Citation: Pınar Zengin Alp, Emrah Evren Kara. The new class Lz,p,E of s− type operators[J]. AIMS Mathematics, 2019, 4(3): 779-791. doi: 10.3934/math.2019.3.779
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In this study, the set of all natural numbers is represented by N and the set of all nonnegative real numbers is represented by R+.
If the dimension of the range space of a bounded linear operator is finite, it is called a finite rank operator [1].
Throughout this study, X and Y denote real or complex Banach spaces. The space of all bounded linear operators from X to Y is denoted by B(X,Y) and the space of all bounded linear operators from an arbitrary Banach space to another arbitrary Banach space is denoted by B.
The theory of operator ideals is a very important field in functional analysis. The theory of normed operator ideals first appeared in 1950's in [2]. In functional analysis, many operator ideals are constructed via different scalar sequence spaces. An s- number sequence is one of the most important examples of this. The definition of s- numbers goes back to E. Schmidt [3], who used this concept in the theory of non-selfadjoint integral equations. In Banach spaces there are many different possibilities of defining some equivalents of s- numbers, namely Kolmogorov numbers, Gelfand numbers, approximation numbers, and several others. In the following years, Pietsch give the notion of s- number sequence to combine all s- numbers in one definition [4, 5, 6].
A map
S:K→(sr(K)) |
which assigns a non-negative scalar sequence to each operator is called an s-number sequence if for all Banach spaces X,Y,X0 and Y0 the following conditions are satisfied:
(i) ‖K‖=s1(K)≥s2(K)≥…≥0, for every K∈B(X,Y),
(ii) sp+r−1(L+K)≤sp(L)+sr(K) for every L,K∈B(X,Y) and p,r∈N,
(iii) sr(MLK)≤‖M‖sr(L)‖K‖ for all M∈B(Y,Y0), L∈B(X,Y) and K∈B(X0,X), where X0,Y0 are arbitrary Banach spaces,
(iv) If rank(K)≤r, then sr(K)=0,
(v) sn−1(In)=1, where In is the identity map of n-dimensional Hilbert space ln2 to itself [7].
sr(K) denotes the r−th s−number of the operator K.
Approximation numbers are frequently used examples of s-number sequence which is defined by Pietsch. ar(K), the r-th approximation number of a bounded linear operator is defined as
ar(K)=inf{ ‖K−A‖:A∈B(X,Y), rank(A)<r}, |
where K∈B(X,Y) and r∈N [4]. Let K∈B(X,Y) and r∈N. The other examples of s-number sequences are given in the following, namely Gel′fand number (cr(K)), Kolmogorov number (dr(K)), Weyl number (xr(K)), Chang number (yr(K)), Hilbert number (hr(K)), etc. For the definitions of these sequences we refer to [1].
In the sequel there are some properties of s−number sequences.
When any isometric embedding J∈B(Y,Y0) is given and an s-number sequence s=(sr) satisfies sr(K)=sr(JK) for all K∈B(X,Y) the s-number sequence is called injective [8,p.90].
Proposition 1. [8,p.90-94] The number sequences (cr(K)) and (xr(K)) are injective.
When any quotient map S∈B(X0,X) is given and an s-number sequence s=(sr) satisfies sr(K)=sr(KS) for all K∈B(X,Y) the s-number sequence is called surjective [8,p.95].
Proposition 2. [8,p.95] The number sequences (dr(K)) and (yr(K)) are surjective.
Proposition 3. [8,p.115] Let K∈B(X,Y). Then the following inequalities hold:
i) hr(K)≤xr(K)≤cr(K)≤ar(K) and
ii) hr(K)≤yr(K)≤dr(K)≤ar(K).
Lemma 1. [5] Let S,K∈B(X,Y), then |sr(K)−sr(S)|≤‖K−S‖ for r=1,2,….
Let ω be the space of all real valued sequences. Any vector subspace of ω is called a sequence space.
In [9] the space Z(u,v;lp) is defined by Malkowsky and Savaş as follows:
Z(u,v;lp)={x∈ω:∞∑n=1|unn∑k=1vkxk|p<∞} |
where 1<p<∞ and u=(un) and v=(vn) are positive real numbers.
