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Compact operators on some Fibonacci difference sequence spaces
Journal of Inequalities and Applicationsvolume 2015, Article number: 203 (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.
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 nonzero 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 nsection. Moreover, we write bs and cs for the sets of sequences with bounded and convergent partial sums, respectively.
A Bspace 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 Kspace. A BKspace is defined as a Kspace which is also a Bspace, that is, a BKspace is a Banach space with continuous coordinates. A BKspace \(X\supset\varphi\) 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 BKspace with \(\Vert x\Vert _{p}= ( \sum_{k}\vert x_{k}\vert ^{p} ) ^{1/p}\) and \(c_{0}\), c, and \(\ell_{\infty}\) are BKspaces with \(\Vert x\Vert _{\infty}=\sup_{k}\vert x_{k}\vert \). Further, the BKspaces \(c_{0}\) 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., \(\lim_{m}\Vert x\sum_{n=0}^{m}\alpha_{n}b_{n}\Vert =0\).
The βdual of a sequence space X is defined by
Let \(A=(a_{nk})_{n,k=0}^{\infty }\) be an infinite matrix of real numbers \(a_{nk}\), where \(n,k\in \mathbb{N}\). We write \(A_{n}\) for the sequence in the nth row of A, that is, \(A_{n}=(a_{nk})_{k=0}^{\infty }\) for every \(n\in \mathbb{N}\). In addition, if \(x=(x_{k})_{k=0}^{\infty }\in \omega \) then we define the Atransform of x as the sequence \(Ax= \{ A_{n}(x) \} _{n=0}^{\infty }\), where
provided the series on the right side converges for each \(n\in \mathbb{N}\).
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 matrix domain \(X_{A}\) of an infinite matrix A in sequence space X is defined by
which is a sequence space.
The following results are very important in our study [2, 3].
Lemma 1.1
Let \(X\supset \phi \) and Y be BKspaces.

(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 \(x\in X\).
The Fibonacci numbers are the sequence of numbers \(\{f_{n}\}_{n=0}^{\infty }\) defined by the linear recurrence equations
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, \(q=p/(p1)\) for \(1< p<\infty\), \(q=\infty\) for \(p=1\) or \(q=1\) for \(p=\infty\).
Recently, in [5] and [6], the Fibonacci difference sequence spaces \(\ell _{p}(\widehat{F})\) and \(\ell _{\infty }(\widehat{F})\) have been defined as follows:
and
Using the notation of the matrix domain, the sequence spaces \(\ell _{p}(\widehat{F})\) and \(\ell _{\infty }(\widehat{F})\) may be redefined by
where the matrix \(\widehat{F}=(\widehat{f}_{nk})\) is defined by
Further, it is clear that the spaces \(\ell _{p}(\widehat{F})\) and \(\ell _{\infty }(\widehat{F})\) are BK spaces with the norms given by
where the sequence \(y=(y_{n})=(\widehat{F}_{n}(x))\) is the \(\widehat{F}\)transform of a sequence \(x=(x_{n})\), i.e.,
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 socalled 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 \(\mathcal{C}(X,Y)\).
Let \((X,d)\) be a metric space, \(x_{0}\in X\) and \(r>0\). Then we write, as usual, \(B(x,r)=\{x\in X:d(x,x_{0})< r\}\) for the open ball of radius r and center \(x_{0}\). Let \(\mathcal{M}_{X}\) denote the class of all bounded subsets of X. If \(Q\in\mathcal{M}_{X}\), then the Hausdorff measure of noncompactness of the set Q, denoted by \(\chi(Q)\), is defined by
The function \(\chi:\mathcal{M}_{X}\rightarrow{}[0,\infty)\) is called the Hausdorff measure of noncompactness.
The basic properties of the Hausdorff measure of noncompactness can be found in [3, 11–14].
Now we recall the definition of the Hausdorff measure of noncompactness operators between Banach spaces. Let X and Y be Banach spaces and \(\chi_{1}\) and \(\chi_{2}\) be the Hausdorff measures of noncompactness on X and Y, respectively. An operator \(L:X\rightarrow Y\) is said to be \((\chi_{1}, \chi_{2})\) bounded if \(L(Q)\in\mathcal{M}_{Y}\) for all \(Q\in \mathcal{M}_{X}\) and there exists a constant \(C\geq0\) such that \(\chi _{2}(L(Q))\leq C\chi_{1}(Q)\) for all \(Q\in\mathcal{M}_{X}\). If an operator L is \((\chi_{1},\chi_{2})\)bounded then the number
is called the \((\chi_{1},\chi_{2})\) measure of noncompactness of L. If \(\chi_{1}=\chi_{2}=\chi\), then we write \(\Vert L\Vert _{(\chi_{1},\chi_{2})}=\Vert L\Vert _{\chi}\).
Now we outline the applications of the Hausdorff measure of noncompactness to the characterization of compact operators between Banach spaces. Let X and Y be Banach spaces and \(L\in \mathcal{B}(X,Y)\). Then the Hausdorff measure of noncompactness of L is given by ([14], Theorem 2.25)
and L is compact if and only if ([14], Corollary 2.26 (2.58))
The identities in (2.1) and (2.2) reduce the characterization of compact operators \(L\in \mathcal{B}(X,Y)\) to the determination of the Hausdorff measure of noncompactness \(\chi (Q)\) of bounded sets Q in a Banach space X. If X has a Schauder basis, then there exist estimates or even identities for \(\chi (Q)\).
Theorem 2.1
Let X be a Banach space with a Schauder basis \((b_{k})_{k=0}^{\infty}\), \(Q\in \mathcal{M}_{X}\), \(\mathcal{P}_{n}:X\rightarrow X\) be the projectors onto the linear span of \(\{b_{0},b_{1},\dots,b_{n}\}\) and \(\mathcal{R}_{n}=I\mathcal{P}_{n}\) for \(n=0,1,\dots\), where I denotes the identity map on X. Then we have
where \(a=\limsup_{n\rightarrow\infty}\Vert\mathcal{R}_{n}\Vert\).
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 BKspaces with AK.
Theorem 2.2
([14], Theorem 2.15)
Let Q be a bounded subset of the normed space X, where X is \(\ell _{p}\) for \(1\leq p<\infty \) or \(c_{0}\). If \(P_{n}:X\rightarrow X\) is the operator defined by \(\mathcal{P}_{n}(x)=x^{[n]}\) for all \(x=(x_{k})_{k=0}^{\infty }\in X\) and \(\mathcal{R}_{n}=I\mathcal{P}_{n}\) for \(n=0,1,\dots \), then we have
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.
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)
Let T be a triangle and X and Y be arbitrary subsets of ω.

