R E S E A R C H
Open Access
Compact operators on some Fibonacci
difference sequence spaces
Abdullah Alotaibi
1, Mohammad Mursaleen
1,2*, Badriah AS Alamri
1and Syed Abdul Mohiuddine
1*Correspondence:
[email protected] 1Operator Theory and Applications Research Group, Department of Mathematics, Faculty of Science, King Abdulaziz University, P.O. Box 80203, Jeddah, 21589, Saudi Arabia 2Department of Mathematics,
Aligarh Muslim University, Aligarh, 202 002, India
Abstract
In this paper, we characterize the matrix classes (
1,
p(F)) (1≤p<∞), where
p(F) is
some Fibonacci difference sequence spaces. We also obtain estimates for the norms of the bounded linear operatorsLAdefined by these matrix transformations and find
conditions to obtain the corresponding subclasses of compact matrix operators by using the Hausdorff measure of noncompactness.
MSC: 46A45; 11B39; 46B50
Keywords: sequence spaces; Fibonacci numbers; compact operators; Hausdorff measure of noncompactness
1 Introduction and preliminaries
LetN={, , , . . .}andRbe the set of all real numbers. We shall writelimk,supk,infk, and
kinstead oflimk→∞,supk∈N,infk∈N, and ∞
k=, respectively. Letωbe the vector space
of all real sequencesx= (xk)k∈N. By the termsequence space, we shall mean any linear
subspace ofω. Letϕ,∞,c, andcdenote the sets of all finite, bounded, convergent and
null sequences, respectively. We writep={x∈ω:
k|xk|p<∞}for ≤p<∞. Also, we
shall use the conventions thate= (, , . . .) ande(n)is the sequence whose only non-zero
term is in thenth place for eachn∈N. For any sequencex= (xk), letx[n]= n
k=xke(k)
be itsn-section. Moreover, we writebsandcsfor the sets of sequences with bounded and convergent partial sums, respectively.
AB-space is a complete normed space. A topological sequence space in which all coor-dinate functionalsπk,πk(x) =xk, are continuous is called aK-space. ABK-space is defined
as aK-space which is also aB-space, that is, aBK-space is a Banach space with continu-ous coordinates. ABK-spaceX⊃ϕis said to haveAKif every sequencex= (xk)∈Xhas
a unique representationx=kxke(k). For example, the spacep(≤p<∞) isBK-space
withxp= (
k|xk|p)/pandc,c, and∞areBK-spaces withx∞=supk|xk|. Further,
theBK-spacescandphaveAK, where ≤p<∞(cf.[]).
A sequence (bn) in a normed space X is called a Schauder basisfor X if for every
x∈X there is a unique sequence (αn) of scalars such that x=
nαnbn, i.e., limmx– m
n=αnbn= .
Theβ-dual of a sequence spaceXis defined by
Xβ=a= (ak)∈ω:ax= (akxk)∈csfor allx= (xk)∈X
.
LetA= (ank)∞n,k=be an infinite matrix of real numbersank, wheren,k∈N. We writeAn
for the sequence in thenth row ofA, that is,An= (ank)∞k=for everyn∈N. In addition,
ifx= (xk)∞k=∈ωthen we define theA-transform of xas the sequenceAx={An(x)}∞n=,
where
An(x) = ∞
k=
ankxk (n∈N)
provided the series on the right side converges for eachn∈N.
For arbitrary subsetsXandY ofω, we write (X,Y) for the class of all infinite matrices that mapXintoY. Thus,A∈(X,Y) if and only ifAn∈Xβfor alln∈NandAx∈Yfor all
x∈X.
The matrix domainXAof an infinite matrixAin sequence spaceXis defined by
XA=
x= (xk)∈ω:Ax∈X
which is a sequence space.
The following results are very important in our study [, ].
Lemma . Let X⊃φand Y be BK -spaces.
(a) Then we have(X,Y)⊂B(X,Y),that is,every matrixA∈(X,Y)defines an operator
LA∈B(X,Y)byLA(x) =Axfor allx∈X.
(b) IfXhasAK,thenB(X,Y)⊂(X,Y),that is,for every operatorL∈B(X,Y)there
exists a matrixA∈(X,Y)such thatL(x) =Axfor allx∈X.
TheFibonacci numbersare the sequence of numbers{fn}∞n=defined by the linear
recur-rence equations
f=f= and fn=fn–+fn–; n≥.
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 Fi-bonacci numbers can be found in [].
Throughout, let ≤p≤ ∞andq denote the conjugate ofp, that is,q=p/(p– ) for <p<∞,q=∞forp= orq= forp=∞.
