Hindawi Publishing Corporation
Abstract and Applied Analysis
Volume 2012, Article ID 378250, 10 pages
doi:10.1155/2012/378250
Research Article
Applications of Measure of Noncompactness in
Matrix Operators on Some Sequence Spaces
M. Mursaleen1, 2 and A. Latif3
1
Department of Mathematics, Aligarh Muslim University, Aligarh 202002, India
Department of Mathematics, Tabuk University, Tabuk, Saudi Arabia
3
Department of Mathematics, Faculty of Science, King Abdulaziz University,
Jeddah 21589, Saudi Arabia
2
Correspondence should be addressed to M. Mursaleen, mursaleenm@gmail.com
Received 10 November 2011; Accepted 3 January 2012
Academic Editor: Paul Eloe
Copyright q 2012 M. Mursaleen and A. Latif. This is an open access article distributed under
the Creative Commons Attribution License, which permits unrestricted use, distribution, and
reproduction in any medium, provided the original work is properly cited.
We determine the conditions for some matrix transformations from nφ, where the sequence
space nφ, which is related to the p spaces, was introduced by Sargent 1960. We also obtain
estimates for the norms of the bounded linear operators defined by these matrix transformations
and find conditions to obtain the corresponding subclasses of compact matrix operators by using
the Hausdorff measure of noncompactness.
1. Introduction and Preliminaries
We shall write w for the set of all complex sequences x xk ∞
k0 . Let ϕ, ∞ , c, and c0 denote
the sets of all finite, bounded, convergent, and null sequences, respectively. We write p :
p
n
{x ∈ w : ∞
k0 |xk | < ∞} for 1 ≤ p < ∞. By e and e n ∈ N, we denote the sequences such
n
n
that ek 1 for k 0, 1, . . ., and en 1 and ek 0 k /
n. For any sequence x xk ∞
k0 ,
n
n
k
let x k0 xk e be its n-section. Moreover, we write bs and cs for the sets of sequences
with bounded and convergent partial sums, respectively.
∞
A sequence bn n0 in a linear metric space X is called Schauder basis if for every x ∈ X,
∞
n
there is a unique sequence λn ∞
n0 of scalars such that x n0 λn b . A sequence space
X with a linear topology is called a K-space if each of the maps pi : X → C defined by
pi x xi is continuous for all i ∈ N. A K-space is called an FK-space if X is complete
linear metric space; a BK-space is a normed FK-space. An FK-space X ⊃ φ is said to have
∞
k
AK if every sequence x xk ∞
k0 ∈ X has a unique representation x k0 xk e , that is,
n
n
k
x k0 xk e → x as n → ∞.
2
Abstract and Applied Analysis
The spaces c0 , c and p 1 ≤ p < ∞ all have Schauder bases but the space ∞ has no
Schauder basis. Among the other classical sequence spaces, the spaces c0 and p 1 ≤ p < ∞
have AK.
Let X, · be a normed space. Then the unit sphere and closed unit ball in X are
denoted by SX : {x ∈ X : x 1} and B X : {x ∈ X : x ≤ 1}. If X ⊃ ϕ is a BK-space
and a ak ∈ w, then we define
a∗X
∞
sup ak xk x∈SX k0
1.1
provided the expression on the right-hand side exists and is finite.
The α-, β-, and γ-duals of a subset X of w are, respectively, defined by
X α {a ak ∈ w : ax ak xk ∈ 1 ∀ x xk ∈ X},
1.2
X β {a ak ∈ w : ax ak xk ∈ cs ∀ x xk ∈ X},
X {a ak ∈ w : ax ak xk ∈ bs ∀ x xk ∈ X}.
γ
Throughout this paper, the matrices are infinite matrices of complex numbers. If A is
an infinite matrix with complex entries ank n, k ∈ N, then we write A ank instead of
∞
A ank ∞
n,k0 . Also, we write An for the sequence in the nth row of A, that is, An ank n,k0
for every n ∈ N. In addition, if x xk ∈ w, then we define the A-transform of x as the
sequence Ax An x∞
n0 , where
An x ∞
ank xk ;
n ∈ N
1.3
k0
provided the series on the right converges for each n ∈ N.
