# Compact Operators for Almost Conservative and Strongly Conservative Matrices

Authors: S. A. Mohiuddine; M. Mursaleen; A. Alotaibi
Journal: Abstract and Applied Analysis
 (2014)

---

## Abstract

We obtain the necessary and sufficient conditions for an almost conservative matrix to define a compact operator. We also establish some necessary and sufficient (or only sufficient) conditions for operators to be compact for matrix classes(
f
,
X
), where X
=
c
,
c
0
,
l
∞. These results are achieved by applying the Hausdorff measure of noncompactness.


---

## Body

## 1. Introduction and Preliminaries

For some basic definitions and notations of this section we refer to [1, 2]. Let w denote the space of all complex sequences x
=
(
x
k
), and let ϕ be the set of all sequences that terminate in zeros. Let ℓ
∞, c, and c
0 denote the spaces of all bounded, convergent, and null sequences, respectively. We will write c
s and ℓ
1 for the spaces of all convergent and absolutely convergent series, respectively. Further, we will use the conventions that e
=
(
1,1
,
1
,
…
) and e
(
n
)
=
(
0,0
,
…
,
1,0
,
…
) where 1 at the nth place for each n
∈
ℕ
=
{
1,2
,
3
,
…
}.For the sequence spacesX and Y, we write
(1)
M
(
X
,
Y
)
=
{
a
=
(
a
k
)
∈
w
:
a
x
=
(
a
k
x
k
)
∈
Y
∀
x
=
(
x
k
)
∈
X
}
which is called the multiplier space of X and Y. The α-, β-, and γ-duals of a sequence space X, which are respectively denoted by X
α, X
β, and X
γ, are defined by
(2)
X
α
=
M
(
X
,
ℓ
1
)
,
X
β
=
M
(
X
,
c
s
)
,
X
γ
=
M
(
X
,
b
s
)
.Throughout this paper, the matrices are infinite matrices of complex numbers. IfA is an infinite matrix with complex entries a
n
k
(
n
,
k
∈
ℕ
), then we write A
=
(
a
n
k
) instead of A
=
(
a
n
k
)
n
,
k
=
1
∞. Also, we write A
n for the sequence in the nth row of A; that is, A
n
=
(
a
n
k
)
k
=
1
∞ for every n
∈
ℕ. In addition, if x
=
(
x
k
)
∈
w, then we define the A-transform of x as the sequence A
x
=
(
A
n
(
x
)
)
n
=
1
∞, where
(3)
A
n
(
x
)
=
∑
k
=
1
∞
‍
a
n
k
x
k
;
(
n
∈
ℕ
)
provided the series on the right converges for each n
∈
ℕ.For arbitrary sequence spacesX and Y, we write (
X
,
Y
) for the class of all infinite matrices that map X into Y. Thus A
∈
(
X
,
Y
) if and only if A
n
∈
X
β for all n
∈
ℕ and A
x
∈
Y for all x
∈
X.The theory ofB
K spaces is the most powerful tool in the characterization of matrix transformations between sequence spaces.A sequence spaceX is called a B
K space if it is a Banach space with continuous coordinates p
n
:
X
→
ℂ
(
n
∈
ℕ
), where ℂ denotes the complex field and p
n
(
x
)
=
x
n for all x
=
(
x
k
)
∈
X and every n
∈
ℕ.The sequence spacesc
0, c, and ℓ
∞ are B
K spaces with the usual sup norm given by ∥
x
∥
ℓ
∞
=
sup
⁡
k
|
x
k
|, where the supremum is taken over all k
∈
ℕ. Also, the space ℓ
1 is a B
K space with the usual ℓ
1-norm defined by ∥
x
∥
ℓ
1
=
∑
k
=
1
∞
‍
|
x
k
|.IfX
⊃
ϕ is a B
K space and a
=
(
a
k
)
∈
w, then we write
(4)
∥
a
∥
X
*
=
sup
⁡
x
∈
S
X
|
∑
k
=
1
∞
‍
a
k
x
k
|
provided the expression on the right exists and is finite which is the case whenever a
∈
X
β, where S
X is the unit sphere in X; that is, S
X
=
{
x
∈
X
:
∥
x
∥
=
1
}.A sequence(
b
k
)
k
=
0
∞ in a linear metric space (
X
,
d
) is called aSchauder basis (or brieflybasis) for X if for every x
∈
X there exists a unique sequence (
α
k
)
k
=
0
∞ of scalars such that x
=
∑
k
=
1
∞
‍
α
k
b
k; that is, d
(
x
,
x
[
n
]
)
→
0
(
n
→
∞
), where x
[
n
]
=
∑
k
=
0
n
‍
α
k
b
k is known as the n-section of x. The series ∑
k
‍
α
k
b
k which has the sum x is called theexpansion of x, and (
α
k
) is called thesequence of coefficients of x with respect to the basis (
b
k
).LetX and Y be Banach spaces. Then, we write ℬ
(
X
,
Y
) for the set of all bounded linear operators L
:
X
→
Y, which is a Banach space with the operator norm given by ∥
L
∥
=
sup
⁡
x
∈
S
X
∥
L
(
x
)
∥
Y for all L
∈
ℬ
(
X
,
Y
). A linear operator L
:
X
→
Y is said to becompact if the domain of L is all of X and for every bounded sequence (
x
n
) in X, the sequence (
L
(
x
n
)
) has a subsequence which converges in Y. An operator L
∈
ℬ
(
X
,
Y
) is said to beof finite rank if dim
⁡
R
(
L
)
<
∞, where R
(
L
) denotes the range space of L. An operator of finite rank is clearly compact. Further, we write 𝒞
(
X
,
Y
) for the class of all compact operators from X to Y. Let us remark that every compact operator in 𝒞
(
X
,
Y
) is bounded; that is, 𝒞
(
X
,
Y
)
⊂
ℬ
(
X
,
Y
). More precisely, the class 𝒞
(
X
,
Y
) is a closed subspace of the Banach space ℬ
(
X
,
Y
) with the operator norm.Finally, the following known results are fundamental for our investigation.Lemma 1.
LetX denote any of the spaces c
0, c, or ℓ
∞. Then, one has X
β
=
ℓ
1 and ∥
a
∥
X
*
=
∥
a
∥
ℓ
1 for all a
∈
ℓ
1.Lemma 2.
LetX and Y be B
K spaces. Then, one has (
X
,
Y
)
⊂
ℬ
(
X
,
Y
); that is, every matrix A
∈
(
X
,
Y
) defines an operator L
A
∈
ℬ
(
X
,
Y
) by L
A
(
x
)
=
A
x for all x
∈
X.


