R E S E A R C H
Open Access
Hausdorff measure of noncompactness of
matrix operators on some sequence spaces of
a double sequential band matrix
Elahe Abyar and Mohammad Bagher Ghaemi
**Correspondence:
[email protected] Islamic Azad University, Karaj Branch, Alborz, Iran
Abstract
The sequence spacesl∞(B˜,p),c(B˜,p), andc0(B˜,p) of non-absolute type derived by the double sequential band matrixB(˜r,˜s) have recently been defined. In this work, we establish identities or estimates for the operator norms and the Hausdorff measure of noncompactness of certain matrix operators on these spaces that are paranormed spaces. Further, we find the necessary and sufficient condition for compactness ofLA
in the class (X,l∞(q)) (whereXis any of the spacesl∞(B˜,p),c(B˜,p) orc0(B˜,p)) and characterize some classes of compact operators on these spaces by using the Hausdorff measure of the noncompactness technique.
Keywords: Hausdorff measure of noncompactness; double sequential matrix; sequence space; paranormed space
1 Preliminaries and background
The first measure of noncompactness, the function α, was defined and studied by Kuratowsky [] in . Darbo [], using this measure, generalized both the classical Schauder fixed point principle and (a special variant of ) Banach’s contraction mapping principle for so called condensing operators. The Hausdorff measure of noncompactness
χwas introduced by Goldensteinet al.[] in .
Recently, the Hausdorff measure of noncompactness turned out to be very useful in the classification of compact operators between Banach spaces. Many authors character-ized the classes of compact operators given by infinite matrices on some sequence spaces by using the Hausdorff measure of noncompactness. For example, in [, ] Malkowsky and Djolovic, in [, ] Malkowsky and Rakocevic, in [] Alotaibiet al., in [] Basar and Malkowsky, and in [, ] Mursaleen and Noman have applied the Hausdorff measure of noncompactness to characterize some classes of compact operators given by matrices on the spaces in the literature.
Further, as we know paranormed spaces are another version of the linear metric space but have more general properties than normed spaces []. Hence, some authors have ap-plied these spaces in their research. For example, in [] Basarir and Kara have character-ized some classes of compact operators given by matrices on a normed sequence space, which is a special case of the paranormed RieszBm-difference sequence spacerq(p,Bm)
and for this purpose they have applied the Hausdorff measure of noncompactness. In [, ] Basarir and Kara and in [] Ozger and Basar have studied these spaces.
In [] Kirisci and Basar have studied the domain of generalized difference matrixB(r,s) in the classical spacesl∞,c, andc.
Afterward, Ozger and Basar in [] introduced the paranormed sequence spaces
l∞(B˜,p), c(B˜,p), andc(B˜,p) that are more general and comprehensive than the corre-sponding consequences of the matrix domain ofB(r,s).
In this work, we establish identities or estimates for the operator norms and the Hausdorff measure of noncompactness of certain matrix operators on these spaces that were defined by Ozger and Basar in [].
Letwbe the space of all real or complex valued sequences. Any vector subspace ofwis called a sequence space. We writel∞,candcfor the spaces of all bounded, convergent, and null sequences, respectively. Also, bybs,cs,l, andlp( <p<∞), we denote the spaces
of all bounded, convergent, absolutely, andp-absolutely convergent series, respectively. A sequence space λwith a linear topology is called aK-space if each of the mapsPi:
λ→Cdefined byPi(x) =xiis continuous for alli∈N, whereCis the set of all complex
numbers. AK-spaceλis called anFK-space providedλis a complete linear metric space and aBK-space if it is a normedFK-space. Letφ be the set of all sequences which have finite number of non-zero terms. AnFK-spaceλwhich containsφis said to have theAK
property if every sequencex= (xk)∞k=∈λhas a unique representationx=
∞
k=xke(k)[].
A linear topological spaceXover the real fieldRis said to be a paranormed space if there is a subadditive functionh:X→Rsuch thath(θ) = ,h(–x) =h(x), and scalar mul-tiplication is continuous, that is,|αn–α| → andh(xn–x)→ implyh(αnxn–αx)→
for everyα∈Randx∈X, whereθis the zero vector in the linear spaceX[].