Let X and Y be subsets of w and A ank an infinite matrix. Then, we say that A
defines a matrix mapping from X into Y , and we denote it by writing A : X → Y if Ax exists
and is in Y for all x ∈ X. By X, Y , we denote the class of all infinite matrices that map X into
Y . Thus A ∈ X, Y if and only if An ∈ X β for all n ∈ N and Ax ∈ Y for all x ∈ X.
†
†
Lemma 1.1 see1. Let † denote any of the symbols α, β, or γ. Then, we have c0 c† ∞ 1 ,
†
†
1 ∞ and p q , where 1 < p < ∞ and q p/p − 1.
Lemma 1.2 see1, 2. Let X be any of the spaces c0 , c, ∞ , or p 1 ≤ p < ∞. Then, we have
· ∗X · Xβ on X β , where · Xβ denotes the natural norm on the dual space X β .
Lemma 1.3 see1, 2. Let X ⊃ ϕ and Y be BK-spaces. Then, we have
a X, Y ⊂ BX, Y , that is, every matrix A ∈ X, Y defines an operator LA ∈ BX, Y by
LA x Ax for all x ∈ X;
b if X has AK, then BX, Y ⊂ X, Y , that is, for every operator L ∈ BX, Y there exists a
matrix A ∈ X, Y such that Lx Ax for all x ∈ X.
Furthermore, we have the following results on the operator norms.
Abstract and Applied Analysis
3
Lemma 1.4 see2. Let X ⊃ ϕ be a BK-space and Y any of the spaces c0 , c, or ∞ . If A ∈ X, Y ,
then
LA AX,∞ supAn ∗X < ∞,
1.4
n
where AX,∞ denotes the operator norm for the matrix A ∈ X, ∞ .
Sargent 3 defined the following sequence spaces.
Let C denote the space whose elements are finite sets of distinct positive integers.
Given any element σ of C, we denote by cσ the sequence {cn σ} such that cn σ 1
for n ∈ σ and cn σ 0 otherwise. Further
Cs σ∈C:
∞
cn σ ≤ s ,
1.5
n1
that is, Cs is the set of those σ whose support has cardinality at most s, and we get
Φ φ φk ∈ w : 0 < φ1 ≤ φn ≤ φn1 and n 1φn ≥ nφn1 .
1.6
For φ ∈ Φ, the following sequence spaces were introduced by Sargent 3 and further
studied in 4
1
x xk ∈ w : xmφ supsup
<∞ ,
|xk |
φs k∈σ
s≥1 σ∈Cs
∞
n φ x xk ∈ w : xnφ sup
|uk |Δφk < ∞ ,
m φ u∈Sx
1.7
k1
where Δφk φk − φk−1 and Sx denotes the set of all sequences that are rearrangements of x.
Remark 1.5 3. i The spaces mφ and nφ are BK spaces with their respective norms. ii
If φn 1 for all n ∈ N, then mφ 1 , nφ ∞ ; if φn n for all n ∈ N, then mφ ∞ ,
nφ 1 . iii 1 ⊆ mφ ⊆ ∞ ∞ ⊇ nφ ⊇ 1 for all φ of Φ.iv mφ† nφ and
nφ† mφ , where † is any of the symbols α, β, or γ.
Recently, Makowsky and Mursaleen 5 have characterized the classes of compact
operators on some BK-spaces, namely, Cmφ, p 1 ≤ p ≤ ∞, Cnφ, p 1 ≤ p <
∞, Cp , mφ1 < p ≤ ∞, and Cp , nφ1 ≤ p ≤ ∞. In this paper, we
determine the conditions for the classes of matrix transformations nφ, c0 , nφ, c,
and nφ, ∞ , and establish estimates for the norms of the bounded linear operators
defined by these matrix transformations. Further, we obtain the necessary and sufficient
or only sufficient conditions for the corresponding subclasses of compact matrix operators
Cnφ, c0 , Cnφ, c, and Cnφ, ∞ by using the Hausdorff measure of noncompactness.
4
Abstract and Applied Analysis
2. The Hausdorff Measure of Noncompactness
Let X, · be a normed space. Then the unit sphere and closed unit ball in X are denoted
by SX : {x ∈ X : x 1} and BX : {x ∈ X : x ≤ 1}. If X and Y are Banach spaces
then BX, Y is the set of all bounded linear operators L : X → Y ; BX, Y is a Banach space
with the operator norm given by L supx∈SX Lx for all L ∈ BX, Y . A linear operator
L : X → Y is said to be compact if the domain of L is all of X, and for every bounded sequence
xn in X, the sequence Lxn has a subsequence which converges in Y . We denote the class
of all compact operators in BX, Y by CX, Y . An operator L ∈ BX, Y is said to be of
finite rank if dim RL < ∞, where RL is the range space of L. An operator of finite rank is
clearly compact. In particular, if Y C then we write X ∗ for the set of all continuous linear
functionals on X with the norm f supx∈SX |fx|.
