Download Compact matrix operators on a new sequence space related to ℓp

Alotaibi et al. Journal of Inequalities and Applications (2016) 2016:193
DOI 10.1186/s13660-016-1129-6
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Compact matrix operators on a new
sequence space related to p spaces
Abdullah Alotaibi1 , Mohammad Mursaleen2* and Syed Abdul Mohiuddine1
*
Correspondence:
[email protected]
2
Department 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 p spaces. 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: BK space; matrix transformation; Hausdorff measure of noncompactness;
compact operator; geometric properties
1 Background, notation, and preliminaries
Let w denote the space of all complex sequences x = (xk )∞
k= . By ∞ , c, c , and φ we denote
the sets of all bounded, convergent, null, and finite sequences, respectively. We write cs
for the set of all convergent series and
p = x ∈ ω :
∞
|xk |p < ∞
for  ≤ p < ∞.
k=
By e and e(n) (n = , , . . .), we denote the sequences with ek =  for all k, and e(n)
n =  and
∞
=

for
k
=
n,
respectively.
For
any
sequence
x
=
(x
)
,
let
e(n)
k k=
k
x
[m]
=
m
xk e(k)
k=
be its m-section.
Let x and y be sequences, X and Y be subsets of ω and A = (ank )∞
n,k= be an infinite matrix
of complex numbers. We write xy = (xk yk )∞
,
k=
Z = x– ∗ Y = {x ∈ ω : xz ∈ Y },
xβ = x– ∗ cs,
∞
β
–
X =
x ∗ cs = a ∈ ω :
ak xk converges for all x ∈ X
x∈X
k=
β
β
β
β
for the β-dual of X. Note that ∞ = cβ = c =  ,  = ∞ and p = q .
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Alotaibi et al. Journal of Inequalities and Applications (2016) 2016:193
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∞
k
By An = (ank )∞
k= and A = (ank )n= we denote the sequences in the nth row and the kth
column of A, and we write
An (x) =
∞
ank xk
(n = , , . . .)
(.)
k=
β
and A(x) = (An (x))∞
n= , provided An ∈ X for all n. The set XA = X(A) = {z ∈ ω : A(z) ∈ X} is
called the matrix domain of A in X. Finally (X, Y ) denotes the class of all matrices A that
map X into Y , that is for which An ∈ X β for all n and A(x) ∈ Y for all x ∈ X, or equivalently
A ∈ (X, Y ) if and only if X ⊂ XA .
The theory of BK spaces is the most powerful tool in the characterization of matrix
transformations between sequence spaces.
A sequence space X is called a BK space if it is a Banach space with continuous coordinates pn : X → C (n ∈ N), where C denotes the complex field and pn (x) = xn for all
x = (xk ) ∈ X and every n ∈ N. A BK space X ⊃ φ is said to have AK if every sequence
(k)
x = (xk ) ∈ X has a unique representation x = ∞
k= xk e .
The sequence spaces ∞ , c, and c are BK spaces with the same sup-norm given by
x
∞ = supk |xk |, where the supremum is taken over all k ∈ N. Further, the space p is
p /p
a BK space with the usual p -norm defined by x
p = ( ∞
k= |xk | ) , where  ≤ p < ∞.
Moreover, the BK spaces c and p ( ≤ p < ∞) have AK [], Examples ., ..
Let X and Y be Banach spaces. Then B (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
= sup L(x)/
x
≤ 
L ∈ B (X, Y ) .
If Y = C then we write X ∗ for the space of all continuous linear functionals on X with the
norm defined by
f = sup f (x) : x
≤ 
f ∈ X∗ .
If X ⊂ w is a normed space and a ∈ w then we write
∗
a
= a
∗X
∞
= sup ak xk : x
≤  ,
(.)
k=
provided the expression on the right-hand side exists and is finite which is the case whenever X ⊃ φ is a BK space and a ∈ X β [], p..
For any subset X of w, the matrix domain of an infinite matrix A in X is defined by
XA = {x ∈ w : Ax ∈ X}.
An infinite matrix T = (tnk ) is called a triangle if tnn =  and tnk =  for all k > 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, if X is a BK space then XT is also
a BK space with the norm given by x
XT = Tx
X for all x ∈ XT [], Theorem ...
The following known results are fundamental for our investigation [], Theorem ...
