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## difference sequence spaces

1

1,2*

1

### and Syed Abdul Mohiuddine

1

*Correspondence:

mursaleenm@gmail.com 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<∞), wherep(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

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, andcdenote the sets of all ﬁnite, bounded, convergent and

null sequences, respectively. We writep={xω:

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-spaceXϕis 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-spacescandphaveAK, where ≤p<∞(cf.[]).

A sequence (bn) in a normed space X is called a Schauder basisfor X if for every

xX there is a unique sequence (αn) of scalars such that x=

nαnbn, i.e., limmxm

n=αnbn= .

Theβ-dual of a sequence spaceXis deﬁned by

=a= (ak)∈ω:ax= (akxk)∈csfor allx= (xk)∈X

.

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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 ifAnfor alln∈NandAxYfor all

xX.

The matrix domainXAof an inﬁnite matrixAin sequence spaceXis deﬁned by

XA=

x= (xk)∈ω:AxX

which is a sequence space.

The following results are very important in our study [, ].

Lemma . Let Xφand Y be BK -spaces.

(a) Then we have(X,Y)⊂B(X,Y),that is,every matrixA∈(X,Y)deﬁnes an operator

LAB(X,Y)byLA(x) =Axfor allxX.

(b) IfXhasAK,thenB(X,Y)⊂(X,Y),that is,for every operatorLB(X,Y)there

exists a matrixA∈(X,Y)such thatL(x) =Axfor allxX.

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

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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. IfQMX, then theHausdorﬀ measure of noncompactness of the

set Q, denoted byχ(Q), is deﬁned by

χ(Q) =inf

>  :Q

n

k=

B(xk,rk),xkX,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 [, –].

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measures of noncompactness onXandY, respectively. An operatorL:XYis said to be (χ,χ)-boundedifL(Q)∈MYfor allQMXand there exists a constantC≥ such that

χ(L(Q))≤(Q) for allQMX. If an operatorLis (χ,χ)-bounded then the number L(χ,χ)=inf

C≥ :χ

L(Q)≤(Q) for allQMX

is called the (χ,χ)-measure of noncompactness of L. If χ =χ =χ, then we write L(χ,χ)=.

Now we outline the applications of the Hausdorﬀ measure of noncompactness to the characterization of compact operators between Banach spaces. LetXandY be Banach spaces andLB(X,Y). Then the Hausdorﬀ measure of noncompactness ofLis given by ([], Theorem .)

=χ

L(SX)

(.)

andLis compact if and only if ([], Corollary . (.))

= . (.)

The identities in (.) and (.) reduce the characterization of compact operators LB(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=,QMX,Pn:XX be the projectors onto the linear span of{b,b, . . . ,bn}and

Rn=IPnfor 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 spacescandp(≤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:XX is the operator deﬁned byPn(x) =x[n]for

all x= (xk)∞k=X andRn=IPnfor n= , , . . . ,then we have

χ(Q) = lim n→∞

sup x∈Q

Rn(x). (.)

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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 haveLB(,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) IfLB(,p(F))then

L=A. (.)

Proof Since is aBK space withAKit follows from Lemma . thatLB(,p(F)) for

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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) IfLB(,λp) then it follows from (.) thatL=LC, whereLCB(,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

LA. (.)

We also obtain fore(k)∈S(k∈N)

LC

e(k)=

n= |cnk|p

p

,

and soLA. 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 QMXT.

Theorem . Let LB(,p(F)) (≤p<∞)and A denote the matrix which represents L.

Then we have

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](mN) for the matrix with the rowsC[m] n = 

for ≤nmandC[m]

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and (.) that

p(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 LB(,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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