We study the spaces w0p, wp, and w∞p of sequences that are strongly summable to 0, summable, and bounded with index p≥1 by the Cesàro method of order 1 and establish the representations of the general bounded linear operators from the spaces wp into the spaces w∞1, w1, and w01. We also give estimates for the operator norm and the Hausdorff measure of noncompactness of such operators. Finally we apply our results to characterize the classes of compact bounded linear operators from w0p and wp into w01 and w1.

1. Introduction

The spaces w0p, wp, and w∞p of all complex sequences that are strongly summable to zero, strongly summable, and strongly bounded, with index p≥1 by the Cesàro method of order 1, were first introduced and studied by Maddox [1, 2]. Further recent studies on the spaces w0(p) where the constant index p in w0p is replaced by the terms of a positive sequence p=(pk) can be found in [3, 4]. Extensive studies of generalizations of the spaces w0p, wp, and w∞p to spaces of sequences of strong weighted means can be found in [5, 6].

In [7], a complete list was given of the characterizations of all matrix transformations from Maddox's spaces into the classical spaces ℓ∞, c, c0, and ℓ1 of all bounded, convergent, and null sequences and of all absolutely convergent series. The characterizations of matrix transformations from the classical sequence spaces into Maddox's spaces with index p=1 were established in [8]. Furthermore, some classes of compact bounded linear operators between those spaces were characterized.

Recently, several authors applied the Hausdorff measure of noncompactness to characterize matrix transformations between sequence spaces that are matrix domains of triangles in the classical sequence spaces, for instance, in [9–14].

In this paper, we extend our studies from the normally considered matrix transformations to the general bounded linear operators from wp into w∞1, w1, and w01. We establish the representations of those operators, deduce estimates for their operator norms and Hausdorff measures of noncompactness, and characterize the corresponding classes of compact bounded linear operators.

2. Notations and Basic Results

In this section we list the notations, concepts, and basic results needed in the paper.

As usual, we denote by ω and ϕ the sets of all complex sequences x=(xk)k=1∞ and of all sequences that terminate in zeros; also let e and e(n) for all n∈ℕ be the sequences with ek=1 for all k and en(n)=0 and ek(n)=0 for k≠n.

A Banach space X⊂ω is a BK space if each coordinate Pn∶X→ℂ with Pnx=xn for x=(xk)k=1∞∈X is continuous. A BK space X⊃ϕ is said to have AK if x[m]=∑k=1mxke(k)→x(m→∞) for every sequence x=(xk)k=1∞∈X.

Let (X,∥·∥) be a normed space and SX={x∈X∶∥x∥=1} and B¯X={x∈X∶∥x∥≤1} denote the unit sphere and closed unit ball in X, respectively. If X and Y are Banach spaces, then we write ℬ(X,Y) for the space of all bounded linear operators L∶X→Y with the operator norm ∥L∥=sup{∥L(x)∥∶x∈SX}; we write X*=ℬ(X,ℂ) for the continuous dual of X, that is, the space of all continuous linear functionals on X with the norm ∥f∥=sup{|f(x)|∶x∈SX}. Furthermore, if (X,∥·∥) is a normed sequence space, then we write ∥a∥X*=supx∈SX|∑k=1∞akxk| for a∈ω provided the expression on the right-hand side exists and is finite which is the case whenever X is a BK space and a∈Xβ [15, Theorem 7.2.9].

For any subset X of ω, the set Xβ={a∈ω∶∑k=1∞akxkconvergesforallx∈X} is called the β-dual of X.