Alotaibi et al. Journal of Inequalities and Applications (2016) 2016:193
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Lemma .
(a) Let X denote any of the spaces c , c, ∞ ,  or p . Then we have a
∗X = a
X β for all
a ∈ X β , where · X β is the natural norm on the dual space X β .
(b) Let X and Y be BK spaces. Then every matrix A ∈ (X, Y ) defines an operator
LA ∈ B(X, Y ) by LA (x) = Ax for all x ∈ X; we denote this by (X, Y ) ⊂ B(X, Y ).
(c) Let X ⊃ φ be a BK space and Y be any of the spaces c , c or ∞ . If A ∈ (X, Y ), then
LA = A
(X,∞ ) = sup An ∗X < ∞.
n
Also, let F be the collection of all non-empty and finite subsets of N = {, , , . . .},
throughout. Then we have the following result.
Lemma . Let X ⊃ φ be a BK space. If A ∈ (X,  ), then
A
(X, ) ≤ LA ≤  · A
(X, ) ,
where A
(X, ) = supN∈F n∈N
An ∗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 [].
Let S and M be subsets of a metric space (X, d) and ε > . Then S is called an ε-net of M
in X if for every x ∈ M there exists s ∈ S such that d(x, s) < ε. Further, if the set S is finite,
then the ε-net S of M is called a finite ε-net of M, and we say that M has a finite ε-net in X.
A subset M of a metric space X is said to be totally bounded if it has a finite ε-net for every
ε > . If X is complete, then M is totally bounded if and only if M is relatively compact (its
closure M̄ is a compact set). Let X and Y be Banach spaces. A linear operator L : X → Y
is called compact if D(L) = X for the domain of L and, for every bounded sequence (xn )∞
n=
in X, the sequence (L(xn ))∞
n= has a convergent subsequence in Y .
By MX , we denote the collection of all bounded subsets of a metric space (X, d). If Q ∈
MX , then the Hausdorff measure of noncompactness of Q, is defined by
χ(Q) = inf{ε >  : Q has a finite ε-net in X}.
The function χ : MX → [, ∞) is called the Hausdorff measure of noncompactness.
It is well known that if Q, Q , and Q are bounded subsets of a metric space X, then we
have
χ(Q) =  if and only if Q is totally bounded,
Q ⊂ Q
implies
χ(Q ) ≤ χ(Q ),
χ(Q) = χ(Q)
for the closure Q of Q,
χ(Q ∪ Q ) = max χ(Q ), χ(Q ) ,
χ(Q ∩ Q ) ≤ min χ(Q ), χ(Q ) .
Further, if X is a normed space then we also have
χ(Q + Q ) ≤ χ(Q ) + χ(Q ),
Alotaibi et al. Journal of Inequalities and Applications (2016) 2016:193
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χ(αQ) = |α|χ(Q) for all α ∈ C.
Let X and Y be Banach spaces and L ∈ B (X, Y ). Then the Hausdorff measure of noncompactness of the operator L, denoted by L
χ , is defined by
L
χ = χ L(SX ) ,
(.)
where S = {x ∈ X : x
≤ } is the unit ball in X. Also we have
L is compact if and only if
L
χ = .
(.)
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 the BK space p ( ≤ 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 sup(I – Pn )x .
n→∞ x∈Q
For some recent related work on this topic, we refer to [–], and []. For some applications in differential and integral equations, we refer to [, –], and [].
2 Results and discussion
Throughout this paper, let r, t ∈ U and s ∈ Uo , where
U = u = (uk ) ∈ w : uk =  for all k and Uo = u = (uk ) ∈ w : u =  .
For any sequence x = (xn ) ∈ w, we define the sequence x̄ = (x̄n ) of generalized means of
x by
x̄n =
n
 sn–k tk xk
rn
(n ∈ N).
(.)
k=
Further, we define the infinite matrix Ā(r, s, t) of generalized means by
Ā(r, s, t)
nk
=
⎧
⎨s
n–k tk /rn
⎩
( ≤ k ≤ n),
(.)
(k > n)
for all n, k ∈ N. Then, by using the notation of (.), it follows by (.) that x̄ is the Ā(r, s, t)transform of x, that is, x̄ = (Ā(r, s, t))x for all x ∈ w.
Alotaibi et al. Journal of Inequalities and Applications (2016) 2016:193
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Moreover, it is obvious by (.) that Ā(r, s, t) is a triangle. Thus, it has a unique inverse
(Ā(r, s, t))– which is also a triangle. More precisely, we put D(s)
 = /s and
s
s