## 2. The Hausdorff Measure of Noncompactness

Most of the definitions, notations, and basic results of this section are taken from [3]. Throughout, we will write ℳ
X for the collection of all bounded subsets of a metric space (
X
,
d
). If Q
∈
ℳ
X, then theHausdorff measure of noncompactness of the set Q, denoted by χ
(
Q
), is defined to be the infimum of the set of all reals ϵ
>
0 such that Q can be covered by a finite number of balls of radii <
ϵ and centers in X. This can equivalently be redefined as follows:
(5)
χ
(
Q
)
=
inf
⁡
{
ϵ
>
0
:
Q
has
a
finite
ϵ
-
net
}
.The functionχ
:
ℳ
X
→
[
0
,
∞
) is called theHausdorff measure of noncompactness.IfQ, Q
1, and Q
2 are bounded subsets of a metric space X, then we have
(6)
χ
(
Q
)
=
0
if
and
only
if
Q
is
totally
bounded
,
Q
1
⊂
Q
2
implies
χ
(
Q
1
)
≤
χ
(
Q
2
)
.
Further, if X is a normed space, then the function χ has some additional properties connected with the linear structure; for example,
(7)
χ
(
Q
1
+
Q
2
)
≤
χ
(
Q
1
)
+
χ
(
Q
2
)
,
χ
(
α
Q
)
=
|
α
|
χ
(
Q
)
∀
α
∈
ℂ
.
Let X and Y be Banach spaces and χ
1 and χ
2 be the Hausdorff measures of noncompactness on X and Y, respectively. An operator L
:
X
→
Y is said to be (
χ
1,χ
2
)-bounded if L
(
Q
)
∈
ℳ
Y for all Q
∈
ℳ
X and there exists a constant C
≥
0 such that χ
2
(
L
(
Q
)
)
≤
C
χ
1
(
Q
) for all Q
∈
ℳ
X. If an operator L is (
χ
1,χ
2
)-bounded then the number ∥
L
∥
(
χ
1
,
χ
2
)
:
=
inf
⁡
{
C
≥
0
:
χ
2
(
L
(
Q
)
)
≤
C
χ
1
(
Q
) for all Q
∈
ℳ
X
} is called the (
χ
1
,
χ
2
)-measure of noncompactness of  L. If χ
1
=
χ
2
=
χ, then we write ∥
L
∥
(
χ
1
,
χ
2
)
=
∥
L
∥
χ.LetX and Y be Banach spaces and L
∈
ℬ
(
X
,
Y
). Then, the Hausdorff measure of noncompactness of L, denoted by ∥
L
∥
χ, can be determined by
(8)
∥
L
∥
χ
=
χ
(
L
(
S
X
)
)
,
and we have that
(9)
L
is
compact
if
and
only
if
∥
L
∥
χ
=
0
.
Furthermore, the function χ is more applicable when X is a Banach space. The most effective way in the characterization of compact operators between the Banach spaces is by applying the Hausdorff measure of noncompactness. The following result of Goldenštein et al. [4, Theorem 1] gives an estimate for the Hausdorff measure of noncompactness in Banach spaces with Schauder bases.Lemma 3.