Recently, in [] and [], the Fibonacci difference sequence spacesp(F) and∞(F) have
been defined as follows:
p(F) =
x= (xn)∈ω:
n fn
fn+
xn–
fn+
fn
xn–
p<∞ ; ≤p<∞
and
∞(F) =
x= (xn)∈ω:sup n∈N
fn
fn+
xn–
fn+
fn
Using the notation of the matrix domain, the sequence spacesp(F) and∞(F) may be
redefined by
p(F) = (p)F (≤p<∞) and ∞(F) = (∞)F, (.)
where the matrixF= (fnk) is defined by
fnk= ⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩
–fn+
fn (k=n– ),
fn
fn+ (k=n),
(≤k<n– ) or (k>n)
(n,k∈N). (.)
Further, it is clear that the spacesp(F) and∞(F) areBKspaces with the norms given by
xp(F)=
n yn(x)
p/p
; (≤p<∞) andx∞(F)=sup n∈N
yn(x), (.)
where the sequencey= (yn) = (Fn(x)) is theF-transform of a sequencex= (xn),i.e.,
yn=Fn(x) = ⎧ ⎨ ⎩ f
fx=x (n= ),
fn
fn+xn–
fn+
fn xn– (n≥)
(n∈N). (.)
2 Methods of measure of noncompactness
The first measure of noncompactness, the functionα, was defined and studied by Kura-towski [] in . Darbo [] 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 noncompactnessχ
was introduced by Goldenšteinet al.[] in , and later studied by Goldenštein and Markus [].
LetXandY be infinite dimensional Banach spaces. We recall that a linear operatorL
fromXintoYis calledcompactif its domain is all ofXand, for every bounded sequence (xn) inX, the sequence the sequence (L(xn)) has a convergent subsequence. We denote the
class of all compact operators inB(X,Y) byC(X,Y).
Let (X,d) be a metric space,x∈X andr> . Then we write, as usual,B(x,r) ={x∈
X:d(x,x) <r}for the open ball of radiusrand centerx. LetMXdenote the class of all
bounded subsets ofX. IfQ∈MX, then theHausdorff measure of noncompactness of the
set Q, denoted byχ(Q), is defined by
χ(Q) =inf
> :Q⊂
n
k=
B(xk,rk),xk∈X,rk<(k= , , . . .),n∈N
.
The functionχ:MX→[,∞) is called theHausdorff measure of noncompactness.
The basic properties of the Hausdorff measure of noncompactness can be found in [, –].
measures of noncompactness onXandY, respectively. An operatorL:X→Yis said to be (χ,χ)-boundedifL(Q)∈MYfor allQ∈MXand there exists a constantC≥ such that
χ(L(Q))≤Cχ(Q) for allQ∈MX. If an operatorLis (χ,χ)-bounded then the number L(χ,χ)=inf
C≥ :χ
L(Q)≤Cχ(Q) for allQ∈MX
is called the (χ,χ)-measure of noncompactness of L. If χ =χ =χ, then we write L(χ,χ)=Lχ.
Now we outline the applications of the Hausdorff measure of noncompactness to the characterization of compact operators between Banach spaces. LetXandY be Banach spaces andL∈B(X,Y). Then the Hausdorff measure of noncompactness ofLis given by ([], Theorem .)
Lχ=χ
L(SX)
(.)
andLis compact if and only if ([], Corollary . (.))
Lχ= . (.)
The identities in (.) and (.) reduce the characterization of compact operators L∈ B(X,Y) to the determination of the Hausdorff measure of noncompactness χ(Q) of bounded setsQin a Banach spaceX. IfXhas a Schauder basis, then there exist estimates or even identities forχ(Q).
Theorem .([] or [], Theorem .) Let X be a Banach space with a Schauder basis
(bk)∞k=,Q∈MX,Pn:X→X be the projectors onto the linear span of{b,b, . . . ,bn}and
Rn=I–Pnfor n= , , . . . ,where I denotes the identity map on X.Then we have
a·lim supn→∞
sup x∈Q
Rn(x)≤χ(Q)≤lim sup n→∞
sup x∈Q
Rn(x),
where a=lim supn→∞Rn.
In particular, the following result shows how to compute the Hausdorff measure of non-compactness in the spacescandp(≤p<∞), which areBK-spaces withAK.
