. For example, the ratio sequences of Fibonacci numbers converge to the golden ratio which is important in sciences and arts. Also, some basic properties of Fibonacci numbers can be found in [4].
Throughout, let \(1\leq p\leq\infty\) and q denote the conjugate of p, that is, \(q=p/(p-1)\) for \(1< p<\infty\), \(q=\infty\) for \(p=1\) or \(q=1\) for \(p=\infty\).
2 Methods of measure of noncompactness
The first measure of noncompactness, the function α, was defined and studied by Kuratowski [7] in 1930. Darbo [8] used this measure to generalize both the classical Schauder fixed point principle and (a special variant of) the Banach contraction mapping principle for so-called condensing operators. The Hausdorff or ball measure of noncompactness χ was introduced by Goldenštein et al. [9] in 1957, and later studied by Goldenštein and Markus [10].
Let X and Y be infinite dimensional Banach spaces. We recall that a linear operator L from X into Y is called compact if its domain is all of X and, for every bounded sequence \((x_{n})\) in X, the sequence the sequence \((L(x_{n}))\) has a convergent subsequence. We denote the class of all compact operators in \(\mathcal{B}(X,Y)\) by \(\mathcal{C}(X,Y)\).
The basic properties of the Hausdorff measure of noncompactness can be found in [3, 11–14].
Theorem 2.1
In particular, the following result shows how to compute the Hausdorff measure of noncompactness in the spaces \(c_{0}\) and \(\ell_{p}\) (\(1\leq p<\infty\)), which are BK-spaces with AK.
Theorem 2.2
([14], Theorem 2.15)
Since matrix mappings between BK spaces define bounded linear operators between these spaces which are Banach spaces, it is natural to use the above results and the Hausdorff measure of noncompactness to obtain necessary and sufficient conditions for matrix operators between BK spaces with a Schauder basis or AK to be compact operators. This technique has recently been used by several authors in many research papers (see for instance [15–25]). In this paper, we characterize the matrix classes \((\ell _{1},\ell _{p}(\widehat{F}))\) (\(1\leq p<\infty \)) and obtain an identity for the norms of the bounded linear operators \(L_{A}\) defined by these matrix transformations. We also find conditions to obtain the corresponding subclasses of compact matrix operators by using the Hausdorff measure of noncompactness.
3 Main results
In [26], the classes \((\ell_{p}(\widehat{F}),Y)\), \((\ell_{\infty }(\widehat{F}),Y)\), \((\ell_{1}(\widehat{F}),Y)\), \(Y=\ell_{\infty},c_{0},c\), and \((\ell_{p}(\widehat{F}),\ell_{1})\), \((\ell_{1}(\widehat{F}),\ell_{p})\) of compact operators were characterized. In this paper, we characterize the classes \(\mathcal{B}(\ell_{1},\ell_{p}^{\lambda})\) for (\(1\leq p<\infty\)) and compute the norm of operators in \(\mathcal{B}(\ell_{1},\ell_{p}^{\lambda})\). We also apply the results of the previous section to determine the Hausdorff measure of noncompactness of operators in \(\mathcal{B}(\ell_{1},\ell _{p}^{\lambda})\), and to characterize the classes \(\mathcal{C}(\ell_{1},\ell_{p})\) for \(1\leq p<\infty\).
Here we characterize the classes \(\mathcal{B}(\ell_{1},\ell_{p}(\widehat{F})) \) for (\(1\leq p<\infty\)) and compute the norm of operators in \(\mathcal{B}(\ell_{1},\ell_{p}(\widehat{F}))\). We also apply the results of the previous section to determine the Hausdorff measure of noncompactness of operators in \(\mathcal{B}(\ell_{1},\ell_{p}(\widehat{F}))\) and to characterize the classes \(\mathcal{C}(\ell_{1},\ell_{p}(\widehat{F}))\) for \(1\leq p<\infty\).
The following result is useful.
Lemma 3.1
([14], Theorem 3.8)
- (a)
Then we have \(A\in (X,Y_{T})\) if and only if \(C=T\cdot A\in (X,Y)\), where C denotes the matrix product of T and A.
- (b)If X and Y are B spaces and \(A\in (X,Y_{T})\) then$$ \Vert L_{A}\Vert =\Vert L_{C}\Vert . $$(3.1)
First we establish the characterizations of the classes \(\mathcal{B}(\ell _{1},\ell_{p}(\widehat{F}))\) for (\(1\leq p<\infty\)) and an identity for the operator norm.
Theorem 3.1
- (a)We have \(L\in \mathcal{B}(\ell _{1},\ell _{p}(\widehat{F}))\) if and only if there exists an infinite matrix \(A\in (\ell _{1},\ell _{p}(\widehat{F}))\) such thatand$$ \Vert A\Vert =\sup_{k} \biggl( \sum _{n}\biggl\vert \frac{f_{n}}{f_{n+1}}a_{nk}- \frac{f_{n+1}}{f_{n}}a_{n-1,k}\biggr\vert ^{p} \biggr) ^{1/p}< \infty $$(3.2)$$ L(x)=Ax\quad \textit{for all }x\in \ell _{1}. $$(3.3)
- (b)If \(L\in \mathcal{B}(\ell _{1},\ell _{p}(\widehat{F}))\) then$$ \Vert L\Vert =\Vert A\Vert . $$(3.4)
Proof
This completes the proof. □
Now we are going to establish a formula for the Hausdorff measure of noncompactness of operators in \(\mathcal{B}(\ell_{1},\ell_{p}(\widehat{F}))\). We need the following result.
Lemma 3.2
([1], Theorem 4.2)
Let X be a linear metric space with a translation invariant metric, T be a triangle and χ, and \(\chi _{T}\) denote the Hausdorff measures of noncompactness on \(\mathcal{M}_{X}\) and \(\mathcal{M}_{X_{T}}\), respectively. Then \(\chi _{T}(Q)=\chi (TQ)\) for all \(Q\in \mathcal{M}_{X_{T}}\).
Theorem 3.2
Proof
This completes the proof. □
Finally, the characterization of \(\mathcal{C}(\ell _{1},\ell _{p}(\widehat{F}))\) is an immediate consequence of Theorem 3.2 and (2.2).
Theorem 3.3
4 Conclusions
The Hausdorff measure of noncompactness can be most effectively used to characterize compact operators between Banach spaces. We have shown how the Hausdorff measure of noncompactness could be applied in the characterization of compact matrix operators between BK spaces. Here we characterized the classes \(\mathcal{B}(\ell _{1},\ell _{p}(\widehat{F}))\) for (\(1\leq p<\infty \)) and computed the norm of operators in \(\mathcal{B}(\ell _{1},\ell _{p}(\widehat{F}))\). We also determined the Hausdorff measure of noncompactness of operators in \(\mathcal{B}(\ell _{1},\ell _{p}(\widehat{F}))\) and then characterized the classes \(\mathcal{C}(\ell _{1},\ell _{p}(\widehat{F}))\) for \(1\leq p<\infty \).
Declarations
Acknowledgements
This project was funded by the Deanship of Scientific Research (DSR) at King Abdulaziz University, Jeddah, under grant no. (69-130-35-RG). The authors, therefore, acknowledge with thanks DSR technical and financial support.
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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