Let A=(ank)n,k=1∞ be an infinite matrix of complex numbers, let X and Y be subsets of ω, and letx∈ω. We write An=(ank)k=1∞ for the sequence in the nth row of A, Anx=∑k=1∞ankxk, Ax=(Anx)n=1∞ (provided all the series converge) and (X,Y) for the class of all matrices A such that An∈Xβ for all n∈ℕ and Ax∈Y for all x∈X. It is known that if X and Y are BK spaces then every matrix A∈(X,Y) defines an operator LA∈ℬ(X,Y) by LA(x)=Ax for all x∈X [15, Theorem 4.2.8] and if, in addition, X has AK then every operator L∈ℬ(X,Y) is given by a matrix A∈(X,Y) such that L(x)=Ax for all x∈X [16, Theorem 1.9].

Throughout, let 1≤p<∞, and let q be the conjugate number of p; that is, q=∞ for p=1 and q=p/(p-1) for 1<p<∞. We write
(1)w0p={x∈ω∶limn→∞1n∑k=1n|xk|p=0},wp={x∈ω∶x-ξ·e∈w0pforsomeξ∈ℂ},w∞p={x∈ω∶supn1n∑k=1n|xk|p<∞}
for the sets of all sequences that are strongly summable to 0, strongly summable, and strongly bounded, with index p by the Cesàro method of order 1; if p=1, we write w0=w01, w=w1, and w∞=w∞1, for short.

The following results are known and can be found in [1] and, for instance, in [7, Proposition 1.1].

For each sequence x∈wp, the wp-limit ξ for which
(2)x-ξ·e∈w0p
is unique. We write ∑ν=∑k=2ν2ν+1-1 for ν=0,1,… and maxν=max2ν≤k≤2ν+1-1.

The setsw0p, wp, and w∞p are BK space with the equivalent block and sectional norms
(3)∥x∥b=supν∈ℕ0(12ν∑ν|xk|p)1/p,∥x∥s=supn(1n∑k=1n|xk|p)1/p.w0p is a closed subspace of wp and wp is a closed subspace of w∞p; w0p has AK, and every sequence x=(xk)k=1∞∈wp has a unique representation
(4)x=ξ·e+∑k=1∞(xk-ξ)e(k)whereξisthewp-limitofx,
and w∞p has no Schauder basis. We always assume that w0p, wp, and w∞p have the block norm, unless explicitly stated otherwise.

We put
(5)∥a∥ℳp={∑ν=0∞2νmaxν|ak|(p=1)∑ν=0∞2ν/p(∑ν|ak|q)1/q(1<p<∞)
and ℳp={a∈ω∶∥a∥ℳp<∞}. The following results are known and can be found, for instance, in [7, Proposition 2.1]:
(6)(w0p)β=(wp)β=(w∞p)β=ℳp,(7)(w0p)*andℳparenormisomorphic,(8)∥a∥w∞p*=∥a∥ℳp∀a∈w∞p;f∈(wp)* if and only if there are a0∈ℂ and a sequence a=(ak)k=1∞∈ℳp such that
(9)f(x)=ξa0+∑k=1∞akxk∀x∈wpwithξfrom(2).
Moreover
(10)∥f∥=|a0|+∥a∥ℳp∀f∈(wp)*.

We need the following result where we assume that the initial and final spaces have the block and sectional norms, respectively.

Proposition 1.

One has the following.

A∈(w∞p,w∞)if and only if(11)∥A∥(w∞p,w∞)=supm(1mmaxNm⊂{1,2,…,m}∥∑n∈NmAn∥ℳp)<∞.Also (w∞p,w∞)=(wp,w∞)=(w0p,w∞).

A∈(w0p,w0)if and only if (11) holds and(12)limm→∞1m∑n=1m|ank|=0foreachk∈ℕ.

A∈(w0p,w)if and only if (11) holds and(13)foreachk∈ℕthereexistsαk∈ℂsuchthatlimm→∞1m∑n=1m|ank-αk|=0.

A∈(wp,w0)if and only if (11) and (12) hold and(14)limm→∞1m∑n=1m|∑k=1∞ank|=0.

A∈(wp,w)if and only if (11) and (13) hold and(15)limm→∞1m∑n=1m|∑k=1∞ank-α~|=0forsomeα~∈ℂ.