LetX be a Banach space with a Schauder basis (
b
k
)
k
=
0
∞ and E
∈
ℳ
X and P
n
:
X
→
X
(
n
∈
ℕ
) the projector onto the linear span of {
b
0
,
b
1
,
…
,
b
n
}. Then, one has
(10)
1
a
·
limsup
⁡
n
→
∞
(
sup
⁡
x
∈
Q
∥
(
I
-
P
n
)
(
x
)
∥
)
≤
χ
(
E
)
≤
limsup
⁡
n
→
∞
(
sup
⁡
x
∈
Q
∥
(
I
-
P
n
)
(
x
)
∥
)
,
where a
=
limsup
⁡
n
→
∞
∥
I
-
P
n
∥ and the operator P
r
:
X
→
X, defined for each r
∈
ℕ by P
r
(
x
)
=
∑
k
=
0
r
‍
ϕ
k
(
x
)
b
k
(
x
∈
X
), is called the projector onto the linear span of {
b
0
,
b
1
,
…
,
b
r
}. Besides, all operators P
r and I
-
P
r are equibounded, where I denotes the identity operator on X.In particular, the following result shows how to compute the Hausdorff measure of noncompactness in the spacesc
0 and ℓ
p
(
1
≤
p
<
∞
) which are B
K-spaces with A
K.Lemma 4.
LetE be a bounded subset of the normed space X, where X is ℓ
p for 1
≤
p
<
∞ or c
0. If P
n
:
X
→
X
(
n
∈
ℕ
) is the operator defined by P
n
(
x
)
=
x
[
n
]
=
(
x
0
,
x
1
,
…
,
x
n
,
0,0
,
…
) for all x
=
(
x
k
)
k
=
0
∞
∈
X, then one has
(11)
χ
(
E
)
=
lim
⁡
n
→
∞
(
sup
⁡
x
∈
Q
∥
(
I
-
P
n
)
(
x
)
∥
)
.
It is easy to see that for E
∈
ℳ
ℓ
p
(12)
χ
(
E
)
=
lim
⁡
n
→
∞
(
sup
⁡
x
∈
Q
∑
k
≥
n
‍
|
x
k
|
p
)
.
Also, it is known that (
e
,
e
(
0
)
,
e
(
1
)
,
…
) is a Schauder basis for the space c and every sequence z
=
(
z
n
)
n
=
0
∞
∈
c has a unique representation z
=
z
-
e
+
∑
n
=
0
∞
‍
(
z
n
-
z
-
)
e
(
n
), where z
-
=
lim
⁡
n
→
∞
z
n. Thus, one defines the projector P
r
:
c
→
c
(
r
∈
ℕ
), onto the linear span of {
e
,
e
(
0
)
,
e
(
1
)
,
…
,
e
(
r
)
}, by
(13)
P
r
(
z
)
=
z
-
e
+
∑
n
=
0
r
‍
(
z
n
-
z
-
)
e
(
n
)
;
(
r
∈
ℕ
)
for all z
=
(
z
n
)
∈
c with z
-
=
lim
⁡
n
→
∞
z
n. In this situation, one has the following.Lemma 5.
LetQ
∈
ℳ
c and P
r
:
c
→
c
(
r
∈
ℕ
) be the projector onto the linear span of {
e
,
e
(
0
)
,
e
(
1
)
,
…
,
e
(
r
)
}. Then, one has
(14)
1
2
·
lim
⁡
r
→
∞
(
sup
⁡
x
∈
Q
∥
(
I
-
P
r
)
(
x
)
∥
ℓ
∞
)
≤
χ
(
Q
)
≤
lim
⁡
r
→
∞
(
sup
⁡
x
∈
Q
∥
(
I
-
P
r
)
(
x
)
∥
ℓ
∞
)
,
where I is the identity operator on c.


