R E S E A R C H
Open Access
Compact matrix operators on a new
sequence space related to
p
spaces
Abdullah Alotaibi
1, Mohammad Mursaleen
2*and Syed Abdul Mohiuddine
1*Correspondence:
2Department of Mathematics,
Aligarh Muslim University, Aligarh, 202002, India
Full list of author information is available at the end of the article
Abstract
In this paper, we derive some identities or estimates for the operator norms and the Hausdorff measures of noncompactness of certain matrix operators on the sequence space
p(r,s,t;B(m)) which is related to
pspaces. By applying the Hausdorff measure of noncompactness, we obtain the necessary and sufficient conditions for such operators to be compact. Further, we study some geometric properties of this space.
MSC: 46B15; 46B45; 46B50
Keywords: BKspace; matrix transformation; Hausdorff measure of noncompactness; compact operator; geometric properties
1 Background, notation, and preliminaries
Letwdenote the space of all complex sequencesx= (xk)∞k=. By∞,c,c, andφwe denote
the sets of all bounded, convergent, null, and finite sequences, respectively. We writecs for the set of all convergent series and
p=
x∈ω:
∞
k=
|xk|p<∞
for ≤p<∞.
Byeande(n)(n= , , . . .), we denote the sequences withe
k= for allk, ande(nn)= and
e(kn)= fork=n, respectively. For any sequencex= (xk)∞k=, let
x[m]=
m
k=
xke(k)
be itsm-section.
Letxandybe sequences,XandYbe subsets ofωandA= (ank)∞n,k=be an infinite matrix
of complex numbers. We writexy= (xkyk)∞k=,
Z=x–∗Y={x∈ω:xz∈Y}, xβ=x–∗cs,
Xβ=
x∈X
x–∗cs=
a∈ω:
∞
k=
akxkconverges for allx∈X
for theβ-dual ofX. Note thatβ∞=cβ=cβ
=,β =∞and
β
p =q.
ByAn= (ank)∞k=andAk= (ank)∞n=we denote the sequences in thenth row and thekth
column ofA, and we write
An(x) =
∞
k=
ankxk (n= , , . . .) (.)
andA(x) = (An(x))∞n=, providedAn∈Xβfor alln. The setXA=X(A) ={z∈ω:A(z)∈X}is
called the matrix domain ofAinX. Finally (X,Y) denotes the class of all matricesAthat mapXintoY, that is for whichAn∈Xβfor allnandA(x)∈Yfor allx∈X, or equivalently
A∈(X,Y) if and only ifX⊂XA.
The theory of BK spaces is the most powerful tool in the characterization of matrix transformations between sequence spaces.
A sequence spaceX is called aBK space if it is a Banach space with continuous co-ordinatespn:X→C(n∈N), whereCdenotes the complex field andpn(x) =xnfor all
x= (xk)∈X and everyn∈N. ABK spaceX⊃φ is said to haveAK if every sequence
x= (xk)∈Xhas a unique representationx= ∞
k=xke(k).
The sequence spaces ∞, c, andc areBK spaces with the same sup-norm given by
x∞=supk|xk|, where the supremum is taken over allk∈N. Further, the spacep is
aBK space with the usualp-norm defined byxp= (
∞
k=|xk|p)/p, where ≤p<∞.
Moreover, theBKspacescandp(≤p<∞) haveAK[], Examples ., ..
LetXandY be Banach spaces. ThenB(X,Y) is the set of all bounded linear operators L:X→Y, a Banach space with the operator norm defined as usual by
L=supL(x)/x ≤ L∈B(X,Y).
IfY=Cthen we writeX∗for the space of all continuous linear functionals onXwith the norm defined by
f=supf(x):x ≤ f ∈X∗.
IfX⊂wis a normed space anda∈wthen we write
a∗=a∗X=sup
∞
k=
akxk
:x ≤
, (.)
provided the expression on the right-hand side exists and is finite which is the case when-everX⊃φis aBK space anda∈Xβ[], p..
For any subsetXofw, the matrix domain of an infinite matrixAinXis defined by
XA={x∈w:Ax∈X}.
An infinite matrixT= (tnk) is called a triangle iftnn= andtnk= for allk>n(n∈N).
The study of matrix domains of triangles in sequence spaces has a special importance due to the various properties which they have. For example, ifXis aBKspace thenXTis also
aBKspace with the norm given byxXT=TxXfor allx∈XT[], Theorem ...
Lemma .
(a) LetXdenote any of the spacesc,c,∞,orp.Then we havea∗X=aXβ for all a∈Xβ,where ·
Xβis the natural norm on the dual spaceXβ.