If A∈(X,Y)in the cases above, then(16)∥A∥(w∞p,w∞)≤∥LA∥≤4·∥A∥(w∞p,w∞).

Proof.

(a) follows from [17, Corollary 1], (6), and (7).

(b) and (c) follow from [15, 8.3.6, p. 123], since w0p and wp are closed subspaces of w∞p.

(d) and (e) follow from Parts (b) and (c) by [15, 8.3.7].

(f) follows from [17, equation (2.8)], (6), and (7).

3. Representation of Bounded Linear Operators

Here we establish the representations of the bounded linear operators in ℬ(wp,Y) for Y=w∞,w,w0 and give estimates for the operator norms in each case. Throughout, we assume that wp and Y have the block and sectional norms, respectively.

We note that, since w0p has AK, every L∈ℬ(w0p,Y) is given by an infinite matrix A∈(w0p,Y), and its operator norm satisfies the inequalities in (16).

Theorem 2.

(a) One has L∈ℬ(wp,w∞) if and only if there exist a matrix A∈(w0p,w∞) and a sequence b∈w∞ such that
(17)L(x)=b·ξ+Ax∀x∈wp,whereξisthewp-limitofx.
Moreover, one has
(18)supm(1mmaxNm⊂{1,…,m}(|∑n∈Nmbn|+∥∑n∈NmAn∥ℳp))≤∥L∥≤4·supm(1mmaxNm⊂{1,…,m}(|∑n∈Nmbn|+∥∑n∈NmAn∥ℳp)).

(b) One has L∈ℬ(wp,w) if and only if there exist a matrix A∈(w0p,w) and a sequence b∈w∞ with
(19)limm→∞1m∑n=1m|bn+Ane-β~|=0forsomeβ~∈ℂ
such that (17) holds; moreover, one has (18).

(c) One has L∈ℬ(wp,w0) if and only if there exist a matrix A∈(w0p,w0) and a sequence b∈w∞ with
(20)limm→∞1m∑n=1m|bn+Ane|=0
such that (17) holds; moreover, one has (18).

Proof.

(a) First we assume L∈ℬ(wp,w∞) and write Ln=Pn∘L for n=1,2,… where Pn denotes the nth coordinate. Since wp is a BK space, it follows that Ln∈(wp)* for each n, and hence we have by (9)
(21)Ln(x)=bn·ξ+Anx∀x∈wp,whereξisthewp-limitofx,(22)bn=Ln(e)-∑k=1∞Ln(e(k)),mmmmank=Ln(e(k))∀nandk,mmnmmAn∈ℳp∀n.
This yields (17); moreover, we have by (10)
(23)∥Ln∥=|bn|+∥An∥ℳpforn=1,2,….
It also follows from (21) and L(e)∈w∞ that
(24)supm1m∑n=1m|Ln(e)|=supm1m∑n=1m|bn+Ae|<∞,
and so b+Ae∈w∞. Furthermore, since L(x(0))=Ax(0) for all x(0)∈w0p, we have A∈(w0p,w∞)=(w∞p,w∞), and so Ae∈w∞ and we obtain b=(b+Ae)-Ae∈w∞.

Now we show (18). We define LNm∶wp→ℂ for all m∈ℕ and for each subset Nm of {1,2,…,m} by
(25)LNm=1m∑m∈NmLn.
Then clearly LNm∈(wp)*, and we obtain by a well-known inequality (cf. [18])
(26)|LNm(x)|≤1m∑n=1m|Ln(x)|≤4·maxNm⊂{1,…,m}|LNm(x)|,
and hence by the first inequality in (26) and by (10)
(27)∥LNm∥=1m∥∑n∈NmLn∥=1m(|∑n∈Nmbn|+∥∑n∈NmAn∥ℳp)≤∥L∥
for all m∈ℕ and all Nm⊂{1,2,…,m}, and so the first inequality in (18) follows. Also, we obtain for all m∈ℕ from the second inequality in (26) and by (10)
(28)1m∑n=1m|Ln(x)|≤4·maxNm⊂{1,…,m}∥LNm∥≤4·supm(1mmaxNm⊂{1,…,m}(|∑n∈Nmbn|+∥∑n∈NmAn∥ℳp)).
This implies the second inequality in (18).