## 3. Almost Conservative Matrices

A continuous linear functionalL on ℓ
∞ is said to be aBanach limit if it has the following properties: (i)  L
(
x
)
=
0 if x
=
0, (ii)  L
(
e
)
=
1, and (iii)  L
(
S
x
)
=
L
(
x
); where S is a shift operator defined by (
S
x
)
n
=
x
n
+
1.A bounded sequencex
=
(
x
k
) is said to bealmost convergent (Lorentz [5]) to the value l if all of its Banach limits coincide; that is, L
(
x
)
=
l for all Banach limits L.Lorentz established the following characterization.A sequencex
=
(
x
k
) is almost convergent to the number l if and only if t
p
n
(
x
)
→
l as p
→
∞ uniformly in n, where
(15)
t
p
n
(
x
)
=
1
p
+
1
∑
m
=
0
p
‍
x
m
+
n
;
(
p
,
n
∈
ℕ
)
.The numberl is called the generalized limit of x, and we write l
=
f
-
lim
⁡
x. We denote the set of all almost convergent sequences by f; that is,
(16)
f
=
{
x
∈
ℓ
∞
:
lim
⁡
p
→
∞
t
p
n
(
x
)
=
L
uniformly
in
n
}
.Remark 6.
Note thatc
⊂
f
⊂
ℓ
∞ and each inclusion is proper.Remark 7.
Sincec
⊂
f
⊂
ℓ
∞, we have ℓ
1
=
ℓ
∞
β
⊂
f
β
⊂
c
β
=
ℓ
1 and hence f
β
=
ℓ
1. Therefore, it is natural by (4) and Lemma 1 that ∥
a
∥
f
*
=
∥
a
∥
ℓ
1 for all a
∈
ℓ
1.Remark 8 (see [6]).
f is a BK-space with ∥
·
∥
∞.Remark 9 (see [6]).
f is a nonseparable closed subspace of (
ℓ
∞
,
∥
·
∥
∞
).
Using the idea of almost convergence, King [7] defined and characterized the almost conservative and almost regular matrices.
An infinite matrixA
=
(
a
n
k
)
n
,
k
=
1
∞ is said to bealmost conservative if A
x
∈
f for all x
∈
c, and we denote it by A
∈
(
c
,
f
). If in addition f
-
lim
⁡
A
x
=
lim
⁡
x, then A is calledalmost regular.Remark 10 (see [7, Theorem  1]).
A matrixA
=
(
a
n
k
)
n
,
k
=
1
∞ is almost conservative if and only if(i)
∥
A
∥
=
sup
⁡
n
(
∑
k
=
1
∞
‍
|
a
n
k
|
)
<
∞,
(ii)
a
(
k
)
=
(
a
n
k
)
n
=
1
∞
∈
f for each k
∈
ℕ,
(iii)
a
=
(
∑
k
=
1
∞
‍
a
n
k
)
n
=
1
∞
∈
f.
Now, we prove the following.Theorem 11.
LetA
=
(
a
n
k
) be an almost conservative matrix. Then, one has
(17)
0
≤
∥
L
A
∥
χ
≤
limsup
⁡
n
→
∞
(
∑
k
=
1
∞
‍
|
a
n
k
|
)
,
(18)
L
A
i
s
c
o
m
p
a
c
t
i
f
lim
⁡
n
→
∞
(