(b) LetXandYbeBK spaces.Then every matrixA∈(X,Y)defines an operator LA∈B(X,Y)byLA(x) =Axfor allx∈X;we denote this by(X,Y)⊂B(X,Y). (c) LetX⊃φbe aBKspace andYbe any of the spacesc,cor∞.IfA∈(X,Y),then
LA=A(X,∞)=sup
n
An∗X<∞.
Also, let F be the collection of all non-empty and finite subsets ofN={, , , . . .}, throughout. Then we have the following result.
Lemma . Let X⊃φbe a BK space.If A∈(X,),then
A(X,)≤ LA ≤· A(X,),
whereA(X,)=supN∈F
n∈NAn∗X<∞.
For the reader’s convenience, we list a few well-known definitions and results concerning the Hausdorff measure of noncompactness which can be found in [, ], and [].
LetSandMbe subsets of a metric space (X,d) andε> . ThenSis called anε-netofM inXif for everyx∈Mthere existss∈Ssuch thatd(x,s) <ε. Further, if the setSis finite, then theε-netSofMis called afiniteε-netofM, and we say thatMhas a finiteε-net inX. A subsetMof a metric spaceXis said to betotally boundedif it has a finiteε-net for every
ε> . IfXis complete, thenMis totally bounded if and only ifMis relatively compact (its closureM¯ is a compact set). LetXandY be Banach spaces. A linear operatorL:X→Y is called compact ifD(L) =Xfor the domain ofLand, for every bounded sequence (xn)∞n=
inX, the sequence (L(xn))∞n=has a convergent subsequence inY.
ByMX, we denote the collection of all bounded subsets of a metric space (X,d). IfQ∈
MX, then theHausdorff measure of noncompactnessofQ, is defined by
χ(Q) =inf{ε> :Qhas a finiteε-net inX}.
The functionχ:MX→[,∞) is called theHausdorff measure of noncompactness.
It is well known that ifQ,Q, andQare bounded subsets of a metric spaceX, then we
have
χ(Q) = if and only if Qis totally bounded,
Q⊂Q implies χ(Q)≤χ(Q),
χ(Q) =χ(Q) for the closureQofQ,
χ(Q∪Q) =max
χ(Q),χ(Q)
,
χ(Q∩Q)≤min
χ(Q),χ(Q)
.
Further, ifXis a normed space then we also have
χ(αQ) =|α|χ(Q) for allα∈C.
LetXandYbe Banach spaces andL∈B(X,Y). Then theHausdorff measure of
noncom-pactnessof the operatorL, denoted byLχ, is defined by
Lχ=χ
L(SX)
, (.)
whereS={x∈X:x ≤}is the unit ball inX. Also we have
Lis compact if and only if Lχ= . (.)
Now we shall point out the well-known result of Goldenštein, Gohberg, and Markus [], Theorem , concerning the Hausdorff measure of noncompactness in Banach spaces. The Hausdorff measure of noncompactness of a bounded subset of theBKspacep(≤p<∞)
is given by the following result.
Lemma . Let X be a BK space with AK and monotone norm,Q∈MX,and Pn:X→X
(n∈N)be the operator(projection)defined by Pn(x,x, . . .) =x[n]= (x,x, . . . ,xn, , , . . .)
for all x= (x,x, . . .)∈X.Then
χ(Q) = lim
n→∞
sup
x∈Q
(I–Pn)x.
For some recent related work on this topic, we refer to [–], and []. For some ap-plications in differential and integral equations, we refer to [, –], and [].
2 Results and discussion
Throughout this paper, letr,t∈Uands∈Uo, where
U=u= (uk)∈w:uk= for allk and Uo=
u= (uk)∈w:u= .
For any sequencex= (xn)∈w, we define the sequencex¯= (x¯n) of generalized means of
xby
¯ xn=
rn
n
k=
sn–ktkxk (n∈N). (.)
Further, we define the infinite matrixA(r,¯ s,t) of generalized means by
¯
A(r,s,t)nk=
⎧ ⎨ ⎩
sn–ktk/rn (≤k≤n),
(k>n) (.)
Moreover, it is obvious by (.) thatA(r,¯ s,t) is a triangle. Thus, it has a unique inverse (A(r,¯ s,t))–which is also a triangle. More precisely, we putD(s)= /sand
D(ns)= sn+
s s · · ·
s s s · · ·
s s s s · · ·
..
. ... ... ... ...
sn– sn– sn– sn– · · · s
sn sn– sn– sn– · · · s
(n= , , , . . .).