Throughout this paper, we assume (pk) is a bounded sequence of strictly positive real
numbers withsuppk=HandM=max{,H}. So, the sequence spacesl∞(p),c(p),c(p), andl(p) (which generalize the classical spacesl∞,c,c, andl, respectively) are defined as follows:
l∞(p) =
x= (xk)∈w:sup k∈N|xk|
pk<∞
,
c(p) =
x= (xk)∈w:∃l∈C lim k→∞|xk–l|
pk=
,
c(p) =
x= (xk)∈w: lim k→∞|xk|
pk=
,
l(p) =
x= (xk)∈w:
k
|xk|pk<∞
( <pk<∞).
Let the functionshandhbe defined on the spacesl∞(p),c(p) orc(p), andl(p) by
h(x) =sup
k∈N
|xk|
pk
M and h(x) =
k
|xk|pk
M
.
The sequence spacesc(p) andc(p) are complete paranormed spaces paranormed byh ifp∈l∞([], Theorem ).l∞(p) is also complete paranormed space byhif and only if
pk> , and alsol(p) is complete paranormed space paranormed byh and{e(k)}k∈N is a
The domain of an infinite matrixAin a sequence spaceλis denoted byλAwhere
λA=
x= (xk)∈w:Ax∈λ
. ()
It is obvious thatλAis a sequence space.
Let (X, · ) be a normed space andX⊃φbe aBK-space anda= (ak)∈w, then theα-,
β-, andγ-dual of a subsetXofware, respectively, defined by
Xα=a= (ak)∈w:ax= (akxk)∈lfor allx= (xk)∈X
,
Xβ=a= (ak)∈w:ax= (akxk)∈csfor allx= (xk)∈X
,
Xγ=a= (ak)∈w:ax= (akxk)∈bsfor allx= (xk)∈X
.
IfAis an infinite matrix with complex entriesank (n,k∈N), then we writeA= (ank). We
define theA-transform ofxas the sequenceAx= (An(x))∞n=, where
An(x) =
∞
k=
ankxk (n∈N),
ifx= (xk)∈wand provided the series on the right-hand side converges for eachn∈N.
IfXandYare subsets ofwandA= (ank) is an infinite matrix, thenAdefines a matrix
mapping fromXintoY, and we denote it byA:X→Y, ifAxexists and is inYfor allx∈X. We denote the class of all infinite matrices that mapXintoY by (X,Y). So,A∈(X,Y) if and only ifAn∈Xβfor allx∈X(we writeAnfor the sequence in thenth row ofA,i.e.,
An= (ank)∞k=for everyn∈N[].
The following results are fundamental for our work.
Remark .
(a) ([]) Let†denote any of the symbolsα,βorγ. Then we havec†=c†=l† ∞=l,
l†=l∞, andlp†=lqwhere <p<∞andq=p/(p– ).
(b) ([]) LetXbe any of the spacesc,c,l∞orlp(≤p<∞). Then · Xβ denotes the natural norm on the dual spaceXβ.
(c) ([]) LetX⊃φandYbe aBK-space. Then we have the following:
(c) (X,Y)⊂B(X,Y), that is, every matrixA∈(X,Y)defines an operatorLA∈B(X,
Y)byLA(x) =Axfor allx∈X.
(c) IfXhasAK, thenB(X,Y)⊂(X,Y), that is, for every operatorLA∈B(X,Y)there
exists a matrixA∈(X,Y)such thatLA(x) =Axfor allx∈X.
Remark .([]) LetX⊃φbe aBK-space. Then:
(a) IfYis any of the spacesc,corl∞, andA∈(X,Y), then
LA=A(X,l∞)=sup
n
An∗X<∞.
(b) IfA∈(X,l), then
where
A(X,l)=sup
N∈F
n∈N
An
∗
X
<∞,
whereFdenotes the collection of all finite subsets ofN, andFr(r∈N) is the
subcollection ofFconsisting of all nonempty subsets ofNwith elements grater than r, that is,
Fr={N∈F:n>rfor alln∈N} (r∈N).
2 The sequence spacesl∞(B˜,p),c(B˜,p), andc0(B˜,p)
In this section we define the sequence spaces of non-absolute type that derived by the dou-ble sequential band matrixB(˜r,˜s). These sequence spaces are the complete paranormed linear spaces.
Letk,n∈Nand˜r= (rk), and˜s= (sk) be the convergent sequences whose entries are
ei-ther constant or distinct non-zero numbers. Then we define the double sequential matrix
B(r˜,s˜) ={bnk(rk,sk)}∞n,k=by
bnk=
⎧ ⎪ ⎨ ⎪ ⎩
rk, k=n,
sk, k=n– ,
, otherwise.
