# Measure of Noncompactness for Compact Matrix Operators on Some BK Spaces

Authors: E. Malkowsky; A. Alotaibi
Journal: Journal of Function Spaces
 (2014)

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## Abstract

We study the spacesw0p, 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.


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## Body

## 1. Introduction

The spacesw0p, 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 fromwp 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 byω 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 spaceX⊂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*=sup⁡x∈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 subsetX of ω, the set Xβ={a∈ω∶∑k=1∞‍akxkconvergesforallx∈X} is called the β-dual of X.LetA=(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, let1≤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∶lim⁡n→∞1n∑k=1n‍|xk|p=0},wp={x∈ω∶x-ξ·e∈w0pforsomeξ∈ℂ},w∞p={x∈ω∶sup⁡n1n∑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 sequencex∈wp, the wp-limitξ for which
(2)x-ξ·e∈w0p
is unique. We write ∑ν=∑k=2ν2ν+1-1‍ for ν=0,1,… and max⁡ν=max⁡2ν≤k≤2ν+1-1.The setsw0p, wp, and w∞p are BK space with the equivalent block and sectional norms
(3)∥x∥b=sup⁡ν∈ℕ0(12ν∑ν|xk|p)1/p,∥x∥s=sup⁡n(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∞‍2νmax⁡ν⁡|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)*andℳparenorm⁡isomorphic,(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)
A∈(w∞p,w∞)   if and only if(11)∥A∥(w∞p,w∞)=sup⁡m(1mmax⁡Nm⊂{1,2,…,m}∥∑n∈Nm‍An∥ℳp)<∞.Also(w∞p,w∞)=(wp,w∞)=(w0p,w∞).(b)
A∈(w0p,w0)   if and only if (11) holds and(12)lim⁡m→∞1m∑n=1m‍|ank|=0foreachk∈ℕ.(c)
A∈(w0p,w)    if and only if (11) holds and(13)foreachk∈ℕthereexistsαk∈ℂsuchthatlim⁡m→∞1m∑n=1m‍|ank-αk|=0.(d)
A∈(wp,w0)   if and only if (11) and (12) hold and(14)lim⁡m→∞1m∑n=1m‍|∑k=1∞‍ank|=0.(e)
A∈(wp,w)   if and only if (11) and (13) hold and(15)lim⁡m→∞1m∑n=1m‍|∑k=1∞‍ank-α~|=0forsomeα~∈ℂ.(f)
IfA∈(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 inℬ(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, sincew0p 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 hasL∈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)sup⁡m(1mmax⁡Nm⊂{1,…,m}(|∑n∈Nm‍bn|+∥∑n∈Nm‍An∥ℳp))≤∥L∥≤4·sup⁡m(1mmax⁡Nm⊂{1,…,m}(|∑n∈Nm‍bn|+∥∑n∈Nm‍An∥ℳp)).
(b) One hasL∈ℬ(wp,w) if and only if there exist a matrix A∈(w0p,w) and a sequence b∈w∞ with
(19)lim⁡m→∞1m∑n=1m‍|bn+Ane-β~|=0forsomeβ~∈ℂ
such that (17) holds; moreover, one has (18).
(c) One hasL∈ℬ(wp,w0) if and only if there exist a matrix A∈(w0p,w0) and a sequence b∈w∞ with
(20)lim⁡m→∞1m∑n=1m‍|bn+Ane|=0
such that (17) holds; moreover, one has (18).Proof.
(a) First we assumeL∈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)sup⁡m1m∑n=1m‍|Ln(e)|=sup⁡m1m∑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·max⁡Nm⊂{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·max⁡Nm⊂{1,…,m}∥LNm∥≤4·sup⁡m(1mmax⁡Nm⊂{1,…,m}(|∑n∈Nm‍bn|+∥∑n∈Nm‍An∥ℳp)).
This implies the second inequality in (18).