Then the entries of (A(r,¯ s,t))–are given by
¯
A(r,s,t)–nk=
⎧ ⎨ ⎩
(–)n–kD(ns–)krk/tn (≤k≤n),
(k>n)
(.)
for alln,k∈N, that is, (A(r,¯ s,t))–=A(t,¯ s,r), wheres= (s
n) such thatsn= (–)nD
(s)
n for all
n∈N[], p.. Therefore, we have by (.) that
xn=
tn
n
k=
(–)n–kD(ns–)krkx¯k (n∈N). (.)
For an arbitrary subsetXofw, the setX(r,s,t) has recently been introduced in [] as the matrix domain of the triangleA(r,¯ s,t) inX, that is,
X(r,s,t) =
x= (xk)∈w:y=
rn
n
k=
sn–ktkxk ∞
n=
∈X
.
It is obvious thatX(r,s,t) is a sequence space wheneverXis a sequence space, and we call it the sequence space of generalized means. Further, ifXis aBK space thenX¯ =X(r,s,t) is also aBK space with the norm given by
xX¯ =yX (x∈ ¯X). (.)
Recently, Maji and Srivastava [] have defined and studied the sequence spaceX(r,s, t;B(m)) forX∈ {
∞,c,c}which is obtained by combining the generalized means and the
mth order generalized difference operatorB(m)(u,v). They characterized some compact
operators on the spacesX(r,s,t;B(m)) forX∈ {
∞,c,c}by using the Hausdorff measure
of noncompactness. In this paper, we derive some identities or estimates for the operator norms and the Hausdorff measures of noncompactness of certain matrix operators on the sequence spacep(r,s,t;B(m)). By applying the Hausdorff measure of noncompactness, we
obtain the necessary and sufficient conditions for such operators to be compact. Further, we study some geometric properties of this space.
The generalized difference matrix of ordermdenoted asB(m)=B(m)(u,v) = (b(m)
nk ),u,v=
(see []) is defined as
b(nkm)=
⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩
m n–k
um–n+kvn–k (max{,n–m} ≤k≤n), (≤k<max{,n–m}),
The sequence spacesX(r,s,t;B(m)) forX∈ {
∞,c,c}are defined as follows:
Xr,s,t;B(m)=x= (xk)∈w: ¯
A(r,s,t)·B(m)x∞n=∈X . By using the matrix domain, we can write
Xr,s,t;B(m)=XA¯(r,s,t;B(m))=
x= (xk)∈w:A¯
r,s,t;B(m)x∈X , whereA(r,¯ s,t;B(m)) =A(r,¯ s,t)·B(m). The sequencex¯= (x¯
n) isA(r,¯ s,t)·B(m)-transform ofx=
(xn),i.e.
¯ xn=
n
j=
n
i=j
m i–j
sn–iti
rn
um+j–ivi–j
xj (n∈N).
In this paper, we are interested in the study of p(r,s,t;B(m)). We have the following
lemma which is immediate by Theorem . in [].
Lemma . If a= (ak)∈(p(r,s,t;B(m)))β,thena˜= (a˜k)∈
β
p (=q)and we have
∞
k=
akxk=
∞
k=
˜
akx¯k (.)
for all x= (xk)∈ ¯pwithx¯= (A(r,¯ s,t;B(m)))x,where
˜ ak=rk
ak
stkum
+
k+
i=k
(–)i–kD
(s)
i–k
ti
∞
j=k+
m+j–i– j–i
(–v)j–i
uj–i+maj
+
∞
i=k+
(–)i–kD
(s)
i–k
ti
∞
j=i
m+j–i– j–i
(–v)j–i uj–i+maj
. (.)
The following results will be needed in our study.
Lemma . We have
a∗¯
p=˜a ∗
p
for all a= (ak)∈ ¯
β
p,wherea˜= (a˜k)is the sequence defined by(.).
Proof Leta= (ak)∈ ¯βp. Then it follows by Lemma . thata˜= (a˜k)∈βp and the equality
(.) holds for all sequencesx= (xk)∈ ¯p andx¯= (x¯k)∈p which are connected by the
relationx¯= (A(r,¯ s,t;B(m)))x. Further, we see thatx∈S¯
pif and only ifx¯∈Sp. Therefore,
we derive from (.) and (.) that
a∗¯
p= sup
x∈S¯p
∞ k=
akxk =x¯sup∈Sp
∞ k= ˜ akx¯k
=˜a∗p.
Remark . By combining Lemmas . and ., we have the following:
(a) Ifa∈((r,s,t;B(m)))β, thena∗¯
=supk|˜ak|<∞.