Recently, Ozger and Basar [] introduced the sequence spaces l∞(B˜,p), c(B˜,p), and
c(B˜,p) as follows:
l∞(B˜,p) =
x= (xk)∈w:sup k∈N
|rkxk+sk–xk–|pk<∞
,
c(B˜,p) =
x= (xk)∈w:∃l∈C lim
k→∞|rkxk+sk–xk––l| pk=
,
c(B˜,p) =
x= (xk)∈w: lim
k→∞|rkxk+sk–xk–| pk=
,
and theB(r˜,˜s)-transforms of these spaces are in the spacesl∞(p),c(p) andc(p), respec-tively.
Because of the notation () we have
l∞(B˜,p) =l∞(p)B˜, c(B˜,p) =
c(p)B˜ and c(B˜,p) =
c(p)
˜
B.
Throughout, we define the sequencey= (yk) by theB(r˜,s˜)-transform of a sequencex= (xk),
that is,
yk=
B(r˜,s˜)xk=rkxk+sk–xk–, ∀k∈N. ()
connected by (). Let (X, · ) be a normed space. IfX⊃φis aBK-space anda= (ak)∈w
then we write
a∗X=sup
x∈SX
∞
k=
akxk
, ()
provided the expression on the right-hand side exists and is finite. Now, we have the following result.
Lemma . The sequence spaces l∞(B˜,p),c(B˜,p),and c(B˜,p)are BK -spaces with the same paranorm given by
xl∞(B˜,p)=B(r˜,˜s)(x)l∞(p)=sup
k∈N
Bk(˜r,s˜)(x)
pk
M, ()
and they are linearly isomorphic to the spaces l∞(p),c(p),and c(p).
Proof See Theorem . in [].
Now, we have the following lemma by the previous result.
Lemma . Let X denote any of the spaces l∞(B˜,p),c(B˜,p)or c(B˜,p).If a= (ak)∈Xβ,then
˜
a= (a˜k)∈l(p)and the equality
∞
k=akxk=
∞
k= ˜
akyk ()
holds for every x= (xk)∈X,where y=B(r˜,˜s)(x)is the associated sequence defined by()
and
˜
ak= k
m= (–)k–m
rk k–
j=m
sj
rj
an. ()
Proof The equality () derived by using equation () from themth partial sum of the series
kankxk, so m
k=
ankxk= m–
k= ˜
ankyk+
∞
k=m
(–)m–k
rm m–
j=k
sj
rj
anj
ym, ()
that the summation running from to m– is equal to zero whenm= . Now, when
m→ ∞we obtain
k
akxk=
k
˜
akyk, ∀n∈N.
Lemma . If X denotes any of the spaces l∞(B˜,p),c(B˜,p)or c(B˜,p),then
a∗X=˜aXβ=˜al(p)= ∞
k= |˜ak|pk
M
<∞, ()
Proof Leta= (ak)∈Xβ. Then, by using Lemma . we havea˜= (a˜k)∈l(p) andy=yk∈Y
(we assume thatYare, respectively, the spacesl∞(p),c(p), andc(p)), which are connected
by equation (). Further, it follows by () thatx∈SXif and only ify∈SY. Therefore, we
derive from () and () that
a∗X=sup
x∈SX
∞
k=
akxk
=ysup∈SY
∞
k= ˜
akyk
=˜a∗Y, ()
and sincea˜∈l(p), we obtain from Remark . (parts (a) and (b))
a∗X=˜a∗Y=˜al(p)<∞. ()
This completes the proof.
Lemma . Let X be any of the spaces l∞(B˜,p),c(B˜,p)or c(B˜,p),and Y be,respectively,the spaces l∞(p),c(p)or c(p)and V be a sequence space and A= (ank)be an infinite matrix.If
A∈(X,V),thenA˜∈(Y,V)such that Ax=Ay for all sequences x˜ ∈X and y∈Y which are connected by equation(),whereA˜ = (a˜nk)is the associated matrix defined by
˜
ank=
∞
i=k
(–)i
ri i–
j=k
sj
rj
ani (∀n∈N). ()
Proof Letx∈X andy∈Y be connected by equation () and suppose thatA∈(X,V). Then, by using Lemma ., we obtainA˜n∈l(p) =Xβfor alln∈Nand the equalityAx=Ay˜
holds, hence Ay˜ ∈V. Since everyy∈Y is the associated sequence of somex∈X, we
deduce thatA˜∈(Y,V). This completes the proof.