· · ·  s
s

··· 
s
s
s
s · · ·   s
(s)
Dn = n+ .
..
..
..
.. (n = , , , . . .).
s ..
.
.
.
.
sn– sn– sn– sn– · · · s sn sn– sn– sn– · · · s Then the entries of (Ā(r, s, t))– are given by
⎧
– ⎨(–)n–k D(s)
( ≤ k ≤ n),
n–k rk /tn
Ā(r, s, t) nk =
⎩
(k > n)
(.)
for all n, k ∈ N, that is, (Ā(r, s, t))– = Ā(t, s , r), where s = (sn ) such that sn = (–)n D(s)
n for all
n ∈ N [], p.. Therefore, we have by (.) that
xn =
n
 (–)n–k D(s)
n–k rk x̄k
tn
(n ∈ N).
(.)
k=
For an arbitrary subset X of w, the set X(r, s, t) has recently been introduced in [] as
the matrix domain of the triangle Ā(r, s, t) in X, that is,
∞
n
 sn–k tk xk
∈X .
X(r, s, t) = x = (xk ) ∈ w : y =
rn
k=
n=
It is obvious that X(r, s, t) is a sequence space whenever X is a sequence space, and we call
it the sequence space of generalized means. Further, if X is a BK space then X̄ = X(r, s, t)
is also a BK space with the norm given by
x
X̄ = y
X
(x ∈ X̄).
(.)
Recently, Maji and Srivastava [] have defined and studied the sequence space X(r, s,
t; B(m) ) for X ∈ {∞ , c, c } which is obtained by combining the generalized means and the
mth order generalized difference operator B(m) (u, v). They characterized some compact
operators on the spaces X(r, s, t; B(m) ) for X ∈ {∞ , 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 space p (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 order m denoted as B(m) = B(m) (u, v) = (b(m)
nk ), u, v = 
(see []) is defined as
⎧
m
m–n+k n–k
⎪
v
(max{, n – m} ≤ k ≤ n),
⎪
⎨ n–k u
b(m)
nk =
⎪
⎪
⎩

( ≤ k < max{, n – m}),

(k > n).
Alotaibi et al. Journal of Inequalities and Applications (2016) 2016:193
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The sequence spaces X(r, s, t; B(m) ) for X ∈ {∞ , c, c } are defined as follows:
∞
X r, s, t; B(m) = x = (xk ) ∈ w : Ā(r, s, t)·B(m) x n= ∈ X .
By using the matrix domain, we can write
X r, s, t; B(m) = XĀ(r,s,t;B(m) ) = x = (xk ) ∈ w : Ā r, s, t; B(m) x ∈ X ,
where Ā(r, s, t; B(m) ) = Ā(r, s, t)·B(m) . The sequence x̄ = (x̄n ) is Ā(r, s, t)·B(m) -transform of x =
(xn ), i.e.
n n
m sn–i ti
m+j–i i–j
x̄n =
xj
u
v
i – j rn
j=
i=j
(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 [].
β
Lemma . If a = (ak ) ∈ (p (r, s, t; B(m) ))β , then ã = (ãk ) ∈ p (= q ) and we have
∞
ak xk =
k=
∞
(.)
ãk x̄k
k=
for all x = (xk ) ∈ ¯p with x̄ = (Ā(r, s, t; B(m) ))x, where
ãk = rk
k+
∞ (s) ak
m + j – i –  (–v)j–i
i–k Di–k
+
(–)
aj
s tk um
ti
uj–i+m
j–i
i=k
j=k+
∞ D(s) m + j – i –  (–v)j–i
+
(–)i–k i–k
aj
j–i
ti j=i
uj–i+m
i=k+
∞
.
(.)
The following results will be needed in our study.
Lemma . We have
a
∗¯p = ã
∗p
for all a = (ak ) ∈ ¯p , where ã = (ãk ) is the sequence defined by (.).
β
β
β
Proof Let a = (ak ) ∈ ¯p . Then it follows by Lemma . that ã = (ãk ) ∈ p and the equality
(.) holds for all sequences x = (xk ) ∈ ¯p and x̄ = (x̄k ) ∈ p which are connected by the
relation x̄ = (Ā(r, s, t; B(m) ))x. Further, we see that x ∈ S¯p if and only if x̄ ∈ Sp . Therefore,
we derive from (.) and (.) that
a
∗¯p
∞
∞
ãk x̄k = ã
∗p .
= sup ak xk = sup x̄∈Sp x∈S¯ p k=
This concludes the proof.
k=
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Remark . By combining Lemmas . and ., we have the following:
(a) If a ∈ ( (r, s, t; B(m) ))β , then a
∗¯ = supk |ãk | < ∞.