Conversely, we assume that A∈(w0p,w∞), that b∈w∞, and that (17) is satisfied. Let ε>0, andx∈wp be given and let ξ be the wp-limit of ξ. Then there exists M∈ℕ such that (1/m)∑n=1∞|xk-ξ|p<εp for all m>M. Thus we have for all m>M(29)|ξ|=(1m∑n=1m|ξ|p)1/p≤(1m∑n=1m|xn|p)1/p+(1m∑n=1m|xn-ξ|p)1/p≤∥x∥b+ε.
Since ε was arbitrary, we have
(30)|ξ|≤∥x∥b.
We define the map g∶wp→w∞ with g(x)=bξ for all x∈wp, where ξ∈ℂ is the wp-limit of x. Then g trivially is linear, and it follows from (30) that
(31)∥g(x)∥b=supm1m∑n=1m|bnξ|≤supm(1m∑n=1m|bn|)·∥x∥b,
and, since b∈w∞, we obtain g∈ℬ(wp,w∞). Furthermore, we have A∈(w0p,w∞)=(wp,w∞), and hence LA∈ℬ(wp,w∞), and so, by (17), L=g+LA∈ℬ(wp,w∞).

(b) First we assume L∈ℬ(wp,w). Then L∈ℬ(wp,w∞), and by Part (a) there are b∈w∞ and A∈(w0p,w∞)=(wp,w∞) such that (17) is satisfied; also clearly (18) is satisfied. It follows from (17), L(e)∈w, and L(e(k))∈w for each k that there exist β~∈ℂ and αk∈ℂ for k=1,2,… such that
(32)limm→∞1m∑n=1m|Ln(e)-β~|=limm→∞1m∑n=1m|bn+Ane-β~|=0,
which is (19), and
(33)limm→∞1m∑n=1m|Ln(e(k))-αk|=limm→∞1m∑n=1m|ank-αk|=0foreachk.
Now it follows from A∈(w0p,w∞) and (33) by Proposition 1(c) that A∈(w0p,w).

Conversely, we assume that A∈(w0p,w), that b∈w∞, and that (17) and (19) are satisfied. Then we have A∈(w0p,w∞), and so L∈ℬ(wp,w∞) by Part (a). Let x∈wp be given and let ξ be the wp-limit of x. Then we have x(0)=x-ξe∈w0p and, by (17),
(34)Ln(x)=bnξ+∑k=1∞ankxk=(bn+Ane)ξ+Anx(0).
Since A∈(w0p,w), the w-limit of η0 of Ax(0) exists and we have by (19) and (34)
(35)0≤limm→∞(1m∑n=1m|Ln(x)-(β~ξ+η0)|)≤limm→∞(1m∑n=1m|(bn+Ane-β~)·ξ|)m+limm→∞(1m∑n=1m|Anx(0)-η0|)=|ξ|·limm→∞(1m∑n=1m|bn+Ane-β~|)=0.
Therefore we have L∈ℬ(wp,w).

(c) The proof of Part (c) is similar to that of Part (b) with β~=0 and αk=0(k=1,2,…).

Remark 3.

It was shown in the proof of [19, Theorem 3.6] that if A∈(w0p,w) then (αk)k=1∞∈ℳp with αk(k=1,2,…) from (33) and in [19, equation (3.14)] that the w-limit of Ax(0) for any sequence x(0) in w0p is given by
(36)η0=∑k=1∞αkxk(0).
Let L∈ℬ(wp,w) and x∈wp and let ξ be the wp-limit of x; then we obtain by (35) and (36) for the w-limit of L(x)(37)η=β~·ξ+η0=β~·ξ+∑k=1∞αk(xk-ξ)=(β~-∑k=1∞αk)·ξ+∑k=1∞αkxkwithβ~from(19).