Theorem . Let A= (ank)be an infinite matrix andA˜ = (a˜nk)be the associated matrix
defined by()and let X be any of the spaces l∞(B˜,p),c(B˜,p)or c(B˜,p).If A is in any of the classes(X,c(p)), (X,c(p))or(X,l∞(p)),then
LA=A(X,l∞(p))=sup
n∈N
∞
n= |˜ank|pk
M
<∞. ()
Proof By combining Remark .(a) and Lemma ., we obtain the equality in ().
3 The Hausdorff measure of noncompactness
Let (X, · ) be a normed space. Then the unit sphere and the closed unit ball inXare denoted bySX={x∈X:x= },BX={x∈X:x ≤}. For the Banach spacesXandY
we denote the set of all bounded linear operatorsL:X→Y byB(X,Y) with the operator norm given byL=supx∈S
XL(x). A linear operatorL:X→Yis said to be compact if
the domain ofLis all ofXand, for every bounded sequence (xn) inX, the sequenceL(xn)
has a subsequence which converges inY. We denote the class of all compact operators inB(X,Y) byC(X,Y). An operatorL∈B(X,Y) is said to be of finite rank ifdimR(L) <∞, whereR(L) denotes the range ofL. An operator of finite rank is clearly compact [].
LetSandMbe subsets of a metric space (X,d) and > . ThenSis called an -net ofM
inXif for everyx∈Mthere existss∈Ssuch thatd(x,s) < . If the setSis finite, then the -netSofMis called a finite -net ofM, and we say thatMhas a finite -net inX. A subset of a metric space is said to be totally bounded if it has a finite -net for every > [].
We denote the collection of all bounded subsets of a metric space (X,d) byMX. IfQ∈
MX, then the Hausdorff measure of noncompactness of the setQ, denoted byχ(Q), is
defined by
χ(Q) =inf{ > :Qhas a finite -net inX}. ()
The functionχ:MX→[,∞) is called the Hausdorff measure of noncompactness.
The basic properties of the Hausdorff measure of noncompactness can be found in [, , ]. For example, ifQ,Q, andQare bounded subsets of a metric space (X,d), then:
() χ(Q) = if and only ifQis totally bounded, () Q⊂Qimpliesχ(Q)≤χ(Q).
Further, ifX is a normed space, then the functionχ has some additional properties connected with the linear structure, that is,
() χ(Q+Q)≤χ(Q) +χ(Q),
() χ(αQ) =|α|χ(Q),∀α∈C.
LetXandY be Banach spaces andL∈B(X,Y). Then the Hausdorff measure of non-compactness ofL, denoted byLχ, is defined by
Lχ=χ
L(SX)
()
and
Lχ= if and only if L∈C(X,Y) []. ()
The following theorem gives an estimate for the Hausdorff measure of noncompactness in Banach spaces with Schauder bases and in this theoremIdenotes the identity operator onX.
Theorem . ([]) Let X be a Banach space with a Schauder basis(bk)∞k=, Q∈MX,
and Pn:X→X(n∈N)be the projector onto the linear span of{b,b, . . . ,bn}.Then we
have
a·lim supn→∞
sup
x∈Q
(I–Pn)(x)≤χ(Q)≤lim sup
n→∞
sup
x∈Q
(I–Pn)(x),
Further, the following result shows how to compute the Hausdorff measure of noncom-pactness in the spacescandlp(≤p<∞) which areBK-spaces withAK.
Theorem .([]) Let X be the normed space that lp for≤p<∞or c and Q be
a bounded subset of X.If Pn:X→X (n∈N)is the operator defined by Pn(x) =x[n]=
(x,x, . . . ,xn, , , . . .)for all x= (xk)∞k=∈X,then we have
χ(Q) =lim sup
n→∞
sup
x∈Q
(I–Pn)(x).
An infinite matrixT= (tnk) is called a triangle iftnn= andtnk= for allk>n(n∈N).
Lemma .([]) Let T be a triangle.Then we have:
(a) For arbitrary subsetsXandY ofw,A∈(X,YT)if and only ifB=TA∈(X,Y).
(b) Further,ifXandYareBK-spaces andA∈(X,YT),thenLA=LB.
4 Compact operators on the spacesl∞(B˜,p),c(B˜,p), andc0(B˜,p)
In this section, we establish some identities or estimates for the Hausdorff measure of noncompactness of certain operators on the spaces l∞(B˜,p), c(B˜,p), and c(B˜,p).
Also, we apply our results to characterize some classes of compact operators on those spaces.