(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
Let \(1\leq p<\infty \).

(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 that
$$ \Vert A\Vert =\sup_{k} \biggl( \sum _{n}\biggl\vert \frac{f_{n}}{f_{n+1}}a_{nk} \frac{f_{n+1}}{f_{n}}a_{n1,k}\biggr\vert ^{p} \biggr) ^{1/p}< \infty $$(3.2)and
$$ L(x)=Ax\quad \textit{for all }x\in \ell _{1}. $$(3.3) 
(b)
If \(L\in \mathcal{B}(\ell _{1},\ell _{p}(\widehat{F}))\) then
$$ \Vert L\Vert =\Vert A\Vert . $$(3.4)
Proof
Since \(\ell _{1}\) is a BK space with AK it follows from Lemma 1.1 that \(L\in \mathcal{B}(\ell _{1},\ell _{p}(\widehat{F}))\) for \(1\leq p<\infty \) if and only if there exists an infinite matrix \(A\in (\ell _{1},\ell _{p}(\widehat{F}))\) such that (3.3) holds. Also we have by Lemma 3.1(a) that \(A\in (\ell _{1},\ell _{p}(\widehat{F}))\) if and only if \(C=\widehat{F}\cdot A\in (\ell _{1},\ell _{p})\), where the matrix \(C=(c_{nk})\) is defined by
Furthermore, we have by [2], Example 8.4.1D, \(C\in (\ell _{1},\ell _{p})\) if and only if
This completes the proof of part (a).
(b) If \(L\in \mathcal{B}(\ell _{1},\ell _{p}^{\lambda })\) then it follows from (3.1) that \(\Vert L\Vert =\Vert L_{C}\Vert \), where \(L_{C}\in \mathcal{B}(\ell _{1},\ell _{p})\) is given by \(L_{C}(x)=Cx\) for all \(x\in \ell _{1}\). It follows by the Minkowski inequality that
and so
We also obtain for \(e^{(k)}\in S_{\ell _{1}}\) (\(k\in \mathbb{N}\))
and so \(\Vert L\Vert \geq \Vert A\Vert \). This and (3.5) yield (3.4).
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
Let \(L\in \mathcal{B}(\ell _{1},\ell _{p}(\widehat{F}))\) (\(1\leq p<\infty \)) and A denote the matrix which represents L. Then we have
Proof
We write \(S=S_{\ell _{1}}\), for short, and \(C^{[m]}\) (\(m\in \mathbb{N}\)) for the matrix with the rows \(C_{n}^{[m]}=0\) for \(0\leq n\leq m\) and \(C_{n}^{[m]}=C_{n}\) for \(n\geq m+1\). It follows from (2.1), Lemma 3.2, (2.3), (3.4), and (3.2) that
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
Let \(L\in \mathcal{B}(\ell _{1},\ell _{p}(\widehat{F})) \) (\(1\leq p<\infty \)) and A denote the matrix which represents L. Then L is compact if and only if
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 \(1\leq p<\infty \).
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Acknowledgements
This project was funded by the Deanship of Scientific Research (DSR) at King Abdulaziz University, Jeddah, under grant no. (6913035RG). The authors, therefore, acknowledge with thanks DSR technical and financial support.
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DOI
MSC
 46A45
 11B39
 46B50
Keywords
 sequence spaces
 Fibonacci numbers
 compact operators
 Hausdorff measure of noncompactness