The Hausdorff measure of noncompactness was defined by Goldenštein et al. in 1957
6.
Let S and M be subsets of a metric space X, d and ε > 0. Then, S is called an ε-net of
M in X if for every x ∈ M there exists s ∈ S such that dx, 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 of a metric space is said to be totally bounded if it has a finite ε-net for every ε > 0.
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 the set Q, denoted by χQ, is defined
by
χQ : inf > 0 : Q ⊂ ∪ni1 Bxi , ri , xi ∈ X, ri < i 1, 2, . . ., n ∈ N .
2.1
The function χ : MX → 0, ∞ is called the Hausdorff measure of noncompactness.
The basic properties of the Hausdorff measure of noncompactness can be found in
2, 7–9 and for recent developments, see 10–18. If Q, Q1 , and Q2 are bounded subsets of a
metric space X, d, then
χQ 0
Q1 ⊂ Q2
iff Q is totally bounded,
implies χQ1 ≤ χQ2 .
2.2
Further, if X is a normed space, then the function χ has some additional properties
connected with the linear structure, for example,
χQ1 Q2 ≤ χQ1 χQ2 ,
χαQ |α|χQ
∀α ∈ C.
2.3
Let X and Y be Banach spaces and χ1 and χ2 the Hausdorff measures of
noncompactness on X and Y , respectively. An operator L : X → Y is said to be χ1 ,χ2 bounded if LQ ∈ MY for all Q ∈ MX and there exist a constant C ≥ 0 such that
χ2 LQ ≤ Cχ1 Q for all Q ∈ MX . If an operator L is χ1 ,χ2 -bounded, then the number
Lχ1 ,χ2 : inf{C ≥ 0 : χ2 LQ ≤ Cχ1 Q for all Q ∈ MX } is called the χ1 , χ2 -measure
of noncompactness ofL. If χ1 χ2 χ, then we write Lχ1 ,χ2 Lχ .
Abstract and Applied Analysis
5
The most effective way in the characterization of compact operators between the
Banach spaces is by applying the Hausdorff measure of noncompactness. This can be
achieved as follows: Let X and Y be Banach spaces and L ∈ BX, Y . Then, the Hausdorff
measure of noncompactness of L, denoted by Lχ , can be determined by
Lχ χLSX ,
2.4
and we have that L is compact if and only if
Lχ 0.
2.5
Now, the following result gives an estimate for the Hausdorff measure of noncompactbasis for
ness in Banach spaces with Schauder bases. It is known that if bk ∞
k0 is a Schauder
α
a Banach space X, then every element x ∈ X has a unique representation x ∞
k0 k x bk ,
where αk k ∈ N are called the basis functionals. Moreover, for each n ∈ N, the operator
Pn : X → X defined by Pn x nk0 αk x bk x ∈ X is called the projector onto the linear
span of {b0 , b1 , . . . , bn }. Besides, all operators Pn and I − Pn are equibounded, where I denotes
the identity operator on X.
Theorem 2.1 see7. Let X be a Banach space with a Schauder basis bk ∞
k0 , Q ∈ MX , and
Pn : X → Xn ∈ N the projector onto the linear span of {b0 , b1 , . . . , bn }. Then, we have
1
· lim sup supI − Pn x ≤ χQ ≤ lim sup supI − Pn x ,
n→∞
a n→∞
x∈Q
x∈Q
2.6
where a lim supn → ∞ I − Pn .
In particular, the following result shows how to compute the Hausdorff measure of
noncompactness in the spaces c0 and p 1 ≤ p < ∞, which are BK-spaces with AK.
Theorem 2.2 see7. Let Q be a bounded subset of the normed space X, where X is p for 1 ≤ p < ∞
or c0 . If Pn : X → Xn ∈ N is the operator defined by Pn x xn x0 , x1 , . . . , xn , 0, 0, . . . for
all x xk ∞
k0 ∈ X, then we have
χQ lim
n→∞
sup I − Pn x .