4. Compact Operators

In this section, we establish estimates for the Hausdorff measures of noncompactness of linear operators and characterize some classes of compact operators from X into Y, where X=w0p,wp and Y=w0,w.

First we recall some useful definitions and results. The reader is referred to the monographs [20–23] for the theory and applications of measures of noncompactness. Let X and Y be Banach spaces and let L∶X→Y be a linear operator. Then L is said to be compact if its domain is all of X and, for every bounded sequence (xn)n=1∞ in X, the sequence (L(xn))n=1∞ has a convergent subsequence in Y. We denote the class of such operators by 𝒞(X,Y).

Let (X,d) be a metric space, B(x,r)={y∈X∶d(y,x)<r} denote the open ball of radius r and centre in x, and ℳX denote the class of bounded subsets of M. Then the map χ∶ℳX→[0,∞) with
(38)χ(Q)=inf{ε>0:Q⊂⋃k=1nB(xk,rk),mxk∈X,rk<ε(k=1,…,n;n∈ℕ)⋃k=1n}
is called the Hausdorff measure of noncompactness.

Let X and Y be Banach spaces and let χ1 and χ2 be the Hausdorff measures of noncompactness on X and Y. Then the operator L∶X→Y is said to be (χ1,χ2)-bounded if L(Q)∈ℳY for every Q∈ℳX, and there exists a positive constant C such that χ2(L(Q))≤C·χ1(Q) for every Q∈ℳX. If an operator L is (χ1,χ2)-bounded, then
(39)∥L∥(χ1,χ2)=inf{C>0:χ2(L(Q))≤C·χ1(Q)mmmmmmmi∀Q∈ℳX}
is called the (χ1,χ2)-measure of noncompactness of L. In particular, if χ1=χ2=χ, then we write ∥L∥χ=∥L∥(χ,χ).

The following useful results are well known.

Proposition 4.

Let X and Y be Banach spaces and L∈ℬ(X,Y). Then one has
(40)∥L∥χ=χ(L(B¯X))=χ(L(SX))
(see [23, Theorem 2.25]),
(41)L∈𝒞(X,Y)iff∥L∥χ=0
(see [23, Corollary 2.26, equation (2.58)]).

We also need the following results which are an immediate consequence of [8, Proposition 3.2 and Lemma 3.5].

Proposition 5.

(a) Let 𝒫n∶w→w be the projectors onto the linear span of {e,e(1),…,e(n)}, I∶w→w the identity, and ℛn=I-𝒫n for n=0,1,…. Then one has for all Q∈ℳw(42)12·limn→∞(supx∈Q∥ℛn(x)∥)≤χ(Q)≤limn→∞(supx∈Q∥ℛn(x)∥).

(b) Let 𝒫n∶w0→w0 be the projectors onto the linear span of {e(1),e(2),…,e(n)}, I∶w0→w0 the identity, and ℛn=I-𝒫n for n=0,1,…. Then one has for all Q∈ℳw0(43)χ(Q)=limn→∞(supx∈Q∥ℛn(x)∥).

Proof.

(a) The inequalities in (42) follow from [8, Proposition 3.2 and Lemma 3.5].

(b) The identity in (42) follows from [8, Proposition 3.2 and Lemma 3.3(a)].

Now we give estimates for the Hausdorff measures of noncompactness of the general operators L∈ℬ(wp,w) and L∈ℬ(wp,w0). Let m,r∈ℕ and m>r. Then we write N(m,r) for any subset of the set {r+1,r+2,…,m}.

Theorem 6.