Conversely, we assume thatA∈(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=sup⁡m1m∑n=1m‍|bnξ|≤sup⁡m(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 assumeL∈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)lim⁡m→∞1m∑n=1m‍|Ln(e)-β~|=lim⁡m→∞1m∑n=1m‍|bn+Ane-β~|=0,
which is (19), and
(33)lim⁡m→∞1m∑n=1m‍|Ln(e(k))-αk|=lim⁡m→∞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 thatA∈(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≤lim⁡m→∞(1m∑n=1m‍|Ln(x)-(β~ξ+η0)|)≤lim⁡m→∞(1m∑n=1m‍|(bn+Ane-β~)·ξ|)m+lim⁡m→∞(1m∑n=1m‍|Anx(0)-η0|)=|ξ|·lim⁡m→∞(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) withβ~=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 fromX 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.LetX 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)-boundedif 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.
LetX 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) Let𝒫n∶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·lim⁡n→∞(sup⁡x∈Q∥ℛn(x)∥)≤χ(Q)≤lim⁡n→∞(sup⁡x∈Q∥ℛn(x)∥).
(b) Let𝒫n∶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)=lim⁡n→∞(sup⁡x∈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 operatorsL∈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) LetL∈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)12lim⁡r→∞(sup⁡m(1mmax⁡N(m,r)(|∑n∈N(m,r)‍γn|+∥∑n∈N(m,r)‍Cn∥ℳp)))≤∥L∥χ≤4·lim⁡r→∞(sup⁡m(1mmax⁡N(m,r)(∥∑n∈N(m,r)‍Cn∥ℳp|∑n∈N(m,r)‍γn|mmmmmmmmmmmmmmmmmmmi+∥∑n∈N(m,r)‍Cn∥ℳp))).
(b) LetL∈ℬ(wp,w0). Then one has
(45)lim⁡r→∞(sup⁡m(1mmax⁡N(m,r)(|∑n∈N(m,r)‍bn|+∥∑n∈N(m,r)‍An∥ℳp)))≤∥L∥χ≤4·lim⁡r→∞(sup⁡m(1mmax⁡N(m,r)(+∥∑n∈N(m,r)‍An∥ℳp|∑n∈N(m,r)‍bn|mmmmmmmmmmmmmmmmmmmi+∥∑n∈N(m,r)‍An∥ℳp))).Proof.
(a) We assumeL∈ℬ(wp,w).
Letx∈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)sup⁡m(1mmax⁡N(m,r)(|∑n∈N(m,r)‍γn|+∥∑n∈N(m,r)‍Cn∥ℳp))≤sup⁡x∈Swp∥ℛr(L(x))∥≤4·sup⁡m(1mmax⁡N(m,r)(|∑n∈N(m,r)‍γn|+∥∑n∈N(m,r)‍Cn∥ℳp)).
Now the inequalities in (44) follow from (40) and (42).
(b) Nowβ~=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) LetL∈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)12lim⁡r→∞(sup⁡m(1mmax⁡N(m,r)∥∑n∈N(m,r)‍Cn∥ℳp))≤∥L∥χ≤4·lim⁡r→∞(sup⁡m(1mmax⁡N(m,r)∥∑n∈N(m,r)‍Cn∥ℳp)).
(b) LetL∈ℬ(w0p,w0). Then one has
(50)lim⁡r→∞(sup⁡m(1mmax⁡N(m,r)∥∑n∈N(m,r)‍An∥ℳp))≤∥L∥χ≤4·lim⁡r→∞(sup⁡m(1mmax⁡N(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 classes𝒞(X,Y) for X=wp,w0p and Y=w,w0.Corollary 8.
LetL∈ℬ(X,Y). Then the necessary and sufficient conditions for L∈𝒞(X,Y) can be read from Table 1 where
(51)(1)lim⁡r→∞⁡(sup⁡m(1mmax⁡N(m,r)⁡∥∑n∈N(m,r)An∥ℳp))=0,(2)lim⁡r→∞⁡((((∥∑n∈N(m,r)An∥ℳp)))sup⁡m(((∥∑n∈N(m,r)An∥ℳp))1mmax⁡N(m,r)⁡(∥∑n∈N(m,r)An∥ℳp|∑n∈N(m,r)bn|mmmmmmmmmmmmmim+∥∑n∈N(m,r)An∥ℳp)))=0,(3)lim⁡r→∞⁡(sup⁡m(1mmax⁡N(m,r)⁡∥∑n∈N(m,r)Cn∥ℳp))=0,(4)lim⁡r→∞⁡(sup⁡m(1mmax⁡N(m,r)⁡(∥∑n∈N(m,r)Cn∥ℳp|∑n∈N(m,r)γn|mmmmmmmmmimnmmm+∥∑n∈N(m,r)Cn∥ℳp)))=0.Table 1
To
From
w
0
p
w
p
w
0
(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.



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*Source: 196489-2014-02-10.xml*