(b) Ifa∈(p(r,s,t;B(m)))β, thena∗¯p= (
∞
k=|˜ak|q)/q<∞, whereq=p/(p– )and
<p<∞.
Throughout this paper, ifA= (ank) is an infinite matrix, we define theassociated matrix
˜
A= (a˜nk) by
˜ ank=rk
ank
stkum
+
k+
i=k
(–)i–kD
(s)
i–k
ti
∞
j=k+
m+j–i– j–i
(–v)j–i uj–i+manj
+
∞
i=k+
(–)i–kD
(s)
i–k
ti
∞
j=i
m+j–i– j–i
(–v)j–i uj–i+manj
(.)
provided the series on the right converge for alln,k∈N, which is the case wheneverAn∈
(p(r,s,t))βfor alln∈N[], Theorem .. Then we have the following.
Lemma . Let Y be a sequence space and A= (ank) be an infinite matrix. If A ∈
(p(r,s,t;B(m)),Y), then A˜ ∈(p,Y) such that Ax =A˜x for all x¯ ∈p(r,s,t) with x¯ =
(A(r,¯ s,t))x,whereA˜ = (a˜nk)is the associated matrix defined by(.).
Proof Suppose thatA∈(p(r,s,t;B(m)),Y) and letx∈p(r,s,t;B(m)). ThenAn∈(p(r,s,t;
B(m)))β for alln∈N. Thus, it follows by Lemma . thatA˜
n∈
β
p for alln∈Nand the
equalityAx=A˜x¯holds which yields thatA˜x¯∈Y, wherex¯is the sequence of generalized means of x,i.e.,x¯= (A(r,¯ s,t;B(m)))x. Further, it is obvious by (.) and Remark . that
everyx¯∈p is the sequence of generalized means of somex∈p(r,s,t;B(m)). Hence, we
deduce thatA˜∈(p,Y). This completes the proof.
Finally, we conclude this section by the following results on operator norms.
Theorem . Let A= (ank)an infinite matrix andA˜= (a˜nk)the associated matrix.If A is
in any of the classes(p(r,s,t;B(m)),∞), (p(r,s,t;B(m)),c)or(p(r,s,t;B(m)),c),then
LA=A(p(r,s,t;B(m)),∞)=sup
n ˜
An∗p<∞.
Proof This is immediate by combining Lemmas . and ..
Theorem . If A∈(p(r,s,t;B(m)),),then
A(p(r,s,t;B(m)),)≤ LA ≤· A(p(r,s,t;B(m)),),
where
A(p(r,s,t;B(m)), )=sup
N∈F
n∈N
˜ An
∗
p
<∞.
Theorem . If A∈((r,s,t;B(m)),p),then
LA=A((r,s,t;B(m)),p)=sup
k ∞
n=
|˜ank|p /p
<∞.
Proof The proof is elementary and left to the reader.
Remark . The characterizations of matrix classes considered in this paper can easily be obtained as in Corollaries . and . of []. Thus, we shall omit these characterizations and only deal with the operator norms and the Hausdorff measures of noncompactness of some matrix operators which are given by infinite matrices in such classes.
3 Main results
In this section, we derive some identities or estimates for the Hausdorff measures of non-compactness of certain matrix operators on the spaces of generalized means. Further, we apply our results to obtain the necessary and sufficient (or only sufficient) conditions for such operators to be compact.
We may begin with quoting the following lemma [], Theorem ..
Lemma . Let X⊃φbe a BK space.Then we have:
(a) IfA∈(X,∞),then
≤ LAχ≤lim sup
n→∞
An∗X.
(b) IfA∈(X,c),then
LAχ=lim sup
n→∞
An∗X.
(c) IfXhasAKorX=∞andA∈(X,c),then
·lim supn→∞
An–α∗X≤ LAχ≤lim sup
n→∞
An–α∗X,
whereα= (αk)withαk=limn→∞ankfor allk∈N.
Now, letA= (ank) be an infinite matrix andA˜ = (a˜nk) the associated matrix defined by
(.). Then, by combining Lemmas ., . and ., we have the following result.
Theorem . We have:
(a) IfA∈(p(r,s,t;B(m)),∞),then
≤ LAχ≤lim sup
n→∞ ˜An ∗
p (.)
and
LAis compact if lim n→∞ ˜An
∗
(b) IfA∈(p(r,s,t;B(m)),c),then
LAχ=lim sup
n→∞ ˜
An∗p (.)
and
LAis compact if and only if lim n→∞ ˜An
∗
p= . (.)