2.7
x∈Q
The Haudorff measure of noncompactness for nφ has recently been determined in
19 as follows.
Theorem 2.3. Let Q be a bounded subset of nφ. Then
χQ lim sup
k → ∞ x∈Q
∞
sup
.
|un |Δφn
u∈Sx
nk
2.8
6
Abstract and Applied Analysis
3. Main Results
First we prove the following basic lemma.
Lemma 3.1. If A ∈ nφ, c, then the following hold
αk lim ank
n→∞
exists for every k ∈ N,
α αk ∈ m φ ,
supAn − α∗
n
lim An x n→∞
3.1
∞
3.2
< ∞,
3.3
nφ
αk xk
∀x xk ∈ n φ .
3.4
k0
Proof. We write x xnφ , for short. Since A ∈ nφ, c, we have
LA sup An ∗nφ < ∞.
∗
n
Further, since ek ∈ nφ and hence Aek ∈ c for all k ∈ N. Consequently, the limits αk in
3.1 exist for all k ∈ N.
Now, let x ∈ nφ be given. Then there is a positive constant K such that xp ≤ K x
for all p ∈ N. Thus we have
p
ank xk An xp ≤ Axp LA xp ≤ KLA x
k0
l∞
l∞
3.5
for all p, n ∈ N. Hence, we obtain from 3.1 that
p
p
αk xk lim ank xk ≤ K LA x;
n → ∞ k0
k0
p∈N .
3.6
This implies that αx αk xk ∈ bs, and since x ∈ X was arbitrary, we deduce that
α ∈ nφγ . But nφγ nφβ and hence 3.2 holds. Moreover, since nφ ⊃ ϕ is a BK space,
3.2 implies α∗nφ < ∞ by Wilansky 19, Theorem 7.2.9. Therefore, we get 3.3 from 15
by using 1.3.
Now, define the matrix B bnk by bnk ank − αk for all n, k ∈ N. Then, it is obvious that
Bn ∈ nφβ mφ for all n ∈ N. Also, it follows by 3.3 that
supBn ∗nφ supAn − α∗nφ < ∞.
n
3.7
n
Furthermore, we have from 3.1 that
lim Bn ek lim bnk 0
n→∞
n→∞
k ∈ N,
3.8
Abstract and Applied Analysis
7
that is, Bek ∈ c0 for all k ∈ N. This leads us to the consequence that B ∈ nφ, c0 by
Malkowsky-Rakocevic 2, Theorem 1.23c. Hence, limn → ∞ Bn x 0 for all x ∈ nφ,
which yields 3.4.
This completes the proof of the lemma.
Theorem 3.2. (a) If A ∈ nφ, c0 , then
LA χ lim supAn ∗nφ
r →∞
3.9
n>r
(b) If A ∈ nφ, c, then
1
∗
∗
· lim supAn − αn φ
≤ LA χ ≤ lim supAn − αnφ ,
2 r → ∞ n≥r
r →∞
n≥r
3.10
where α αk with αk lim ank for all k ∈ N.
r →∞
(c) If A ∈ nφ, ∞ , then
0 ≤ LA χ ≤ lim
sup
r →∞
n≥r
An ∗nφ
3.11
.
Proof. We write S Snφ , for short. Then, we have by 2.4 and Lemma 1.3 a that
LA χ χAS.
3.12
For a, we have AS ∈ Mc0 . Thus, it follows by Theorem 2.2 that
LA χ χAS lim
r →∞
supI − Pr Ax∞
,
3.13
x∈S
where Pr : c0 → c0 r ∈ N is the operator defined by Pr x x0 , x1 , . . . , xr , 0, 0, . . . for all
x xk ∈ c0 . This yields that I − Pr Ax∞ supn>r |An x| for all x ∈ nφ and every
r ∈ N. Thus, by combining 1.1 and 1.3, we have for every r ∈ N that
supI − Pr Ax∞ supAn ∗nφ .
x∈S
3.14
n>r
Hence, by 3.13 we get 3.9.