q /q
(b) If a ∈ (p (r, s, t; B(m) ))β , then a
∗¯ = ( ∞
< ∞, where q = p/(p – ) and
k= |ãk | )
p
 < p < ∞.
Throughout this paper, if A = (ank ) is an infinite matrix, we define the associated matrix
à = (ãnk ) by
ãnk = rk
k+
∞ (s) ank
m + j – i –  (–v)j–i
i–k Di–k
+
(–)
anj
s tk um
ti
uj–i+m
j–i
i=k
j=k+
∞ D(s) m + j – i –  (–v)j–i
+
(–)i–k i–k
anj
j–i
ti j=i
uj–i+m
i=k+
∞
(.)
provided the series on the right converge for all n, k ∈ N, which is the case whenever An ∈
(p (r, s, t))β for all n ∈ 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 à ∈ (p , Y ) such that Ax = Ãx̄ for all x ∈ p (r, s, t) with x̄ =
(Ā(r, s, t))x, where à = (ãnk ) is the associated matrix defined by (.).
Proof Suppose that A ∈ (p (r, s, t; B(m) ), Y ) and let x ∈ p (r, s, t; B(m) ). Then An ∈ (p (r, s, t;
β
B(m) ))β for all n ∈ N. Thus, it follows by Lemma . that Ãn ∈ p for all n ∈ N and the
equality Ax = Ãx̄ holds which yields that Ãx̄ ∈ Y , where x̄ is the sequence of generalized
means of x, i.e., x̄ = (Ā(r, s, t; B(m) ))x. Further, it is obvious by (.) and Remark . that
every x̄ ∈ p is the sequence of generalized means of some x ∈ p (r, s, t; B(m) ). Hence, we
deduce that à ∈ (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 and à = (ãnk ) 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 Ãn ∗p < ∞.
n
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
∗
Ãn A
(p (r,s,t;B(m) ), ) = sup < ∞.
N∈F n∈N
p
Proof This result follows from Lemmas . and ..
Alotaibi et al. Journal of Inequalities and Applications (2016) 2016:193
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Theorem . If A ∈ ( (r, s, t; B(m) ), p ), then
LA = A
( (r,s,t;B(m) ),p ) = sup
∞
k
/p
|ãnk |
p
< ∞.
n=
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 noncompactness 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 ⊃ φ be a BK space. Then we have:
(a) If A ∈ (X, ∞ ), then
 ≤ LA χ ≤ lim sup An ∗X .
n→∞
(b) If A ∈ (X, c ), then
LA χ = lim sup An ∗X .
n→∞
(c) If X has AK or X = ∞ and A ∈ (X, c), then

· lim sup An – α
∗X ≤ LA χ ≤ lim sup An – α
∗X ,
 n→∞
n→∞
where α = (αk ) with αk = limn→∞ ank for all k ∈ N.
Now, let A = (ank ) be an infinite matrix and à = (ãnk ) the associated matrix defined by
(.). Then, by combining Lemmas ., . and ., we have the following result.
Theorem . We have:
(a) If A ∈ (p (r, s, t; B(m) ), ∞ ), then
 ≤ LA χ ≤ lim sup Ãn ∗p
(.)
n→∞
and
LA is compact
if lim Ãn ∗p = .
n→∞
(.)
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(b) If A ∈ (p (r, s, t; B(m) ), c ), then
LA χ = lim sup Ãn ∗p
(.)
n→∞
and
LA is compact
if and only if
lim Ãn ∗p = .
n→∞
(.)
(c) If A ∈ (p (r, s, t; B(m) ), c), then

· lim sup Ãn – α̃
∗p ≤ LA χ ≤ lim sup Ãn – α̃
∗p
 n→∞
n→∞
(.)
lim Ãn – α̃
∗p = ,
(.)
and
LA is compact
if and only if
n→∞
where α̃ = (α̃k ) with α̃k = limn→∞ ãnk for all k ∈ 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 (.).
Since p (r, s, t; B(m) ) is a BK space, we deduce by means of Lemma . that (.) and (.)
are immediate by parts (a) and (b) of Lemma ., respectively.
To prove (.), we have A ∈ (p (r, s, t; B(m) ), c) and hence à ∈ (X, c) by Lemma .. Therefore, it follows by part (c) of Lemma . that