(c) IfA∈(p(r,s,t;B(m)),c),then
·lim supn→∞ ˜
An–α˜∗p≤ LAχ≤lim sup
n→∞ ˜
An–α˜∗p (.)
and
LAis compact if and only if lim
n→∞ ˜An–α˜
∗
p= , (.)
whereα˜= (α˜k)withα˜k=limn→∞a˜nkfor allk∈N.
Proof Note that parts (a) and (b) are proved in []. Further it is obvious that (.), (.)
and (.) are, respectively, obtained from (.), (.), and (.) by using (.). Thus, we have to prove (.), (.) and (.).
Sincep(r,s,t;B(m)) is aBKspace, we deduce by means of Lemma . that (.) and (.)
are immediate by parts (a) and (b) of Lemma ., respectively.
To prove (.), we haveA∈(p(r,s,t;B(m)),c) and henceA˜∈(X,c) by Lemma ..
There-fore, it follows by part (c) of Lemma . that
·lim supn→∞ ˜An–α˜ ∗
X≤ LA˜χ≤lim sup
n→∞ ˜An–α˜ ∗
X, (.)
whereα˜= (α˜k) andα˜k=limn→∞a˜nkfor allk∈N.
Now, let us write S=SX andS¯ =Sp(r,s,t;B(m)), for short. Then we obtain by (.) and
Lemma .
LAχ=χ
LA(S)¯
=χ(AS)¯ (.)
and
LA˜χ=χ
LA˜(S)
=χ(AS).˜ (.)
Further, we see thatx∈ ¯Sif and only ifx¯∈S, and sinceAx=A˜x¯by Lemma ., we deduce thatAS¯=AS. This leads us with (.) and (.) to the consequence that˜ LAχ=LA˜χ.
Hence, we get (.) from (.). This completes the proof.
It is worth mentioning that the condition in (.) is only a sufficient condition for the operatorLAto be compact, whereA∈(p(r,s,t;B(m)),∞) andXis aBK space withAK
orX=∞. More precisely, the following example will show that it is possible forLAto
be compact whilelimn→∞ ˜An∗X= . Hence, in general, we have just ‘if ’ in (.) of
Example . Let us define the matrixA= (ank) byan=st/r andank= fork≥
(n∈N). LetB(m)=I, the identity matrix. Then we have for everyx= (x
k)∈p(r,s,t;B(m)),
Ax= (stx/r)eand henceA∈(p(r,s,t;B(m)),∞). Further, it is obvious thatLA is of
finite rank and soLAis compact. On the other hand, by using (.), it can easily be seen
thatA˜n=e() for alln∈N. Thus, we obtain by Lemma . that ˜An∗X= for alln∈N,
which implies thatlimn→∞ ˜An∗X= .
Moreover, as an immediate consequence of Theorem ., we have the following corol-lary.
Corollary . If either A∈(∞(r,s,t;B(m)),c)or A∈(∞(r,s,t;B(m)),c),then the operator
LAis compact.
Proof LetA∈(¯∞,c). Then we have by Lemma . thatA˜ ∈(∞,c) which implies that
limn→∞(∞k=|˜ank|) = , that is,limn→∞ ˜An∗∞= by Lemma .. This leads us with
The-orem .(b) to the consequence thatLAis compact. Similarly, ifA∈(∞(r,s,t;B(m)),c) then
˜
A∈(∞,c) and hencelimn→∞( ∞
k=|˜ank–α˜k|) = , which can be written aslimn→∞ ˜An–
˜
α∗∞ = , whereα˜ = (α˜k) andα˜k=limn→∞a˜nk for allk∈N. Therefore, we deduce from
Theorem .(c) thatLAis compact.
Throughout, letFr (r∈N) be the subcollection ofF consisting of all non-empty and
finite subsets ofNwith elements that are greater thanr, that is,
Fr={N∈F:n>rfor alln∈N} (r∈N).
Then we have the following.
Lemma .(Theorem . in []) Let X⊃φbe a BK space.If A∈(X,),then
lim
r→∞
sup
N∈Fr
n∈N
An ∗
X
≤ LAχ≤· lim
r→∞
sup
N∈Fr
n∈N
An ∗
X
.
Theorem . If A∈(p(r,s,t;B(m)),),then
lim
r→∞A
(r)
(p(r,s,t;B(m)),)≤ LAχ≤·rlim→∞A
(r)
(p(r,s,t;B(m)),) (.) and
LAis compact if and only if lim
r→∞A
(r)
(p(r,s,t;B(m)),)= , (.) where
A(r)
(p(r,s,t;B(m)),)=Nsup∈Fr
n∈N
˜ An
∗
X
(r∈N).