The Cesaro sequence space cesp is defined as ([10,11,19])
cesp={x=(xk)∈ω:∞∑n=1(1nn∑k=1|xk|)p<∞}, 1<p<∞. |
If an operator K∈B(X,Y) satisfies ∞∑n=1(an(K))p<∞ for 0<p<∞, K is defined as an lp type operator in [4] by Pietsch. Afterwards ces-p type operators which is a new class obtained via Cesaro sequence space is introduced by Constantin [12]. Later on Tita in [14] proved that the class of lp type operators and ces-p type operators coincide.
In [15], ς(s)p, the class of s−type Z(u,v;lp) operators is given. For more information about sequence spaces and operator ideals we refer to [1,13,16,18,20].
Let X′, the dual of X, be the set of continuous linear functionals on X. The map x∗⊗y:X→Y is defined by
(x∗⊗y)(x)=x∗(x)y |
where x∈X, x∗∈ X′ and y∈Y.
A subcollection ℑ of B is said to be an operator ideal if for each component ℑ(X,Y)= ℑ∩B(X,Y) the following conditions hold:
(i) if x∗∈ X′, y∈Y, then x∗⊗y∈ℑ(X,Y),
(ii) if L,K∈ℑ(X,Y), then L+K∈ℑ(X,Y),
(iii) if L∈ℑ(X,Y), K∈B(X0,X) and M∈B(Y,Y0), then MLK∈ℑ(X0,Y0) [6].
Let ℑ be an operator ideal and ρ:ℑ→ R+ be a function on ℑ. Then, if the following conditions hold:
(i) if x∗∈ X′, y∈Y, then ρ(x∗⊗y)=‖x∗‖‖y‖;
(ii) if ∃C≥1 such that ρ(L+K)≤C[ρ(L)+ρ(K)];
(iii) if L∈ℑ(X,Y),K∈B(X0,X) and M∈B(Y,Y0), then ρ(MLK)≤‖M‖ρ(L)‖K‖,
ρ is said to be a quasi-norm on the operator ideal ℑ [6].
For special case C=1, ρ is a norm on the operator ideal ℑ.
If ρ is a quasi-norm on an operator ideal ℑ, it is denoted by [ℑ,ρ]. Also if every component ℑ(X,Y) is complete with respect to the quasi-norm ρ, [ℑ,ρ] is called a quasi-Banach operator ideal.
Let [ℑ,ρ] be a quasi-normed operator ideal and J∈B(Y,Y0) be a isometric embedding. If for every operator K∈B(X,Y) and JK∈ℑ(X,Y0) we have K∈ℑ(X,Y) and ρ(JK)=ρ(K), [ℑ,ρ] is called an injective quasi-normed operator ideal. Furthermore, let [ℑ,ρ] be a quasi-normed operator ideal and S∈B(X0,X) be a quotient map. If for every operator K∈B(X,Y) and KS∈ℑ(X0,Y) we have K∈ℑ(X,Y) and ρ(KS)=ρ(K), [ℑ,ρ] is called an surjective quasi-normed operator ideal [6].
Let K′ be the dual of K. An s− number sequence is called symmetric (respectively, completely symmetric) if for all K∈B, sr(K)≥sr(K′) (respectively, sr(K)=sr(K′)) [6].
Lemma 2. [6] The approximation numbers are symmetric, i.e., ar(K′)≤ar(K) for K∈B.
Lemma 3. [6] Let K∈B. Then
cr(K)=dr(K′) and cr(K′)≤dr(K). |
In addition, if K is a compact operator then cr(K′)=dr(K).
Lemma 4. [8] Let K∈B. Then
xr(K)=yr(K′) and yr(K′)=xr(K) |
.
The dual of an operator ideal ℑ is denoted by ℑ′ and it is defined as [6]
ℑ′(X,Y)={K∈B(X,Y):K′∈ℑ(Y′,X′)} |
.
An operator ideal ℑ is called symmetric if ℑ⊂ℑ′ and is called completely symmetric if ℑ=ℑ′ [6].