∑
k
=
1
∞
‍
|
a
n
k
|
)
=
0
.Proof.
Let us remark that the expression on the right of (17) exists and is finite by Remark 10(i). We write S
=
S
c, for short. Since A
∈
(
c
,
f
), we have by Lemma 2 that L
A
(
S
)
=
A
S
∈
ℳ
f. Thus, we obtain by (8) that
(19)
∥
L
A
∥
χ
=
χ
(
A
S
)
.
We define the operatorsP
r
:
f
→
f
(
r
∈
ℕ
) by P
r
(
x
)
=
(
x
1
,
x
2
,
…
,
x
r
,
0,0
,
…
) for all x
=
(
x
n
)
n
=
1
∞
∈
f. Then, we have
(20)
A
S
⊂
P
r
(
A
S
)
+
(
I
-
P
r
)
(
A
S
)
;
(
r
∈
ℕ
)
,
where I is the identity operator on f. Thus, it follows by the elementary properties of the function χ that
(21)
0
≤
χ
(
A
S
)
≤
χ
(
P
r
(
A
S
)
)
+
χ
(
(
I
-
P
r
)
(
A
S
)
)
=
χ
(
(
I
-
P
r
)
(
A
S
)
)
≤
sup
⁡
x
∈
S
∥
(
I
-
P
r
)
(
A
x
)
∥
ℓ
∞
for all r
∈
ℕ. Further, we have for every r
∈
ℕ that ∥
(
I
-
P
r
)
(
A
x
)
∥
ℓ
∞
=
sup
⁡
n
>
r
|
A
n
(
x
)
| for all x
∈
c. Therefore, by using (3), (4), and Lemma 2, we derive that
(22)
sup
⁡
x
∈
S
∥
(
I
-
P
r
)
(
A
x
)
∥
ℓ
∞
=
sup
⁡
n
>
r
∥
A
n
∥
c
*
=
sup
⁡
n
>
r
∥
A
n
∥
ℓ
1
;
(
r
∈
ℕ
)
.
Thus, we obtain that(23)
0
≤
χ
(
A
S
)
≤
sup
⁡
n
>
r
∥
A
n
∥
ℓ
1
;
(
r
∈
ℕ
)
and hence
(24)
0
≤
χ
(
A
S
)
≤
lim
⁡
r
→
∞
(
sup
⁡
n
>
r
∥
A
n
∥
ℓ
1
)
=
limsup
⁡
n
→
∞
∥
A
n
∥
ℓ
1
.
This and (19) yield (17). Finally, we get (18) from (9) and (17).
This completes the proof.It is worth mentioning that the condition in (18) is only a sufficient condition for the operator L
A to be compact, where A is an almost conservative matrix. More precisely, the following example will show that it is possible for L
A to be compact while lim
⁡
n
→
∞
(
∑
k
=
1
∞
‍
|
a
n
k
|
)
≠
0. Hence, in general, we have just “if” in (18) of Theorem 11.Example 12.
Define the matrixA
=
(
a
n
k
) by a
n
1
=
1 and a
n
k
=
0 for k
>
1
(
n
∈
ℕ
). Then, we have A
x
=
x
1
e
∈
f for all x
=
(
x
k
)
k
=
1
∞
∈
c and hence A
∈
(
c
,
f
); that is, A is almost conservative. Also, it is obvious that L
A is of finite rank and so L
A is compact. On the other hand, we have A
n
=
e
(
1
) and hence ∥
A
n
∥
ℓ
1
=
1 for all n
∈
ℕ. This implies that lim
⁡
n
→
∞
∥
A
n
∥
ℓ
1
=
1.