We mentioned the following lemmas, which will be used in proving our results.
Lemma .([]) Let X⊃φ be a BK -space with AK or X=l∞.If A∈(X,C),then the following hold:
αk= lim
n→∞ank exists for every k∈N, ()
α= (αk)∈Xβ, ()
sup
n
An–α∗X<∞,
lim
n→∞An(x) = ∞
k=
αkxk for all x= (xk)∈X. ()
Lemma .([]) Let X⊃φbe a BK -space.Then we have:
(a) IfA∈(X,c),then
LAχ= lim
r→∞
sup
n>r
An∗X
. ()
(b) IfXhasAKorX=l∞andA∈(X,c),then
·rlim→∞
sup
n≥r
An–α∗X
≤ LAχ≤ lim
r→∞
sup
n≥r
An–α∗X
, ()
whereα= (αk)withαk=limn→∞ankfor allk∈N.
(c) IfA∈(X,l∞),then
≤ LAχ≤ lim
r→∞
sup
n>r
An∗X
Theorem . Let X denote any of the spaces l∞(B˜,p),c(B˜,p)or c(B˜,p),and q= (qk)be a
bounded sequence of strictly positive real numbers.Then we have:
(a) IfA∈(X,c(q)),then
LAχ=lim sup
n→∞ ∞
k= |˜ank|pk
M
()
and
LAis compact if and only if lim n→∞
∞
k= |˜ank|pk
M
= . ()
(b) IfA∈(X,c(q)),then we have
·lim supn→∞ ∞
k=
|˜ank–α˜k|pk
M
≤ LAχ≤lim sup
n→∞ ∞
k=
|˜ank–α˜k|pk
M
()
and
LAis compact if and only if lim n→∞
∞
k=
|˜ank–α˜k|pk
M
= , ()
wherelimn→∞a˜nk=α˜k.
(c) IfA∈(X,l∞(q)),then
≤ LAχ≤lim sup
n→∞
∞
k= |˜ank|pk
M
()
and
LAis compact if and only if lim n→∞
∞
k= |˜ank|pk
M
= . ()
Proof We begin with the proof of parts (a) and (c). We know,A∈(X,c(q)) if and only if
An∈Xβ(∀n∈N) andAx∈c(p) for allx∈X. So, from Lemma .
An∗X= ˜AnXβ= ˜Anl(q)= ∞
k=
|˜ank| ()
for allk∈N. Thus, we get () from () and Lemma .(a). Similarly, we can get () from () and Lemma .(c).
Further, ifAis an infinite matrix defined by
ank=
⎧ ⎪ ⎨ ⎪ ⎩
rk, k=n,
sk, k=n– ,
then
Ax=ann–xn–en–+annxnen=sn–xn–en–+rnxnen.
So, by () it is easy to see thata˜nk= . ThenA˜n= and
lim
n→∞ ˜Anl(q)=nlim→∞
∞
k= |˜ank|pk
M
= .
Hence, by combining the latter and () we get () and ().
To prove (b) we combine Lemma . and Lemma .(b) and we have (). We writeS=SX, for short. Then we obtain by () and Remark .(c)
LAχ=χ(AS). ()
So we haveAS∈Mc(p)(whereMcis the class of all bounded subsets ofc(q)). Thus, we are
going to apply Theorem . to get an estimate for the value ofχ(AS) in (). To this aim, we define the projectorsPr:c(q)→c(q) (r∈N) byP(z) =zeandPr(z) =ze+
r–
n=(zn–
z)e(n)forr≥ (wherez= (z
k)∈c(q) andz=limn→∞zn). Then we have for everyr∈N,
(I–Pr)(z) =
∞
n=r(zn–z)e(n)and hence
(I–Pr)(z)
l∞(p)=sup
n≥r|
zn–z|
pk
M ()
for allz∈c(q) and everyr∈N. Thus, from () and Theorem ., we get
·rlim→∞
sup
x∈S
(I–Pr)(Ax)l∞(p)
≤ LAχ≤ lim
r→∞
sup
x∈S
(I–Pr)(Ax)l∞(p)
. ()
Now, for every given x∈X, lety∈Y be an associated sequence space defined by (), whereYis, respectively, the spacesl∞(q),c(q) orc(q). It is given thatA∈(X,c(q)), then by
Lemma . we haveA˜ ∈(Y,c(q)) andAx=Ay˜ . Further, it follows from Lemma . that the limitsα˜k=limn→∞a˜nk exist for allk,α˜= (α˜k)∈l(q) =Yβandlimn→∞A˜n(Y) =
∞
k=α˜kyk.