Proof It is obvious that (.) is obtained by combining Lemmas . and .. Also, by using
Corollary . Let <p<∞,q=p/(p– ).If A∈(p(r,s,t;B(m)),),then
lim
r→∞A
(r)
(p(r,s,t;B(m)),)≤ LAχ≤·rlim→∞A
(r)
(p(r,s,t;B(m)),) and
LAis compact if and only if lim
r→∞A
(r)
(p(r,s,t;B(m)),)= , where
A(r)
(p(r,s,t;B(m)), )=Nsup∈Fr
∞
k=
n∈N
˜ ank
q
/q
(r∈N).
Now, we prove the following result.
Theorem . Let≤p<∞.If A∈((r,s,t;B(m)),p),then
LAχ= lim
r→∞
sup
k ∞
n=r
|˜ank|p /p
(.)
and
LAis compact if and only if lim
r→∞
sup
k ∞
n=r
|˜ank|p
= . (.)
Proof We writeS¯=S¯. Then we see by Lemma . thatLA(S) =¯ AS¯∈Mp. Thus, it follows
from (.) and Lemma . that
LAχ=χ(AS) =¯ lim
r→∞
sup
x∈¯S
(I–Pr)(Ax)p
, (.)
wherePr:p→p (r∈N) is the operator defined byPr(x) = (x,x, . . . ,xr, , , . . .) for all
x= (xk)∈pandIis the identity operator onp.
On the other hand, letx∈(r,s,t;B(m)) be given. Thenx¯∈ and sinceA∈((r,s,t;
B(m)),
p), we obtain from Lemma . thatA˜ ∈(,p) andAx=A˜x. Thus, we have for¯
everyr∈N,
(I–Pr)(Ax)p=(I–Pr)(A˜x)¯ p
=
∞
n=r+
A˜n(x)¯p /p
=
∞
n=r+
∞
k=
˜ ankx¯k
p/p
≤ ∞
k=
∞
n=r+
≤ ¯x
sup
k ∞
n=r+
|˜ank|p /p
=x(r,s,t)
sup
k ∞
n=r+
|˜ank|p /p
.
This yields
sup
x∈¯S
(I–Pr)(Ax)p≤sup k
∞
n=r+
|˜ank|p /p
(r∈N).
Therefore, we deduce from (.) that
LAχ≤ lim
r→∞
sup
k ∞
n=r+
|˜ank|p /p
. (.)
To prove the converse inequality, letb(k)∈
(r,s,t;B(m)) be such that (A(r,¯ s,t;B(m)))b(k)=
e(k) (k∈N), that is,e(k)is the sequence of generalized means ofb(k) for eachk∈N(see
Corollary . in []). Then we have by Lemma . thatAb(k)=Ae˜ (k)= (a˜
nk)∞n=for every
k∈N.
Now, letB={b(k):k∈N}. ThenB⊂ ¯Sand henceAB⊂AS, which implies that¯ χ(AB)≤
χ(AS) =¯ LAχ.
Further, it follows by applying Lemma . that
χ(AB) = lim
r→∞
sup
k ∞
n=r+
An
b(k)p
/p
= lim
r→∞
sup
k ∞
n=r+
|˜ank|p /p
.
Thus, we obtain
lim
r→∞
sup
k ∞
n=r+
|˜ank|p /p
≤ LAχ. (.)
Hence, we get (.) by combining (.) and (.). This completes the proof, since
(.) is immediate by (.) and (.).
Finally, we end this section with the following example, which shows that the limit in (.) may not be zero, that is, there exist matrix operators in the classB(¯,p) which are
not compact, where ≤p<∞.
Example . LetA= (ank) be the infinite matrix defined byA=A(r,¯ s,t;B(m)). LetB(m)=
I, the identity matrix. Since(r,s,t;B(m)) is the matrix domain ofAin, we haveA∈
((r,s,t;B(m)),) and henceA∈(¯,p) for ≤p<∞. Further, it is trivial to see that the
Now, letr∈Nbe given. Then we have, for everyk∈N,
∞
n=r
|˜ank|p= ⎧ ⎨ ⎩
(k≥r),
(k<r).
This implies that
sup
k ∞
n=r
|˜ank|p /p
= (r∈N)
which leads us with (.) of Theorem . to the conclusion thatLAχ= and henceLA
is not compact.
4 Geometric properties
Recently there has been a lot of interest in investigating geometric properties of sequence spaces besides topological and some other usual properties. In the literature, there are many papers concerning geometric properties of various Banach sequence spaces.