· lim sup Ãn – α̃
∗X ≤ LÃ χ ≤ lim sup Ãn – α̃
∗X ,
 n→∞
n→∞
(.)
where α̃ = (α̃k ) and α̃k = limn→∞ ãnk for all k ∈ N.
Now, let us write S = SX and S̄ = Sp (r,s,t;B(m) ) , for short. Then we obtain by (.) and
Lemma .
LA χ = χ LA (S̄) = χ(AS̄)
(.)
LÃ χ = χ LÃ (S) = χ(ÃS).
(.)
and
Further, we see that x ∈ S̄ if and only if x̄ ∈ S, and since Ax = Ãx̄ by Lemma ., we deduce
that AS̄ = ÃS. This leads us with (.) and (.) to the consequence that LA χ = LÃ χ .
Hence, we get (.) from (.). This completes the proof.
It is worth mentioning that the condition in (.) is only a sufficient condition for the
operator LA to be compact, where A ∈ (p (r, s, t; B(m) ), ∞ ) and X is a BK space with AK
or X = ∞ . More precisely, the following example will show that it is possible for LA to
be compact while limn→∞ Ãn ∗X = . Hence, in general, we have just ‘if ’ in (.) of Theorem .(a).
Alotaibi et al. Journal of Inequalities and Applications (2016) 2016:193
Page 10 of 19
Example . Let us define the matrix A = (ank ) by an = s t /r and ank =  for k ≥ 
(n ∈ N). Let B(m) = I, the identity matrix. Then we have for every x = (xk ) ∈ p (r, s, t; B(m) ),
Ax = (s t x /r )e and hence A ∈ (p (r, s, t; B(m) ), ∞ ). Further, it is obvious that LA is of
finite rank and so LA is compact. On the other hand, by using (.), it can easily be seen
that Ãn = e() for all n ∈ N. Thus, we obtain by Lemma . that Ãn ∗X =  for all n ∈ N,
which implies that limn→∞ Ãn ∗X = .
Moreover, as an immediate consequence of Theorem ., we have the following corollary.
Corollary . If either A ∈ (∞ (r, s, t; B(m) ), c ) or A ∈ (∞ (r, s, t; B(m) ), c), then the operator
LA is compact.
Proof Let A ∈ (¯∞ , c ). Then we have by Lemma . that à ∈ (∞ , c ) which implies that
∗
limn→∞ ( ∞
k= |ãnk |) = , that is, limn→∞ Ãn ∞ =  by Lemma .. This leads us with Theorem .(b) to the consequence that LA is compact. Similarly, if A ∈ (∞ (r, s, t; B(m) ), c) then
à ∈ (∞ , c) and hence limn→∞ ( ∞
k= |ãnk – α̃k |) = , which can be written as limn→∞ Ãn –
α̃
∗∞ = , where α̃ = (α̃k ) and α̃k = limn→∞ ãnk for all k ∈ N. Therefore, we deduce from
Theorem .(c) that LA is compact.
Throughout, let Fr (r ∈ N) be the subcollection of F consisting of all non-empty and
finite subsets of N with elements that are greater than r, that is,
Fr = {N ∈ F : n > r for all n ∈ N} (r ∈ N).
Then we have the following.
Lemma . (Theorem . in []) Let X ⊃ φ be a BK space. If A ∈ (X,  ), then
lim
∗
∗
sup An ≤ LA χ ≤  · lim sup An .
r→∞
r→∞ N∈F
r n∈N
N∈Fr n∈N
X
X
Theorem . If A ∈ (p (r, s, t; B(m) ),  ), then
≤ LA χ ≤  · lim A
(r)
lim A
(r)
( (r,s,t;B(m) ), )
( (r,s,t;B(m) ),
r→∞
p
r→∞

p
)
(.)
and
LA is compact
if and only if
lim A
(r)
= ,
( (r,s,t;B(m) ), )
r→∞
p

(.)
where
A
(r)
(p (r,s,t;B(m) ), )
∗
Ãn = sup N∈Fr n∈N
(r ∈ N).
X
Proof It is obvious that (.) is obtained by combining Lemmas . and .. Also, by using
(.), we get (.) from (.).
Alotaibi et al. Journal of Inequalities and Applications (2016) 2016:193
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Corollary . Let  < p < ∞, q = p/(p – ). If A ∈ (p (r, s, t; B(m) ),  ), then
≤ LA χ ≤  · lim A
(r)
lim A
(r)
( (r,s,t;B(m) ), )
( (r,s,t;B(m) ),
r→∞
p
r→∞