Therefore, we derive from () that
(I–Pr)(Ax)l∞(q)=(I–Pr)(Ay˜ )l∞(q) ()
=sup
n
A˜n(y) –
∞
k= ˜
αkyk
pk M
()
=sup
n
∞
k=
(a˜nk–α˜k)yk
pk M
()
forr∈N. Sincex∈S=SXif and only ify∈SY, we obtain by () and Remark .(c)
sup(I–Pr)(Ax)l∞(q)=sup n>r
sup
∞
k=
(a˜nk–α˜k)yk
pk M
=sup
n ˜
An–α˜∗Y=sup n ˜
for allr∈N. So, we obtain () and () by () and (), respectively, and this completes
the proof.
The following lemmas are necessary for the next theorem.
Lemma .([]) Let x= (xn)∈l.Then the inequalities
sup
N∈Fr
n∈N
xn
≤
∞
n=r+
|xn| ≤·sup N∈Fr
n∈N
xn
()
hold for every r∈N.
Lemma .([]) Let X⊃φbe a BK -space.If A∈(X,l),then
lim
r→∞
sup
N∈Fr
n∈N
An
∗
X
≤ LAχ≤· lim
r→∞
sup
N∈Fr
n∈N
An ∗ X () and
LAis compact if and only if lim r→∞
sup
N∈Fr
n∈N
An ∗ X = . ()
Theorem . Let X denote any of the spaces l∞(B˜,p),c(B˜,p)or c(B˜,p).If A∈(X,l(q)),
then
lim
r→∞
A((rX),l(q))≤ LAχ≤· lim
r→∞
A((rX),l(q)), ()
where
A((rX),l(q))=sup
N∈Fr ∞
k=
n∈N ˜ ank pk M
(r∈N)
and
LAis compact if and only if lim r→∞
A((rX),l(q))= .
Proof SinceF⊃F⊃F⊃. . . , by using Remark .(b), the sequence (A((rX),l(q)))
∞
r=of non-negative reals is nonincreasing and bounded. So, the limit in () exists. Now, we write
S=SXfor short. Then we haveLA(S) =AS∈Ml(q) by Remark .(c). So, it follows from () and Theorem . that
LAχ=χ(AS) = lim
r→∞
sup
x∈S
∞
n=r+
An(x) pk
. ()
By Lemma ., we have
sup
N∈Fr
n∈N
An(x)
pk≤
∞
n=r+
An(x) pk≤
·sup
N∈Fr
n∈N
An(x)
for allx∈Xand everyr∈N, becauseA∈(X,l(q)). SinceAn∈Xβfor alln∈N, we derive
from () and Lemma . that
sup
x∈S
n∈N
An(x)
pk M
=sup
x∈S
∞ k=
n∈N
ank xk
pk M =
n∈N
An pk M ∗ X =
n∈N
˜
An
pk M
l(q)
for allN∈Fr(r∈N). These relations, together with (), imply
sup
N∈Fr
n∈N ˜
An
pk M
l(q)
≤sup
x∈S
∞
n=r+
An(x)
pk M
≤·sup
N∈Fr
n∈N
˜
An
pk M
l(q)
()
for everyr∈N. Thus, by passing to the limits in () asr→ ∞and using (), we get ().
This completes the proof.
5 Some applications
In this section we obtain some estimates or identities for the operator norms of certain matrix operators and we deduce the necessary and sufficient conditions for such operators to be compact. Furthermore, we obtain these results by using the Hausdorff measure of the noncompactness technique.
Corollary . Let X denote any of the spaces l∞(B˜,p),c(B˜,p)or c(B˜,p),and A be an infinite matrix.If A is in any of the classes(X,cs), (X,cs)or(X,bs),then we have
LA=sup n ∞ k= n m= ˜ amk
pkM
<∞ (n∈N). ()
Further:
(a) IfA∈(X,cs),then
LAχ=lim sup
n→∞ ∞ k= n m= ˜ amk
pkM
()
and
LAis compact if and only if lim n→∞ ∞ k= n m= ˜ amk
pkM
(b) IfA∈(X,cs),then
·lim supn→∞
∞ k= n m= ˜
amk–a˜k
pkM
≤ LAχ≤lim sup
n→∞ ∞ k= n m= ˜
amk–a˜k
pkM
, ()
where
˜
a= (a˜k) witha˜k= lim n→∞ n m= ˜ amk
for allk∈N
and
LAis compact if and only if lim n→∞ ∞ k= n m= ˜
amk–a˜k
pkM
= . ()
(c) IfA∈(X,bs),then
≤ LAχ≤lim sup
n→∞ ∞ k= n m= ˜ amk
pkM
()
and
LAis compact if and only if lim n→∞ ∞ k= n m= ˜ amk
pkM
= . ()
Proof The equality in () is immediate by Theorem .. By using Theorem . in [] in the new spacesl∞(B˜,p),c(B˜,p), andc(B˜,p) (forX) and by applying Lemma . and these
relations withcs= (c)s,cs= (c)s, and (bs) = (l∞)swe obtain (), (), and (). Then by
using Theorem . in [] and Lemma . we obtain (), (), and ().