(a) Let L∈ℬ(wp,w). One uses the notations of Theorem 2 and writes γn=bn-β~+∑k=1∞αk for n=1,2,… and C=(cnk)n,k=1∞ for the matrix with cnk=ank-αk for all n and k. Then one has
(44)12limr→∞(supm(1mmaxN(m,r)(|∑n∈N(m,r)γn|+∥∑n∈N(m,r)Cn∥ℳp)))≤∥L∥χ≤4·limr→∞(supm(1mmaxN(m,r)(∥∑n∈N(m,r)Cn∥ℳp|∑n∈N(m,r)γn|mmmmmmmmmmmmmmmmmmmi+∥∑n∈N(m,r)Cn∥ℳp))).

(b) Let L∈ℬ(wp,w0). Then one has
(45)limr→∞(supm(1mmaxN(m,r)(|∑n∈N(m,r)bn|+∥∑n∈N(m,r)An∥ℳp)))≤∥L∥χ≤4·limr→∞(supm(1mmaxN(m,r)(+∥∑n∈N(m,r)An∥ℳp|∑n∈N(m,r)bn|mmmmmmmmmmmmmmmmmmmi+∥∑n∈N(m,r)An∥ℳp))).

Proof.

(a) We assume L∈ℬ(wp,w).

Let x∈wp be given, ξ be the wp-limit of x, and y=L(x). Then we have from Theorem 2(b) and (17) that y=b·ξ+Ax, where A∈(w0p,w) and b∈w∞, and it follows that
(46)yn=bn·ξ+Anx=(bn+∑k=1∞ank)·ξ+An(x-ξ·e)∀n∈ℕ.
Furthermore, the complex numbers β~ and αk(k=1,2,…) satisfying (19) and (33) exist by Theorem 2(b), and (αk)k=1∞∈ℳp by Remark 3, so η∈ℂ defined in (37) is the w-limit of y=L(x) by Remark 3. Now let 𝒫r∶w→w be the projector onto the linear span of {e,e(1),…,e(r)} and ℛr=I-𝒫r for r=0,1,…, where I is the identity on w; hence ℛr(y)=∑n=r+1∞(yn-η)e(n) by (4). Let r∈ℕ be given. We write fn,r(x)=(ℛr(L(x)))n for all n and obtain fn,r(x)=0 for n≤r and, for n>r, from (46) and (37)
(47)fn,r(x)=yn-η=bn·ξ+Anx-(β~·ξ+∑k=1∞αk(xk-ξ))=(bn-β~+∑k=1∞αk)·ξ+∑k=1∞(ank-αk)xk=γn·ξ+Cnx.
Since fn,r∈(wp)*, we obtain by the same kind of argument as in the proof of (18) that
(48)supm(1mmaxN(m,r)(|∑n∈N(m,r)γn|+∥∑n∈N(m,r)Cn∥ℳp))≤supx∈Swp∥ℛr(L(x))∥≤4·supm(1mmaxN(m,r)(|∑n∈N(m,r)γn|+∥∑n∈N(m,r)Cn∥ℳp)).
Now the inequalities in (44) follow from (40) and (42).

(b) Now β~=αk=0 for k=1,2,…, and as in the proof of Part (a) we obtain (48) with γn and Cn replaced by bn and An, respectively, and the inequalities in (45) follow from (40) and (43).

Corollary 7.

(a) Let L∈ℬ(w0p,w). Then L is given by a matrix A∈(w0p,w) and one has, writing C=(cnk)n,k=1∞ for the matrix with cnk=ank-αk for all n and k, where αk is given by (33),
(49)12limr→∞(supm(1mmaxN(m,r)∥∑n∈N(m,r)Cn∥ℳp))≤∥L∥χ≤4·limr→∞(supm(1mmaxN(m,r)∥∑n∈N(m,r)Cn∥ℳp)).

(b) Let L∈ℬ(w0p,w0). Then one has
(50)limr→∞(supm(1mmaxN(m,r)∥∑n∈N(m,r)An∥ℳp))≤∥L∥χ≤4·limr→∞(supm(1mmaxN(m,r)∥∑n∈N(m,r)An∥ℳp)).