p
)
and
LA is compact
if and only if
lim A
(r)
= ,
( (r,s,t;B(m) ), )
r→∞
p

where
q /q
∞ ãnk = sup
A
(r)
(p (r,s,t;B(m) ), )
N∈Fr
(r ∈ N).
k= n∈N
Now, we prove the following result.
Theorem . Let  ≤ p < ∞. If A ∈ ( (r, s, t; B(m) ), p ), then
LA χ = lim sup
r→∞
∞
k
/p p
|ãnk |
(.)
n=r
and
LA is compact
if and only if
lim sup
r→∞
k
∞
p
|ãnk |
= .
(.)
n=r
Proof We write S̄ = S¯ . Then we see by Lemma . that LA (S̄) = AS̄ ∈ Mp . Thus, it follows
from (.) and Lemma . that
LA χ = χ(AS̄) = lim sup(I – Pr )(Ax)p ,
r→∞
(.)
x∈S̄
where Pr : p → p (r ∈ N) is the operator defined by Pr (x) = (x , x , . . . , xr , , , . . .) for all
x = (xk ) ∈ p and I is the identity operator on p .
On the other hand, let x ∈  (r, s, t; B(m) ) be given. Then x̄ ∈  and since A ∈ ( (r, s, t;
(m)
B ), p ), we obtain from Lemma . that à ∈ ( , p ) and Ax = Ãx̄. Thus, we have for
every r ∈ N,
(I – Pr )(Ax) = (I – Pr )(Ãx̄)
p
p
=
=
∞
Ãn (x̄)p
/p
n=r+
∞
p /p
∞ ãnk x̄k n=r+ k=
≤
∞
∞
k=
n=r+
/p
|ãnk x̄k |
p
Alotaibi et al. Journal of Inequalities and Applications (2016) 2016:193
k
/p ∞
≤ x̄
 sup
Page 12 of 19
|ãnk |
p
n=r+
= x
 (r,s,t) sup
k
∞
/p |ãnk |p
.
n=r+
This yields
∞
/p
p
|ãnk |
sup (I – Pr )(Ax) p ≤ sup
k
x∈S̄
(r ∈ N).
n=r+
Therefore, we deduce from (.) that
LA χ ≤ lim sup
r→∞
k
/p ∞
|ãnk |
p
.
(.)
n=r+
To prove the converse inequality, let b(k) ∈  (r, s, t; B(m) ) be such that (Ā(r, s, t; B(m) ))b(k) =
e (k ∈ N), that is, e(k) is the sequence of generalized means of b(k) for each k ∈ N (see
Corollary . in []). Then we have by Lemma . that Ab(k) = Ãe(k) = (ãnk )∞
n= for every
k ∈ N.
Now, let B = {b(k) : k ∈ N}. Then B ⊂ S̄ and hence AB ⊂ AS̄, which implies that χ(AB) ≤
χ(AS̄) = LA χ .
Further, it follows by applying Lemma . that
(k)
∞
(k) p
An b χ(AB) = lim sup
r→∞
k
= lim sup
r→∞
k
n=r+
∞
/p /p |ãnk |
p
.
n=r+
Thus, we obtain
lim sup
r→∞
k
∞
/p |ãnk |
p
≤ LA χ .
(.)
n=r+
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 class B (¯ , p ) which are
not compact, where  ≤ p < ∞.
Example . Let A = (ank ) be the infinite matrix defined by A = Ā(r, s, t; B(m) ). Let B(m) =
I, the identity matrix. Since  (r, s, t; B(m) ) is the matrix domain of A in  , we have A ∈
( (r, s, t; B(m) ),  ) and hence A ∈ (¯ , p ) for  ≤ p < ∞. Further, it is trivial to see that the
associated matrix à is the identity matrix, that is, ãnn =  and ãnk =  for k = n (n ∈ N).
Alotaibi et al. Journal of Inequalities and Applications (2016) 2016:193
Page 13 of 19
Now, let r ∈ N be given. Then we have, for every k ∈ N,
∞
|ãnk |p =
n=r
⎧
⎨
(k ≥ r),
⎩ (k < r).
This implies that
sup
k
∞
/p
|ãnk |
p
=
(r ∈ N)
n=r
which leads us with (.) of Theorem . to the conclusion that LA χ =  and hence LA
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 space X is said to be a Köthe sequence space (see [, ], and []) if X is a
subspace of w such that
(i) if x ∈ w, y ∈ X, and |x(i)| ≤ |y(i)| for all i ∈ N, then x ∈ X and x
≤ y
;
(ii) there exists an element x ∈ X such that x(i) >  for all i ∈ N.
A Köthe sequence space X is said to have the Fatou property if for any real sequence x
and any {xn } in X such that xn → x coordinatewise and supn xn < ∞, we have x ∈ X and
xn → x
.
A Banach space X is said to have the Banach-Saks property if every bounded sequence
{xn } in X admits a subsequence {zn } such that the sequence {tk (z)} is convergent in X with
respect to the norm, where