Let E=(En) be a partition of finite subsets of the positive integers which satisfies
maxEn<minEn+1 |
for n∈N+. In [21] Foroutannia defined the sequence space lp(E) by
lp(E)={x=(xn)∈ω:∞∑n=1|∑j∈Enxj|p<∞}, (1≤p<∞) |
with the seminorm ‖|⋅|‖p,E, which defined as:
‖|x|‖p,E=(∞∑n=1|∑j∈Enxj|p)1p. |
For example, if En={3n−2,3n−1,3n} for all n, then x=(xn)∈lp(E) if and only if ∞∑n=1|x3n−2+x3n−1+x3n|p<∞. It is obvious that ‖|⋅|‖p,E is not a norm, since we have ‖|x|‖p,E=0 while x=(−1,1,0,0,…) and En={3n−2,3n−1,3n} for all n. For the particular case En={n} for n∈N+ we get lp(E)=lp and ‖|x|‖p,E=‖x‖p.
For more information about block sequence spaces, we refer the reader to [17,22,23,24,25].
Let u=(un) and v=(vn) be positive real number sequences. In this section, by replacing lp with lp(E) we get the sequence space Z(u,v;lp(E)) defined as follows:
Z(u,v;lp(E))={x∈ω:∞∑n=1|unn∑k=1∑j∈Ekvjxj|p<∞}. |
An operator K∈B(X,Y) is in the class of s-type Z(u,v;lp(E)) if
∞∑n=1(unn∑k=1∑j∈Ekvjsj(K))p<∞, (1<p<∞). |
The class of all s-type Z(u,v;lp(E)) operators is denoted by Lz,p,E(X,Y).
In particular case if En={n} for n=1,2,…, then the class Lz,p,E(X,Y) reduces to the class ς(s)p.
Conditions used in Theorem 1 hold throughout the remainder of the paper.
Theorem 1. Fix 1<p<∞. If ∞∑n=1(un)p<∞ and M>0 is such that v2k−1+v2k≤Mvk,M>0 for all k∈N, then Lz,p,E is an operator ideal.
Proof. Let x∗∈X′ and y∈Y. Since the rank of the operator x∗⊗y is one, sn(x∗⊗y)=0 for n≥2. By using this fact
∞∑n=1(unn∑k=1∑j∈Ekvjsj(x∗⊗y))p=∞∑n=1(un)p(v1s1(x∗⊗y))p=∞∑n=1(un)p(v1)p‖x∗⊗y‖p=∞∑n=1(un)p(v1)p‖x∗‖p‖y‖p<∞. |
Therefore x∗⊗y∈Lz,p,E(X,Y).
Let L,K∈ Lz,p,E(X,Y). Then
∞∑n=1(unn∑k=1∑j∈Ekvjsj(L))p<∞, ∞∑n=1(unn∑k=1∑j∈Ekvjsj(K))p<∞. |
To show that L+K∈Lz,p,E(X,Y), let us begin with
n∑k=1∑j∈Ekvjsj(L+K)≤n∑k=1(∑j∈Ekv2j−1s2j−1(L+K)+∑j∈Ekv2js2j(L+K))≤n∑k=1∑j∈Ek(v2j−1+v2j)s2j−1(L+K)≤Mn∑k=1∑j∈Ekvj(sj(L)+sj(K))≤Mn∑k=1(∑j∈Ekvjsj(L)+∑j∈Ekvjsj(K)). |
By using Minkowski inequality we get;
(∞∑n=1(unn∑k=1(∑j∈Ekvjsj(L+K)))p)1p≤M(∞∑n=1(unn∑k=1(∑j∈Ekvjsj(L)+∑j∈Ekvjsj(K)))p)1p≤M[(∞∑n=1(unn∑k=1∑j∈Ekvjsj(L))p)1p+(∞∑n=1(unn∑k=1∑j∈Ekvjsj(K))p)1p]<∞. |
Hence L+K∈Lz,p,E(X,Y).
Let M∈B(Y,Y0), L∈Lz,p,E(X,Y) and K∈B(X0,X). Then,
∞∑n=1(unn∑k=1∑j∈Ekvjsj(MLK))p≤∞∑n=1(unn∑k=1∑j∈Ek‖M‖‖K‖vjsj(L))p≤‖M‖p‖K‖p(∞∑n=1(unn∑k=1∑j∈Ekvjsj(L))p)<∞. |
So MLK∈Lz,p,E(X0,Y0).
Therefore Lz,p,E(X,Y) is an operator ideal.
Theorem 2. ‖K‖z,p,E=(∞∑n=1(unn∑k=1∑j∈Ekvjsj(K))p)1p(∞∑n=1(un)p)1pv1 is a quasi-norm on the operator ideal Lz,p,E.