We denote the space of all sequences of bounded variation bybυ, that is,
bυ=x= (xk)∈w: (xk–xk–)∈l
.
Corollary . Let X denote any of the spaces l∞(B˜,p),c(B˜,p)or c(B˜,p),and A be an infi-nite matrix.If A∈(X,bυ),then
sup
N∈F
∞
k=
n∈N
(a˜nk–a˜(n–)(k–))pk
M
≤ LA ≤·sup N∈F
∞
k=
n∈N
(a˜nk–a˜(n–)(k–))pk
M
Furthermore,if A∈(X,bυ),then
lim
r→∞
sup
N∈Fr ∞
k=
n∈N
(a˜nk–a˜(n–)(k–))pk
M
≤ LAχ≤· lim
r→∞
sup
N∈Fr ∞
k=
n∈N
(a˜nk–a˜(n–)(k–))pk
M
and
LAis compact if and only if lim r→∞
sup
N∈Fr ∞
k=
n∈N
(a˜nk–a˜(n–)(k–))
pk
M
= .
Proof See Theorem . in [] and Theorem ..
We writebυpfor the space of all sequences ofp-bounded variation, that is,
bυp=x= (xk)∈w: (xk–xk–)∈lp
( <p<∞).
bυpis aBK-space with its natural norm (cf.[]). For everya=a
k∈(bυp)β, we have
a∗bυp= ∞ k= ∞
j=k
aj
qq
. ()
So, by using () we obtain the following consequence of Theorem . in [] in the spaces
l∞(B˜,p),c(B˜,p), andc(B˜,p).
Corollary . Let X denote any of the spaces l∞(B˜,p)and c(B˜,p), <p<∞,q= (qk)be
a bounded sequence of strictly positive real numbers and A is an infinite matrix.If A∈
(bυp,X),then
LA=A((br)υp,l∞(B˜,p))=sup
n>r
∞ k= ∞
j=k
˜
anj
qkqkpkM
(r∈N).
Further:
(a) IfA∈(bυp,l∞(B˜,p)),then
≤ LAχ≤ lim
r→∞
sup
n>r
∞ k= ∞
j=k
˜
anj
qkqkpkM
and
LAis compact if and only if lim r→∞
sup
n>r
∞ k= ∞
j=k
˜
anj
qkqkpkM
(b) IfA∈(bυp,c(B˜,p)),then
LAχ= lim
r→∞
sup
n>r
∞
k=
∞
j=k
˜
anj
qkqkpkM
and
LAis compact if and only if lim r→∞
sup
n>r
∞
k=
∞
j=k
˜
anj
qkqkpkM
= .
Proof The proof is a special case of Theorem . in [] in the new spacesl∞(B˜,p) and c(B˜,p) whenX=bυp. Now by using Lemma . the proof is complete. Now, we consider some relations betweenX(whereXis any of the spacesl∞(B˜,p),c(B˜,p)
orc(B˜,p)) and some another sequence spaces derived by the domain of the triple band matrix. First we introduce the sequence spacesl∞(B),c(B),c(B), andlp(B) as the set of all
sequences whoseB(r,s,t)-transforms are in the spacesl∞,c,c, andlp, respectively (for
more details refer to []),
l∞(B) =
x= (xk)∈w:sup k∈N
|rxk+sxk–+txk–|<∞
,
c(B) =x= (xk)∈w:∃l∈Clim|rxk+sxk–+txk––l|=
,
c(B) =
x= (xk)∈w:lim|rxk+sxk–+txk–|=
,
lp(B) =
x= (xk)∈w:
k
|rxk+sxk–+txk–|p<∞
.