tk (z) = (z + z + · · · + zk )
k
∀k ∈ N.
A Banach space X is said to have the weak Banach-Saks property whenever given any
weakly null sequence {xn } in X there exists its subsequence {zn } such that the sequence
{tk (z)} converges to zero strongly.
Given any p ∈ (, ∞), we say that a Banach space (X, · ) has the Banach-Saks property
of type p if there exists a constant c >  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 type p ∈ (, ∞) and weak Banach-Saks property for Cesàro
sequence spaces have been considered in [].
We say that a Banach space X has the weak fixed point property if every nonexpansive self-mapping defined on a non-empty weakly compact convex subset of X has a fixed
point.
Alotaibi et al. Journal of Inequalities and Applications (2016) 2016:193
Page 14 of 19
In [], García-Falset introduced the following coefficient for a Banach space (X, · ):
R(X) = sup lim inf xn – x
: x ∈ B(X), {xn } ⊂ B(X) and xn →  weakly
n→∞
and he proved (see [, ]) that a Banach space X with R(X) <  has the weak fixed point
property.
A Köthe sequence space X is said to be order continuous if for any sequence {xn } and
any x in X+ (the positive cone in X) such that xn (i) ≤ x(i) as n → ∞ for all i, n ∈ N and
xn (i) →  for any i ∈ N, we have xn →  as n → ∞.
Clarkson modulus of convexity of a normed space (X, · ) is defined (see work by Clarkson [] and Day []) by the formula
x + y
; x, y ∈ S(X), x – y
= ε
δX (ε) = inf  –

for any ε ∈ [, ]. The inequality δX (ε) >  for all ε ∈ (, ] characterizes the uniform convexity of X and the equality δX () =  characterizes strict convexity (= rotundity) of X.
The Gurariı̆ modulus of convexity of a normed space X is defined (see []) by
βX (ε) = inf  – inf αx + ( – α)y; x, y ∈ δ(X), x – y
= ε
α∈[,]
for any ε ∈ [, ]. It is obvious that δX (ε) ≤ βX (ε) for any Banach space X and any ε ∈ [, ].
It is also known that βX (ε) ≤ δX (ε) for any ε ∈ [, ] and that X is rotund if and only if
βX (ε) =  and X is uniformly convex if and only if βX (ε) >  for any ε ∈ [, ].
Theorem . The Banach-Saks type of the space p (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 in p (r, s, t; B(m) ). Set t = x =  and t = xn = x . Then there exists
r ∈ N such that
∞
(i) t (i)e < ε .
p (r,s,t;B(m) )
i=r +
Since the fact that {xn } is a weakly null sequence implies that xn → , coordinatewise,
there is an n ∈ N such that
r

xn (i)e(i) i=
< ε
p (r,s,t;B(m) )
for all n ≥ n . Set t = xn . Then there exists an r > r such that
∞
(i) t (i)e i=r +
p (r,s,t;B(m) )
< ε .
Alotaibi et al. Journal of Inequalities and Applications (2016) 2016:193
Page 15 of 19
By using the fact that xn →  coordinatewise, there exists an n > n such that
r

(i) xn (i)e < ε
p (r,s,t;B(m) )
i=
for all n ≥ n . Continuing this process, we can find by induction two increasing subsequences (ri ) and (ni ) such that
rj
(i) xn (i)e < εj
p (r,s,t;B(m) )
i=
for all n ≥ nj+ and
∞
(i) tj (i)e < εj ,
p (r,s,t;B(m) )
i=rj +
where tj = xnj . Hence,
n tj j=
p (r,s,t;B(m) )
n rj–
rj
∞
(i)
(i)
(i) =
tj (i)e +
tj (i)e +
tj (i)e j=
i=rj– +
i=
n rj
(i) ≤
tj (i)e j=
p (r,s,t;B(m) )
i=rj– +
j=
n ∞
(i) +
tj (i)e j=
i=rj +
n rj
≤
tj (i)e(i) j=
+
n
j=
and
n rj
p
(i) tj (i)e j=
i=rj– +
p (r,s,t;B(m) )
p
rj i
n
=
ui vk tj (k)
j= i=rj– + k=
i
p
n ∞ ≤
ui vk tj (k)
j= i= k=
≤ (n + ).
Hence we obtain
n rj
n p