Proof. Let x∗∈X′ and y∈Y. Since the rank of the operator x∗⊗y is one, sn(x∗⊗y)=0 for n≥2. Then
(∞∑n=1(unn∑k=1∑j∈Ekvjsj(x∗⊗y))p)1p(∞∑n=1(un)p)1pv1=((∞∑n=1(un)p)vp1‖x∗⊗y‖p)1p(∞∑n=1(un)p)1pv1=‖x∗⊗y‖=‖x∗‖‖y‖ . |
Therefore ‖x∗⊗y‖z,p,E=‖x∗‖‖y‖.
Let L,K∈Lz,p,E(X,Y). Then
n∑k=1∑j∈Ekvjsj(L+K)≤n∑k=1∑j∈Ekv2j−1s2j−1(L+K)+∑j∈Env2js2j(L+K)≤n∑k=1∑j∈Ek(v2j−1+v2j)s2j−1(L+K)≤Mn∑k=1∑j∈Ekvj(sj(L)+sj(K)). |
By using Minkowski inequality we get;
(∞∑n=1(unn∑k=1∑j∈Ekvjsj(L+K))p)1p≤(∞∑n=1(Munn∑k=1∑j∈Ekvj(sj(L)+sj(K)))p)1p≤M[(∞∑n=1(unn∑k=1∑j∈Ekvjsj(L))p)1p+(∞∑n=1(unn∑k=1∑j∈Ekvjsj(K))p)1p]. |
Hence
‖L+K‖z,p,E≤M(‖S‖z,p,E+‖K‖z,p,E). |
Let M∈B(Y,Y0), L∈Lz,p,E(X,Y) and K∈B(X0,X)
(∞∑n=1(unn∑k=1∑j∈Ekvjsj(MLK))p)1p≤(∞∑n=1(unn∑k=1∑j∈Ek‖M‖‖K‖vjsj(L))p)1p≤‖M‖‖K‖(∞∑n=1(unn∑k=1∑j∈Ekvjsj(L))p)1p<∞ |
‖MLK‖z,p,E≤‖M‖‖K‖‖L‖z,p,E. |
Therefore ‖K‖z,p,E is a quasi-norm on Lz,p,E.
Theorem 3. Let 1<p<∞. [Lz,p,E(X,Y),‖K‖z,p,E] is a quasi-Banach operator ideal.
Proof. Let X,Y be any two Banach spaces and 1≤p<∞. The following inequality holds
‖K‖z,p,E=(∞∑n=1(unn∑k=1∑j∈Ekvjsj(K))p)1p(∞∑n=1(un)p)1pv1≥‖K‖ |
for K∈Lz,p,E(X,Y).
Let (Km) be Cauchy in Lz,p,E(X,Y). Then for every ε>0 there exists n0∈N such that
‖Km−Kl‖z,p,E<ε | (2.1) |
for all m,l≥n0. It follows that
‖Km−Kl‖≤‖Km−Kl‖z,p,E<ε. |
Then (Km) is a Cauchy sequence in B(X,Y). B(X,Y) is a Banach space since Y is a Banach space. Therefore ‖Km−K‖→0 as m→∞ for some K∈B(X,Y). Now we show that ‖Km−K‖z,p,E→0 as m→∞ for K∈Lz,p,E(X,Y).
The operators Kl−Km, K−Km are in the class B(X,Y) for Km,Kl,K∈B(X,Y).
|sn(Kl−Km)−sn(K−Km)|≤‖Kl−Km−(K−Km)‖=‖Kl−K‖. |
Since Kl→K as l→∞ we obtain
sn(Kl−Km)→sn(K−Km) | (2.2) |
It follows from (2.1) that the statement
‖Km−Kl‖z,p,E=(∞∑n=1(unn∑k=1∑j∈Ekvjsj(Km−Kl))p)1p(∞∑n=1(un)p)1pv1<ε |
holds for all m,l≥n0. We obtain from (2.2) that
(∞∑n=1(unn∑k=1∑j∈Ekvjsj(Km−K))p)1p(∞∑n=1(un)p)1pv1≤ε. |
Hence we have
‖Km−K‖z,p,E<ε for all m≥n0. |
Finally we show that K∈Lz,p,E(X,Y).