We now consider the special case of Theorems . and . in [] whenT=B(r˜,s˜), and
X=lp(B).
Corollary . Let X denote any of the spaces l∞(B˜,p),c(B˜,p)or c(B˜,p),and A be an infi-nite matrix.If A∈(lq(B),X),then
LA=A(lq(B),l∞(B˜,p))=sup
n
∞
k=
|˜ank|pkqk
qkM
<∞,
where q= (qk)is a bounded sequence of strictly positive real numbers.
Further:
(a) IfA∈(lq(B),c(B˜,p)),then
LAχ=lim sup
n→∞ ∞
k=
|˜ank|pkqk
qkM
<∞
and
LAis compact if and only if lim sup n→∞
∞
k=
|˜ank|pkqk
qkM
(b) IfA∈(lq(B),c(B˜,p)),then
·lim supn→∞
∞
k=
|˜ank–α˜k|pkqk
qkM
≤ LAχ≤lim sup
n→∞ ∞
k=
|˜ank–α˜k|pkqk
qkM
,
whereα˜k= (α˜nk)withlimn→∞a˜nk=α˜kand
LAis compact if and only if lim sup n→∞
∞
k=
|˜ank–α˜k|pkqk
qkM
= .
(c) IfA∈(lq(B),l∞(B˜,p)),then
≤ LAχ≤lim sup
n→∞
∞
k=
|˜ank|pkqk
qkM
<∞
and
LAis compact if and only if lim sup n→∞
∞
k=
|˜ank|pkqk
qkM
= .
Example . LetXbe similar to Corollary . andY be one of the spacesc(B),c(B) or
l∞(B), andAbe an infinite matrix. IfA∈(X,Y), then
LA=A(X,l∞(B))=sup
n
∞
k= |˜ank|pk
<∞,
whereA˜= (a˜nk) is the associated matrix defined by
˜
ank=
∞
j=k k–j
i=
–s+√s– tr r
j–k–i
–s–√s– tr r
i
anj
r .
Further:
(a) IfA∈(X,c(B)), then
LAχ=lim sup
n
∞
k= |˜ank|pk
and
LAis compact if and only if lim sup n
∞
k= |˜ank|pk
(b) IfA∈(X,c(B)), then
·lim supn
∞
k=
|˜ank–α˜k|pk
≤ LAχ≤lim sup
n
∞
k=
|˜ank–α˜k|pk
,
whereα˜k= (α˜nk)withlimn→∞a˜nk=α˜k, and
LAis compact if and only if lim sup n
∞
k=
|˜ank–α˜k|pk
= .
(c) IfA∈(X,l∞(B)), then
≤ LAχ≤lim sup
n
∞
k= |˜ank|pk
and
LAis compact if and only if lim sup n
∞
k= |˜ank|pk
= .
Proof The proof is a consequence of Theorem ..
The final result is a special case of Theorem . in [] in the new spacesl∞(B˜,p),c(B˜,p), andc(B˜,p).
Corollary . Let X and Y be similar to Example.and A be an infinite matrix.If A∈
(Y,X),then
LA=A(Y,l∞(B˜,p))=sup
n
∞
k= |˜ank|pk
M
<∞.
Further:
(a) IfA∈(Y,c(B˜,p)),then
LAχ=lim sup
n
∞
k= |˜ank|pk
M
and
LAis compact if and only if lim sup n
∞
k= |˜ank|pk
M
= .
(b) IfA∈(Y,c(B˜,p)),then
·lim supn
∞
k=
|˜ank–α˜k|pk
M
≤ LAχ≤lim sup
n
∞
k=
|˜ank–α˜k|pk
M
whereα˜k= (α˜nk)withlimn→∞a˜nk=α˜k,and
LAis compact if and only if lim sup n
∞
k=
|˜ank–α˜k|pk
M
= .
(c) IfA∈(Y,l∞(B˜,p)),then
≤ LAχ≤lim sup
n
∞
k= |˜ank|pk
M
and
LAis compact if and only if lim sup n
∞
k= |˜ank|pk
M
= .
Competing interests
The authors declare that they have no competing interests.
Authors’ contributions
All authors of the manuscript have read and agreed to its content and are accountable for all aspects of the accuracy and integrity of the manuscript in accordance with ICMJE criteria.
Acknowledgements
The authors would like to thank the anonymous reviewer for his/her helpful feedback and valuable suggestions, which led to an improvement in the quality of their paper.
Received: 15 August 2015 Accepted: 26 November 2015
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