Theorem .([], Theorem .) Let Q be a bounded subset of the normed space X,
where X ispfor≤p<∞or c.If Pn:X→X is the operator defined byPn(x) =x[n]for
all x= (xk)∞k=∈X andRn=I–Pnfor n= , , . . . ,then we have
χ(Q) = lim n→∞
sup x∈Q
Rn(x). (.)
technique has recently been used by several authors in many research papers (see for in-stance [–]). In this paper, we characterize the matrix classes (,p(F)) (≤p<∞)
and obtain an identity for the norms of the bounded linear operatorsLAdefined 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 [], the classes (p(F),Y), (∞(F),Y), ((F),Y),Y=∞,c,c, and (p(F),), ((F),p)
of compact operators were characterized. In this paper, we characterize the classes
B(,λp) for (≤p<∞) and compute the norm of operators inB(,λp). We also apply the
results of the previous section to determine the Hausdorff measure of noncompactness of operators inB(,λp), and to characterize the classesC(,p) for ≤p<∞.
Here we characterize the classesB(,p(F)) for (≤p<∞) and compute the norm of
operators inB(,p(F)). We also apply the results of the previous section to determine the
Hausdorff measure of noncompactness of operators inB(,p(F)) and to characterize the
classesC(,p(F)) for ≤p<∞.
The following result is useful.
Lemma .([], Theorem .) Let T be a triangle and X and Y be arbitrary subsets ofω.
(a) Then we haveA∈(X,YT)if and only ifC=T·A∈(X,Y),whereCdenotes the
matrix product ofTandA.
(b) IfXandYareBspaces andA∈(X,YT)then
LA=LC. (.)
First we establish the characterizations of the classesB(,p(F)) for (≤p<∞) and an
identity for the operator norm.
Theorem . Let≤p<∞.
(a) We haveL∈B(,p(F))if and only if there exists an infinite matrixA∈(,p(F))
such that
A=sup k
n fn
fn+
ank–
fn+
fn
an–,k
p/p<∞ (.)
and
L(x) =Ax for allx∈. (.)
(b) IfL∈B(,p(F))then
L=A. (.)
Proof Since is aBK space withAKit follows from Lemma . thatL∈B(,p(F)) for
Also we have by Lemma .(a) thatA∈(,p(F)) if and only ifC=F·A∈(,p), where
the matrixC= (cnk) is defined by
cnk=
fn
fn+
ank–
fn+
fn
an–,k.
Furthermore, we have by [], Example ..D,C∈(,p) if and only if
C=sup k
∞
n= |cnk|p
/p
<∞.
This completes the proof of part (a).
(b) IfL∈B(,λp) then it follows from (.) thatL=LC, whereLC∈B(,p) is given
byLC(x) =Cxfor allx∈. It follows by the Minkowski inequality that LC(x)p=
∞
n=
∞
k=
cnkxk
p/p ≤
∞
k= |xk|
∞
n= |cnk|p
/p
≤ C · x=A · x,
and so
L ≤ A. (.)
We also obtain fore(k)∈S(k∈N)
LC
e(k)=
∞
n= |cnk|p
p
,
and soL ≥ A. This and (.) yield (.).
This completes the proof. Now we are going to establish a formula for the Hausdorff measure of noncompactness of operators inB(,p(F)). We need the following result.
Lemma .([], Theorem .) Let X be a linear metric space with a translation invariant metric,T be a triangle andχ,andχTdenote the Hausdorff measures of noncompactness
onMXandMXT,respectively.ThenχT(Q) =χ(TQ)for all Q∈MXT.
Theorem . Let L∈B(,p(F)) (≤p<∞)and A denote the matrix which represents L.
Then we have
Lχ
p(F)=m→∞lim
sup
k ∞
n=m fn
fn+
ajk–
fn+
fn
aj–,k p
/p
. (.)
Proof We writeS=S, for short, andC
[m](m∈N) for the matrix with the rowsC[m] n =
for ≤n≤mandC[m]
and (.) that
Lχp(F)=χp(F)
L(S)=χp
LC(S)
= lim m→∞
sup
x∈S
Rm(Cx)p
= lim m→∞
sup
x∈S
C[m]xp= lim m→∞C
[m]
= lim m→∞
sup
k ∞
n=m+ fn
fn+
ajk–
fn+
fn
aj–,k p
/p
.
This completes the proof. Finally, the characterization ofC(,p(F)) is an immediate consequence of Theorem .
and (.).
Theorem . Let L∈B(,p(F)) (≤p<∞)and A denote the matrix which represents L.
Then L is compact if and only if
lim m→∞
sup
k ∞
n=m fn
fn+
ajk–
fn+
fn
aj–,k p
= .
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 classesB(,p(F)) for (≤p<∞) and
computed the norm of operators inB(,p(F)). We also determined the Hausdorff
mea-sure of noncompactness of operators inB(,p(F)) and then characterized the classes
C(,p(F)) for ≤p<∞.
Competing interests
The authors declare that they have no competing interests.
Authors’ contributions
The authors contributed equally and significantly in writing this paper. All authors read and approved the final manuscript.
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.
Received: 19 March 2015 Accepted: 22 May 2015 References
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