Proof.

(a) The estimates in (49) are easily obtained from [8, Theorem 3.6] with ∥·∥(X,w∞)*=∥·∥(w∞p,w∞) defined in (11).

(b) The estimates in (50) follow from those in (45) with bn=0 for all n.

We apply our results and close with the characterizations of the classes 𝒞(X,Y) for X=wp,w0p and Y=w,w0.

Corollary 8.

Let L∈ℬ(X,Y). Then the necessary and sufficient conditions for L∈𝒞(X,Y) can be read from Table 1 where
(51)(1)limr→∞(supm(1mmaxN(m,r)∥∑n∈N(m,r)An∥ℳp))=0,(2)limr→∞((((∥∑n∈N(m,r)An∥ℳp)))supm(((∥∑n∈N(m,r)An∥ℳp))1mmaxN(m,r)(∥∑n∈N(m,r)An∥ℳp|∑n∈N(m,r)bn|mmmmmmmmmmmmmim+∥∑n∈N(m,r)An∥ℳp)))=0,(3)limr→∞(supm(1mmaxN(m,r)∥∑n∈N(m,r)Cn∥ℳp))=0,(4)limr→∞(supm(1mmaxN(m,r)(∥∑n∈N(m,r)Cn∥ℳp|∑n∈N(m,r)γn|mmmmmmmmmimnmmm+∥∑n∈N(m,r)Cn∥ℳp)))=0.

To

From

w0p

wp

w0

(1)

(2)

w

(3)

(4)

Proof.

The conditions in (1)–(4) are immediate consequences of (41) and the conditions in (50), (45), (49), and (44), respectively.

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

Acknowledgment

The authors gratefully acknowledge the financial support from King Abdulaziz University, Jeddah, Saudi Arabia.

MaddoxI. J.On Kuttner's theoremMaddoxI. J.SpanosP.ThorpeB.The w0(p)-w0(q) mapping problem for factorable matrices. IISpanosP.ThorpeB.The w0(p)-w0(q) mapping problem for factorable matrices. IGrosse-ErdmannK.-G.Grosse-ErdmannK.-G.Strong weighted mean summability and Kuttner's theoremBaşarF.MalkowskyE.AltayB.Matrix transformations on the matrix domains of triangles in the spaces of strongly C1-summable and bounded sequencesDjolovićI.MalkowskyE.Compact operators into the spaces of strongly C1 summable and bounded sequencesBaşarırM.KaraE. E.On the B-difference sequence space derived by generalized weighted mean and compact operatorsBasarirM.KaraE. E.On compact operators on the Riesz Bm-difference sequence spaceDjolovićI.Compact operators on the spaces a0r(Δ) and acr(Δ)de MalafosseB.RakočevićV.Applications of measure of noncompactness in operators on the spaces sα,sα0,sα(c),lαpMursaleenM.KarakayaV.PolatH.SimşekN.Measure of noncompactness of matrix operators on some difference sequence spaces of weighted meansMursaleenM.NomanA. K.Applications of Hausdorff measure of noncompactness in the spaces of generalized meansWilanskyA.JarrahA. M.MalkowskyE.Ordinary, absolute and strong summability and matrix transformationsMalkowskyE.RakočevićV.The measure of noncompactness of linear operators between certain sequence spacesPeyerimhoffA.Über ein Lemma von Herrn H. C. ChowMalkowskyE.Banach algebras of matrix transformations between spaces of strongly bounded and summable sequencesAkhmerovR. R.KamenskiĭM. I.PotapovA. S.RodkinaA. E.SadovskiĭB. N.ToledanoJ. M. A.Dominguez BenavidesT.Lopez AcedoG.BanaśJ.GoeblK.MalkowskyE.RakočevićV.An introduction into the theory of sequence spaces and measures of noncompactness