∞∑n=1(unn∑k=1∑j∈Ekvjsj(K))p≤∞∑n=1(unn∑k=1∑j∈Ekv2j−1s2j−1(K)+unn∑k=1∑j∈Ekv2js2j(K))p≤∞∑n=1(unn∑k=1∑j∈Ek(v2j−1+v2j)s2j−1(K−Km+Km))p≤M∞∑n=1(unn∑k=1∑j∈Ekvj(sj(K−Km)+sj(Km)))p |
By using Minkowski inequality; since Km∈Lz,p,E(X,Y) for all m and ‖Km−K‖z,p,E→0 as m→∞, we have
M(∞∑n=1(unn∑k=1∑j∈Ekvj(sj(K−Km)+sj(Km)))p)1p≤M[(∞∑n=1(unn∑k=1∑j∈Ekvjsj(K−Km))p)1p+(∞∑n=1(unn∑k=1∑j∈Ekvjsj(Km))p)1p]<∞ |
which means K∈Lz,p,E(X,Y).
Definition 1. Let μ=(μi(K)) be one of the sequences s=(sn(K)), c=(cn(K)), d=(dn(K)), x=(xn(K)), y=(yn(K)) and h=(hn(K)). Then the space L(μ)z,p,E generated via μ=(μi(K)) is defined as
L(μ)z,p,E(X,Y)={K∈B(X,Y):∞∑n=1(unn∑k=1∑j∈Ekvjμj(K))p<∞,(1<p<∞)}. |
And the corresponding norm ‖K‖(μ)z,p,E for each class is defined as
‖K‖(μ)z,p,E=(∞∑n=1(unn∑k=1∑j∈Ekvjμj(K))p)1p(∞∑n=1(un)p)1pv1. |
Proposition 4. The inclusion L(a)z,p,E⊆L(a)z,q,E holds for 1<p≤q<∞.
Proof. Since lp⊆lq for 1<p≤q<∞ we have L(a)z,p,E⊆L(a)z,q,E.
Theorem 4. Let 1<p<∞. The quasi-Banach operator ideal [L(s)z,p,E,‖K‖(s)z,p,E] is injective, if the sequence sn(K) is injective.
Proof. Let 1<p<∞ and K∈B(X,Y) and J∈B(Y,Y0) be any isometric embedding. Suppose that JK∈L(s)z,p,E(X,Y0). Then
∞∑n=1(unn∑k=1∑j∈Ekvjsj(JK))p<∞ |
Since s=(sn) is injective, we have
sn(K)=sn(JK) for all K∈B(X,Y),n=1,2,…. | (2.3) |
Hence we get
∞∑n=1(unn∑k=1∑j∈Ekvjsj(K))p=∞∑n=1(unn∑k=1∑j∈Ekvjsj(JK))p<∞ |
Thus K∈L(s)z,p,E(X,Y) and we have from (2.3)
‖JK‖(s)z,p,E=(∞∑n=1(unn∑k=1∑j∈Ekvjsj(JK))p)1p(∞∑n=1(un)p)1pv1=(∞∑n=1(unn∑k=1∑j∈Ekvjsj(K))p)1p(∞∑n=1(un)p)1pv1=‖K‖(s)z,p,E |
Hence the operator ideal [L(s)z,p,E,‖K‖(s)z,p,E] is injective.
Conclusion 1. [8,p.90-94] Since the number sequences (cn(K)) and (xn(K)) are injective, the quasi-Banach operator ideals [L(c)z,p,E,‖K‖(c)z,p,E] and [L(x)z,p,E,‖K‖(x)z,p,E] are injective.
Theorem 5. Let 1<p<∞. The quasi-Banach operator ideal [L(s)z,p,E,‖K‖(s)z,p,E] is surjective, if the sequence (sn(K)) is surjective.
Proof. Let 1<p<∞ and K∈B(X,Y) and S∈B(X0,X) be any quotient map. Suppose that KS∈L(s)z,p,E(X0,Y). Then
∞∑n=1(unn∑k=1∑j∈Ekvjsj(KS))p<∞. |
Since s=(sn) is surjective, we have
sn(K)=sn(KS) for all K∈B(X,Y),n=1,2,…. | (2.4) |
Hence we get
∞∑n=1(unn∑k=1∑j∈Ekvjsj(K))p=∞∑n=1(unn∑k=1∑j∈Ekvjsj(KS))p<∞. |
Thus K∈L(s)z,p,E(X,Y) and we have from (2.4)
‖KS‖(s)z,p,E=(∞∑n=1(unn∑k=1∑j∈Ekvjsj(KS))p)1p(∞∑n=1(un)p)1pv1=(∞∑n=1(unn∑k=1∑j∈Ekvjsj(K))p)1p(∞∑n=1(un)p)1pv1=‖K‖(s)z,p,E. |
Hence the operator ideal [L(s)z,p,E,‖K‖(s)z,p,E]is surjective.