A Banach spaceXis said to be aKöthe sequence space(see [, ], and []) ifXis a subspace ofwsuch that
(i) ifx∈w,y∈X, and|x(i)| ≤ |y(i)|for alli∈N, thenx∈Xandx ≤ y; (ii) there exists an elementx∈Xsuch thatx(i) > for alli∈N.
A Köthe sequence spaceXis said to have theFatou propertyif for any real sequencex and any{xn}inXsuch thatxn→xcoordinatewise andsupnxn<∞, we havex∈Xand
xn → x.
A Banach spaceXis said to have theBanach-Saks propertyif every bounded sequence {xn}inXadmits a subsequence{zn}such that the sequence{tk(z)}is convergent inXwith
respect to the norm, where
tk(z) =
k(z+z+· · ·+zk) ∀k∈N.
A Banach space X is said to have theweak Banach-Saks propertywhenever given any weakly null sequence{xn}inXthere exists its subsequence{zn}such that the sequence
{tk(z)}converges to zero strongly.
Given anyp∈(,∞), we say that a Banach space (X, · ) has theBanach-Saks property
of type pif there exists a constantc> such that every weakly null sequence{xk}has a
subsequence{xk}such that (see [])
n
=
xk
≤cn
p ∀n∈N.
The Banach-Saks property of typep∈(,∞) and weak Banach-Saks property for Cesàro sequence spaces have been considered in [].
In [], García-Falset introduced the following coefficient for a Banach space (X, · ):
R(X) =suplim inf
n→∞ xn–x:x∈B(X),{xn} ⊂B(X) andxn→ weakly
and he proved (see [, ]) that a Banach spaceXwithR(X) < has the weak fixed point property.
A Köthe sequence spaceXis said to beorder continuousif for any sequence{xn}and
anyxinX+ (the positive cone inX) such thatxn(i)≤x(i) asn→ ∞for alli,n∈Nand
xn(i)→ for anyi∈N, we havexn → asn→ ∞.
Clarksonmodulus of convexityof a normed space (X, · ) is defined (see work by Clark-son [] and Day []) by the formula
δX(ε) =inf
–x+y
;x,y∈S(X),x–y=ε
for anyε∈[, ]. The inequalityδX(ε) > for allε∈(, ] characterizes the uniform
con-vexity ofXand the equalityδX() = characterizes strict convexity (= rotundity) ofX.
The Gurari˘ı modulus of convexity of a normed spaceXis defined (see []) by
βX(ε) =inf
– inf
α∈[,]
αx+ ( –α)y;x,y∈δ(X),x–y=ε
for anyε∈[, ]. It is obvious thatδX(ε)≤βX(ε) for any Banach spaceXand anyε∈[, ].
It is also known thatβX(ε)≤δX(ε) for anyε∈[, ] and thatXis rotund if and only if βX(ε) = andXis uniformly convex if and only ifβX(ε) > for anyε∈[, ].
Theorem . The Banach-Saks type of the spacep(r,s,t;B(m))is equal to p.
Proof Let (εn) be a sequence of positive numbers for which
∞
n=εn≤. Let{xn}be a
weakly null sequence inp(r,s,t;B(m)). Sett=x= andt=xn=x. Then there exists r∈Nsuch that
∞
i=r+ t(i)e(i)
p(r,s,t;B(m))
<ε.
Since the fact that{xn} is a weakly null sequence implies thatxn→, coordinatewise,
there is ann∈Nsuch that
r
i=
xn(i)e(i)
p(r,s,t;B(m)) <ε
for alln≥n. Sett=xn. Then there exists anr>rsuch that
∞
i=r+ t(i)e(i)
By using the fact thatxn→ coordinatewise, there exists ann>nsuch that r i=
xn(i)e(i)
p(r,s,t;B(m))
<ε
for alln≥n. Continuing this process, we can find by induction two increasing
subse-quences (ri) and (ni) such that
rj i=
xn(i)e(i)
p(r,s,t;B(m)) <εj
for alln≥nj+and
∞
i=rj+
tj(i)e(i)
p(r,s,t;B(m))
<εj,
wheretj=xnj. Hence,
n j= tj
p(r,s,t;B(m))
= n j=
rj–
i=
tj(i)e(i)+ rj
i=rj–+
tj(i)e(i)+
∞
i=rj+
tj(i)e(i)
p(r,s,t;B(m)) ≤
n
j=
rj
i=rj–+ tj(i)e(i)
p(r,s,t;B(m))
+ n j=
rj–
i=
tj(i)e(i)
p(u,v)
+ n j= ∞
i=rj+
tj(i)e(i)
p(r,s,t;B(m)) ≤
n
j=
rj
i=rj–+ tj(i)e(i)
p(r,s,t;B(m)) + n j= εj and n j= rj
i=rj–+ tj(i)e(i)
p
p(r,s,t;B(m)) = n j= rj
i=rj–+
i k=
uivktj(k) p ≤ n j= ∞ i= i k=
uivktj(k) p
≤(n+ ).