To prove b, we have AS ∈ Mc . Thus, we are going to apply Theorem 2.1 to get an
estimate for the value of χAS in 3.12. For this, we know that every z zn ∈ c has a
n
unique representation z ze ∞
n0 zn − ze , where z limn → ∞ zn . Thus, we define the
8
Abstract and Applied Analysis
n
for r ≥ 1. Then,
projectors Pr : c → cr ∈ N by P0 z ze and Pr z ze r−1
n0 zn − z e
∞
n
we have for every r ∈ N that I − Pr z nr zn − ze and hence
I − Pr z∞ sup|zn − z|
3.15
n>r
for all z ∈ c and every r ∈ N. Obviously I − Pr z∞ ≤ 2z∞ , hence I − Pr ≤ 2 for all
r
r
r
r ∈ N. Further, for each r ∈ N, we define the sequence zr zn ∈ c by zr −1 and zn 1
r
for n / r. Then zr ∞ 1 and limn → ∞ zn 1. Therefore, I − Pr ≥ I − Pr zr ∞ 2
by 3.15. Consequently, we have I − Pr 2 for all r ∈ N. Hence, from 3.12 we obtain by
applying Theorem 2.1 that
1
· μA ≤ LA χ ≤ μA,
2
3.16
where
μA lim sup supI − Pr Ax∞
r →∞
.
3.17
x∈S
Now, it is given that A ∈ nφ, c. Thus, it follows from Lemma 3.1 that the limits αk limn → ∞ ank exist for all k, α αk ∈ nφβ mφ and
lim An x n→∞
∞
3.18
αk xk
k0
for all x xk ∈ nφ. Therefore, we derive from 3.15 that
I − Pr Ax∞ sup|An x −
n≥r
sup|
∞
αk xk |
k0
∞
ank − αk xk |
3.19
n≥r k0
for all x xk ∈ nφ and every r ∈ N. Consequently, we obtain by 1.3 that
supI − Pr Ax∞ supAn − α∗n φ
n≥r
x∈S
r ∈ N.
3.20
Hence, we get 3.10 from 3.16.
Finally, to prove c we define Pr : ∞ → ∞ r ∈ N as in the proof of part a for all
x xk ∈ ∞ .
Then, it is clear that
AS ⊂ Pr AS I − Pr AS
r ∈ N.
3.21
Abstract and Applied Analysis
9
Thus, it follows by the elementary properties of the function χ that
0 ≤ χAS ≤ χPr AS χI − Pr AS
χI − Pr AS
3.22
≤ supx∈S I − Pr Ax∞
supAn ∗n φ
n>r
for all r ∈ N. This and 3.12 together imply 3.11. This completes the proof of the theorem.
As an immediate consequence of Theorem 3.2 and 2.5, we have the following.
Corollary 3.3. (a) If A ∈ nφ, c0 , then
LA is compact iff lim
r →∞
supAn ∗n φ
n>r
0.
3.23
(b) If A ∈ nφ, c, then
LA is compact iff lim supAn −
r →∞
n≥r
α∗n φ
0,
3.24
where α αk with αk limn → ∞ ank for all k ∈ N.
(c) If A ∈ nφ, ∞ , then
LA is compact if
lim supAn ∗n φ 0.
r → ∞ n>r
3.25
Remark 3.4. It is worth mentioning that the condition in 3.25 is only a sufficient condition
for the operator LA to be compact, where A ∈ nφ, ∞ . In the following example, we show
that it is possible for LA to be compact while limr → ∞ supn>r An ∗nφ / 0.
x
∈
S
0
for
some
x
Choose a fixed m ∈ N such that xm
/
nφ . Now, we define the
k
m n ∈ N. Then, we have Ax xm e for all
matrix A ank by anm 1 and ank 0 for all k /
x xk ∈ w, hence A ∈ w, ∞ ⊂ nφ, ∞ . Also, since LA is of finite rank, LA is compact.
On the other hand, we have An em and hence An ∗nφ supx∈Snφ |xm | for all n ∈ N by
1.3. This implies that
> 0.
lim supAn ∗nφ syp |xm | ≥ xm
r →∞
n>r
3.26
x∈Snφ
Acknowledgments
This paper was written when the first author visited Tabuk University, Tabuk, during May 16–
June 13, 2011, and he is very thankful to the administration of Tabuk University for providing
10
Abstract and Applied Analysis
him the hospitalities during the stay. The second author gratefully acknowledges the
partial financial support from the Deanship of Scientific Research DSR at King Abdulaziz
University, Jeddah. The authors thank the referees for their valuable comments.
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Discrete Dynamics in
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Volume 2014
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Optimization
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Volume 2014