Conclusion 2. [8,p.95] Since the number sequences (dn(K)) and (yn(K)) are surjective, the quasi-Banach operator ideals [L(d)z,p,E,‖K‖(d)z,p,E] and [L(y)z,p,E,‖K‖(y)z,p,E] are surjective.
Theorem 6. Let 1<p<∞. Then the following inclusion relations holds:
i L(a)z,p,E⊆L(c)z,p,E⊆L(x)z,p,E⊆L(h)z,p,E
ii L(a)z,p,E⊆L(d)z,p,E⊆L(y)z,p,E⊆L(h)z,p,E.
Proof. Let K∈L(a)z,p,E. Then
∞∑n=1(unn∑k=1∑j∈Ekvjsj(K))p<∞ |
where 1<p<∞. And from Proposition 3, we have;
∞∑n=1(unn∑k=1∑j∈Ekvjhj(K))p≤∞∑n=1(unn∑k=1∑j∈Ekvjxj(K))p≤∞∑n=1(unn∑k=1∑j∈Ekvjcj(K))p≤∞∑n=1(unn∑k=1∑j∈Ekvjaj(K))p<∞ |
and
∞∑n=1(unn∑k=1∑j∈Ekvjhj(K))p≤∞∑n=1(unn∑k=1∑j∈Ekvjyj(K))p≤∞∑n=1(unn∑k=1∑j∈Ekvjdj(K))p≤∞∑n=1(unn∑k=1∑j∈Ekvjaj(K))p<∞. |
So it is shown that the inclusion relations are satisfied.
Theorem 7. For 1<p<∞, L(a)z,p,E is a symmetric operator ideal and L(h)z,p,E is a completely symmetric operator ideal.
Proof. Let 1<p<∞.
Firstly, we show that L(a)z,p,E is symmetric in other words L(a)z,p,E⊆(L(a)z,p,E)′ holds. Let K∈L(a)z,p,E. Then
∞∑n=1(unn∑k=1∑j∈Ekvjaj(K))p<∞. |
It follows from [6, p.152] an(K′)≤an(K) for K∈B. Hence we get
∞∑n=1(unn∑k=1∑j∈Ekvjaj(T′))p≤∞∑n=1(unn∑k=1∑j∈Ekvjaj(K))p<∞. |
Therefore K∈(L(a)z,p,E)′. Thus L(a)z,p,E is symmetric.
Now we prove that the equation L(h)z,p,E=(L(h)z,p,E)′ holds. It follows from [8,p.97] that hn(K′)=hn(K) for K∈B. Then we can write
∞∑n=1(unn∑k=1∑j∈Ekvjhj(K′))p=∞∑n=1(unn∑k=1∑j∈Ekvjhj(K))p. |
∞∑n=1(unn∑k=1∑j∈Ekvjhj(K′))p=∞∑n=1(unn∑k=1∑j∈Ekvjhj(K))p. |
Hence L(h)z,p,E is completely symmetric.
Theorem 8. Let 1<p<∞. The equation L(c)z,p,E=(L(d)z,p,E)′ and the inclusion relation L(d)z,p,E⊆(L(c)z,p,E)′ holds. Also, for a compact operator K, K∈L(d)z,p,E if and only if K′∈(L(c)z,p,E).
Proof. Let 1<p<∞. For K∈B it is known from [8] that cn(K)=dn(K′) and cn(K′)≤dn(K). Also, when K is a compact operator, the equality cn(K′)=dn(K) holds. Thus the proof is clear.
Theorem 9. L(x)z,p,E=(L(y)z,p,E)′ and L(y)z,p,E=(L(x)z,p,E)′ hold for 1<p<∞.
Proof. Let 1<p<∞. For K∈B we have from [8] that xn(K)=yn(K′) and yn(K)=xn(K′). Thus the proof is clear.
The authors would like to thank anonymous referees for their careful corrections and valuable comments on the original version of this paper.
The authors declare no conflict of interest.
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