Hence we obtain
n j= rj
i=rj–+ tj(i)e(i)
≤ n j= p
By using the inequality ≤(n+ )p for alln∈Nand ≤p<∞, we have n j= tj
p(r,s,t;B(m))
≤(n+ )p+ ≤(n+ )p.
Therefore, the spacep(r,s,t;B(m)) has the Banach-Saks typep, which completes the proof
of the theorem.
Theorem . The Gurari˘ı of modulus of convexity for the normed spacep(r,s,t;B(m))
satisfies the inequality
βp(r,s,t;B(m))(ε)≤ –
–
ε
pp
for any≤ε≤.
Proof Letx∈p(r,s,t;B(m)). By using (.), we have
xp(r,s,t;B(m))=A(r,¯ s,t)·B(m)
x
p=
n
A(r,¯ s,t)·B(m)xnp
p
.
Let ≤ε≤ and using (.), let us consider the following sequences:
x= (xn) = H – ε
pp
,H ε , , , . . . ,
t= (tn) = H – ε
pp
,H –ε , , , . . . ,
whereH= (A(r,¯ s,t)·B(m))–
nk. Sinceyn= ((A(r,¯ s,t)·B(m))x)nandzn= ((A(r,¯ s,t)·B(m))t)n, we
have
y= (yn) =
–
ε
pp
, ε , , , . . . ,
z= (zn) =
–
ε
pp
, –ε , , , . . . .
By using the sequences given above, we obtain the following equalities:
xp
p(r,s,t;B(m))=A(r,¯ s,t)·B
(m)xp
p= – ε
pp
p+ε p = – ε p + ε p = ,
tp
p(r,s,t;B(m))=A(r,¯ s,t)·B
(m)tp
p= – ε
pp p+–ε
= –
ε
p
+
ε
p
= ,
x–tp(r,s,t;B(m))=A(r,¯ s,t)·B
(m)x–A(r,¯ s,t)·B(m)t
p
=
–
ε
pp
–
–
ε
pp
p+ε –
–ε
p
p
=ε.
To complete the estimate of the Gurari˘ı on the modulus of convexity, it remains to calcu-late the infimum ofαx+ ( –α)tp
p(r,s,t;B(m))for ≤α≤. We have
inf
≤α≤αx+ ( –α)tp(r,s,t;B(m)) = inf
≤α≤α
¯
A(r,s,t)·B(m)x+ ( –α)A(r,¯ s,t)·B(m)t
p
= inf
≤α≤
α
–
ε
pp
+ ( –α)
–
ε
pp p
+α
ε
+ ( –α)
–ε
p
p
= inf
≤α≤
–
ε
p
+|α– |p
ε
pp
=
–
ε
pp
.
Consequently, we get forp≥ the inequality
βlp(u,v)(ε)≤ –
–
ε
pp
,
which is the desired result.
5 Conclusion
Recently the sequence spaceX(r,s,t;B(m)) forX∈ {
∞,c,c}has been studied by Maji and
Srivastava [] which is obtained by combining the generalized means and themth order generalized difference operatorB(m)(u,v). They characterized some compact operators on
the spacesX(r,s,t;B(m)) forX∈ {
∞,c,c}by using the Hausdorff measure of
noncompact-ness. In this paper, we have derived some identities or estimates for the operator norms and the Hausdorff measures of noncompactness of certain matrix operators on the sequence spacep(r,s,t;B(m)). Further, by applying the Hausdorff measure of noncompactness, we
obtained the necessary and sufficient conditions for such operators to be compact. In the last section, we have studied some geometric properties of this space,e.g.the property Banach-Saks type and Gurari˘ı modulus of convexity.
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.
Author details
1Department of Mathematics, Faculty of Science, King Abdulaziz University, P.O. Box 80203, Jeddah, 21589, Saudi Arabia. 2Department of Mathematics, Aligarh Muslim University, Aligarh, 202002, India.
Acknowledgements
The authors gratefully acknowledge the financial support from King Abdulaziz University, Jeddah, Saudi Arabia.
Received: 22 December 2015 Accepted: 11 July 2016
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