tj (i)e(i) ≤

= (n + ) p .
j=
i=rj– +
j=
i=
p (r,s,t;B(m) )
p (r,s,t;B(m) )
i=rj– +
p (r,s,t;B(m) )
i=rj +
n rj–
(i) +
tj (i)e εj
p (u,v)
Alotaibi et al. Journal of Inequalities and Applications (2016) 2016:193
Page 16 of 19

By using the inequality  ≤ (n + ) p for all n ∈ N and  ≤ p < ∞, we have
n tj j=


≤ (n + ) p +  ≤ (n + ) p .
p (r,s,t;B(m) )
Therefore, the space p (r, s, t; B(m) ) has the Banach-Saks type p, which completes the proof
of the theorem.
Theorem . The Gurariı̆ of modulus of convexity for the normed space p (r, s, t; B(m) )
satisfies the inequality
p p
ε
βp (r,s,t;B(m) ) (ε) ≤  –  –

for any  ≤ ε ≤ .
Proof Let x ∈ p (r, s, t; B(m) ). By using (.), we have

p p
(m) (m)
x
p (r,s,t;B(m) ) = Ā(r, s, t)·B x p =
.
Ā(r, s, t)·B x n
n
Let  ≤ ε ≤  and using (.), let us consider the following sequences:
p p ε
ε
x = (xn ) = H  –
, , , . . . ,
,H


p p ε
ε
, H – , , , . . . ,
t = (tn ) = H  –


(m)
where H = (Ā(r, s, t) · B(m) )–
)x)n and zn = ((Ā(r, s, t) · B(m) )t)n , we
nk . Since yn = ((Ā(r, s, t) · B
have
p p ε
ε
, , , . . . ,
–
,


p p ε
ε
z = (zn ) =
–
, – , , , . . . .


y = (yn ) =
By using the sequences given above, we obtain the following equalities:
p
x
(r,s,t;B(m) )
p
p  p p
p
ε
(m) p
+ ε
= Ā(r, s, t) · B x p =  –


p p
ε
ε
=–
+


= ,
p  p p
p
(m) p
– ε
+ – ε t
=
Ā(r,
s,
t)
·
B
=
(m) )

(r,s,t;B
p
p

p
t
Alotaibi et al. Journal of Inequalities and Applications (2016) 2016:193
Page 17 of 19
p p
ε
ε
=–
+


= ,
x – t
p (r,s,t;B(m) ) = Ā(r, s, t) · B(m) x – Ā(r, s, t)·B(m) t p
p p
p p p p p ε
ε
ε
+ – –ε =  –
– –



 = ε.
To complete the estimate of the Gurariı̆ on the modulus of convexity, it remains to calcup
late the infimum of αx + ( – α)t
(r,s,t;B(m) ) for  ≤ α ≤ . We have
p
inf αx + ( – α)t p (r,s,t;B(m) )
≤α≤
= inf α Ā(r, s, t) · B(m) x + ( – α) Ā(r, s, t) · B(m) t p
≤α≤
p p
p p p
ε
ε
= inf α  –
+ ( – α)  –
≤α≤


p p
ε
ε + α
+ ( – α) – 

p
p p
ε
ε
= inf  –
+ |α – |p
≤α≤


p p
ε
= –
.

Consequently, we get for p ≥  the inequality
p p
ε
,
βlp (u,v) (ε) ≤  –  –

which is the desired result.
5 Conclusion
Recently the sequence space X(r, s, t; B(m) ) for X ∈ {∞ , c, c } has been studied by Maji and
Srivastava [] which is obtained by combining the generalized means and the mth order
generalized difference operator B(m) (u, v). They characterized some compact operators on
the spaces X(r, s, t; B(m) ) for X ∈ {∞ , c, c } by using the Hausdorff measure of noncompactness. 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
space p (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 Gurariı̆ modulus of convexity.
Competing interests
The authors declare that they have no competing interests.
Alotaibi et al. Journal of Inequalities and Applications (2016) 2016:193
Page 18 of 19
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
1
Department of Mathematics, Faculty of Science, King Abdulaziz University, P.O. Box 80203, Jeddah, 21589, Saudi Arabia.
2
Department 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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