## 4. Compact Operators for Strongly Conservative Matrices

An infinite matrixA
=
(
a
n
k
)
n
,
k
=
1
∞ is said to bestrongly conservative if A
x
∈
c for all x
∈
f, and we denote it by A
∈
(
f
,
c
). If in addition f
-
lim
⁡
x
=
lim
⁡
A
x, then A is calledstrongly regular (cf. [5]).In this final section, we establish some necessary and sufficient (or only sufficient) conditions for operators to be compact for matrix classes(
f
,
X
), where X
=
c
,
c
0
,
ℓ
∞.We may begin with the following lemmas which will be needed in the sequel.Lemma 13.
If the matrixA is in any of the classes (
f
,
c
0
), (
f
,
c
), or (
f
,
ℓ
∞
), then
(25)
sup
⁡
n
(
∑
k
=
1
∞
‍
|
a
n
k
|
)
<
∞
.Proof.
This can be seen from the class(
f
,
c
) characterized by Lorentz [5] and by using the fact that (
f
,
c
0
)
⊂
(
c
,
c
0
), (
f
,
c
)
⊂
(
c
,
c
), and (
f
,
ℓ
∞
)
⊂
(
c
,
ℓ
∞
).
This completes the proof of the theorem.Lemma 14.
IfA
∈
(
f
,
c
), then one has
(26)
α
k
=
lim
⁡
n
→
∞
a
n
k
e
x
i
s
t
s
f
o
r
e
v
e
r
y
k
∈
ℕ
,
(27)
α
=
(
α
k
)
k
=
1
∞
∈
ℓ
1
,
(28)
sup
⁡
n
(
∑
k
=
1
∞
‍
|
a
n
k
-
α
k
|
)
<
∞
.Proof.
It is trivial that (26) holds, since e
(
k
)
∈
f for all k
∈
ℕ. Further, by combining (26) and Lemma 13, we have for every m
∈
ℕ that ∑
k
=
1
m
‍
|
α
k
|
≤
sup
⁡
n
(
∑
k
=
1
∞
‍
|
a
n
k
|
)
<
∞ which implies that (27) holds. Finally, it follows by (27) and Lemma 13 that (28) holds.
This completes the proof of the theorem.Now, we prove the following result on the Hausdorff measure of noncompactness.Theorem 15.
LetA
=
(
a
n
k
) be an infinite matrix. Then, one has the following.(i)
IfA
∈
(
f
,
c
0
), then
(29)
∥
L
A
∥
χ
=
limsup
⁡
n
→
∞
(
∑
k
=
1
∞
‍
|
a
n
k
|
)
.
(ii)
IfA
∈
(
f
,
c
), then
(30)
1
2
·
limsup
⁡
n
→
∞
(
∑
k
=
1
∞
‍
|
a
n
k
-
α
k
|
)
≤
∥
L
A
∥
χ
≤
limsup
⁡
n
→
∞
(
∑
k
=
1
∞
‍
|
a
n
k
-
α
k
|
)
,
where α
k
=
lim
⁡
n
→
∞
a
n
k for all k
∈
ℕ.
(iii)
IfA
∈
(
f
,
ℓ
∞
), then
(31)
0
≤
∥
L
A
∥
χ
≤
limsup
⁡
n
→
∞
(
∑
k
=
1
∞
‍
|
a
n
k
|
)
.Proof.
Let us remark that the expressions in (29), (30), and (31) exist by Lemmas 13 and 14.
We writeS
=
S
f. Then, we obtain by (8) and Lemma 2 that
(32)
∥
L
A
∥
χ
=
χ
(
A
S
)
.
For (i), we haveA
S
∈
ℳ
c
0. Thus, it follows by applying Lemma 3 that
(33)
χ
(
A
S
)
=
lim
⁡
r
→
∞
(
sup
⁡
x
∈
S
∥
(
I
-
P
r
)
(
A
x
)
∥
ℓ
∞
)
,
where P
r
:
c
0
→
c
0
(
r
∈
ℕ
) is the operator defined by P
r
(
x
)
=
x
[
r
] for all x
∈
c
0. This yields that ∥
(
I
-
P
r
)
(
A
x
)
∥
ℓ
∞
=
sup
⁡
n
>
r
|
A
n
(
x
)
| for all x
∈
f and every r
∈
ℕ. Therefore, by using (3), (4), and Remark 7, we have for every r
∈
ℕ that
(34)
sup
⁡
x
∈
S
∥
(
I
-
P
r
)
(
A
x
)
∥
ℓ
∞
=
sup
⁡
n
>
r
∥
A
n
∥
f
*
=
sup
⁡
n
>
r
∥
A
n
∥
ℓ
1
.
This and (33) imply that
(35)
χ
(
A
S
)
=
lim
⁡
r
→
∞
(
sup
⁡
n
>
r
∥
A
n
∥
ℓ
1
)
=
limsup
⁡
n
→
∞
∥
A
n
∥
ℓ
1
.
Hence, we get (29) by (32).
To prove (ii), we haveA
S
∈
ℳ
c. Thus, we are going to apply Lemma 4 to get an estimate for the value of χ
(
A
S
) in (32). For this, let P
r
:
c
→
c
(
r
∈
ℕ
) be the projectors defined by (13). Then, we have for every r
∈
ℕ
0 that (
I
-
P
r
)
(
z
)
=
∑
n
=
r
+
1
∞
‍
(
z
n
-
z
-
)
e
(
n
) and hence
(36)
∥
(
I
-
P
r
)
(
z
)
∥
ℓ
∞
=
sup
⁡
n
>
r
|
z
n
-
z
-
|
for all z
=
(
z
n
)
∈
c and every r
∈
ℕ
0, where z
-
=
lim
⁡
n
→
∞
z
n and I is the identity operator on c.
Now, by using (32), we obtain by applying Lemma 5 that
(37)
1
2
·
lim
⁡
r
→
∞
(
sup
⁡
x
∈
S
∥
(
I
-
P
r
)
(
A
x
)
∥
ℓ
∞
)
≤
∥
L
A
∥
χ
≤
lim
⁡
r
→
∞
(
sup
⁡
x
∈
S
∥
(
I
-
P
r
)
(
A
x
)
∥
ℓ
∞
)
.
Further, sinceA
∈
(
f
,
c
), we have by combining Lemmas 13 and 14 that α
=
(
α
k
)
∈
ℓ
1 and lim
⁡
n
→
∞
A
n
(
x
)
=
∑
k
=
1
∞
‍
α
k
x
k for all x
=
(
x
k
)
∈
f. Consequently, we derive from (36) that
(38)
∥
(
I
-
P
r
)
(
A
x
)
∥
ℓ
∞
=
sup
⁡
n
>
r
|
A
n
(
x
)
-
∑
k
=
1
∞
‍
α
k
x
k
|
=
sup
⁡
n
>
r
|
∑
k
=
1
∞
‍
(
a
n
k
-
α
k
)
x
k
|
for all x
=
(
x
k
)
∈
f and every r
∈
ℕ
0. Therefore, it follows by (4) that
(39)
sup
⁡
x
∈
S
∥
(
I
-
P
r
)
(
A
x
)
∥
ℓ
∞
=
sup
⁡
n
>
r
∥
A
n
-
α
∥
f
*
=
sup
⁡
n
>
r
∥
A
n
-
α
∥
ℓ
1
;
(
r
∈
ℕ
0
)
.
Hence, from (37) we get (30).
For (iii), we haveA
S
∈
ℳ
ℓ
∞. Thus, we define P
r
:
ℓ
∞
→
ℓ
∞
(
r
∈
ℕ
) by P
r
(
x
)
=
(
x
1
,
x
2
,
…
,
x
r
,
0,0
,
…
) for all x
=
(
x
k
)
∈
ℓ
∞. Then, the proof can be achieved similarly as the proof of Theorem 11.
This completes the proof of the theorem.Finally, we conclude our work by the following corollary.Corollary 16.
LetA
=
(
a
n
k
) be an infinite matrix. Then, one has the following.(i)
IfA
∈
(
f
,
c
0
), then
(40)
L
A
i
s
c
o
m
p
a
c
t
i
f
a
n
d
o
n
l
y
i
f
lim
⁡
n
→
∞
(
∑
k
=
1
∞
‍
|
a
n
k
|
)
=
0
.
(ii)
IfA
∈
(
f
,
c
), then
(41)
L
A
i
s
c
o
m
p
a
c
t
i
f
a
n
d
o
n
l
y
i
f
lim
⁡
n
→
∞
(
∑
k
=
1
∞
‍
|
a
n
k
-
α
k
|
)
=
0
,
where α
k
=
lim
⁡
n
→
∞
a
n
k for all k
∈
ℕ.
(iii)
IfA
∈
(
f
,
ℓ
∞
), then
(42)
L
A
i
s
c
o
m
p
a
c
t
i
f
lim
⁡
n
→
∞
(
∑
k
=
1
∞
‍
|
a
n
k
|
)
=
0
.Proof.
This result follows from Theorem15 by using (9).Remark 17.
As in our Example12, it can be shown similarly that the equivalence in (42) of Corollary 16 does not hold.Remark 18.
Since matrix mappings between BK spaces define bounded linear operators between these spaces which are Banach spaces, it is natural to use the Hausdorff measure of noncompactness to obtain necessary and sufficient conditions for matrix operators between BK spaces to be compact operators. This technique has recently been used by several authors in many research papers (cf. [8–14]). Further the technique of measures of noncompactness has also been used in solving the infinite system of differential equations in some sequence spaces (see [15–17]).



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*Source: 567317-2014-03-24.xml*