http://www.aimspress.com/journal/Math DOI:10.3934/math.2019.3.779 Received: 09 April 2019 Accepted: 02 June 2019 Published: 05 July 2019 Research article
The new class L
z,p,Eof s− type operators
Pınar Zengin Alp and Emrah Evren Kara∗
Department of Mathematics, D¨uzce University, Konuralp, Duzce, Turkey * Correspondence: Email: karaeevren@gmail.com.
Abstract: The purpose of this study is to introduce the class of s-type Zu, v; lp(E)
operators, which we denote by Lz,p,E(X, Y), we prove that this class is an operator ideal and quasi-Banach operator
ideal by a quasi-norm defined on this class. Then we define classes using other examples of s-number sequences. We conclude by investigating which of these classes are injective, surjective or symmetric. Keywords: block sequence space; operator ideal; s-numbers; quasi-norm
Mathematics Subject Classification: 47B06, 47B37, 47L20.
1. Introduction
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 [4–6].
A map
which assigns a non-negative scalar sequence to each operator is called an s-number sequence if for all Banach spaces X, Y, X0and Y0the following conditions are satisfied:
(i) kKk= 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) ≤ kMk sr(L) kKk 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 ln2to 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 { kK − Ak : 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 Gel0f andnumber (c
r(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(J K) 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 )| ≤ kK − S k 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 Zu, v; lp
is defined by Malkowsky and Savas¸ as follows:
Zu, v; lp = x ∈ω : ∞ X n=1 un n X k=1 vkxk p < ∞
where 1 < p < ∞ and u= (un) and v= (vn) are positive real numbers.
The Cesaro sequence space cespis defined as ( [10, 11, 19])
cesp = x= (xk) ∈ω : ∞ X n=1 1 n n X k=1 |xk| p < ∞ , 1 < p < ∞.
If an operator K ∈ B (X, Y) satisfies ∞ P n=1 (an(K)) p < ∞ for 0 < p < ∞, K is defined as an l p 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 lptype
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 X0, 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∗∈ X0 and y ∈ Y.
A subcollection = of B is said to be an operator ideal if for each component =(X, Y) = = ∩ B (X, Y) the following conditions hold:
(i) if x∗∈ X0, y ∈ Y, then x∗⊗ y ∈ =(X, Y) , (ii) if L, K ∈ = (X, Y) , then L+ K ∈ = (X, Y) ,
(iii) if L ∈ =(X, Y) , K ∈ B(X0, X) and M ∈ B(Y, Y0), then
MLK ∈ =(X0, Y0) [6].
Let = be an operator ideal and ρ : = → R+be a function on =. Then, if the following conditions hold:
(i) if x∗∈ X0, y ∈ Y, then ρ (x∗⊗ y)= kx∗
k kyk ; (ii) if ∃C ≥ 1 such that ρ (L+ K) ≤ C ρ (L) + ρ (K) ;
(iii) if L ∈ =(X, Y) , K ∈ B(X0, X) and M ∈ B(Y, Y0), then
ρ (MLK) ≤ kMk ρ (L) kKk,
ρ is said to be a quasi-norm on the operator ideal = [6]. For special case C = 1, ρ is a norm on the operator ideal =.
If ρ is a quasi-norm on an operator ideal =, it is denoted by=, ρ. Also if every component = (X, Y) is complete with respect to the quasi-norm ρ,=, ρ is called a quasi-Banach operator ideal.
Let =, 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 J K ∈ = (X, Y0) we have K ∈ = (X, Y) and ρ (J K) = ρ (K), =, ρ is
called an injective quasi-normed operator ideal. Furthermore, let=, 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 ∈ = (X0, Y) we
have K ∈ = (X, Y) and ρ (KS)= ρ (K), =, ρ is called an surjective quasi-normed operator ideal [6]. Let K0 be the dual of K. An s− number sequence is called symmetric (respectively, completely symmetric) if for all K ∈ B, sr(K) ≥ sr(K
0
) (respectively,sr(K)= sr(K
0
)) [6]. Lemma 2. [6] The approximation numbers are symmetric, i.e., ar(K
0
) ≤ ar(K) for K ∈ B .
Lemma 3. [6] Let K ∈ B. Then
cr(K)= dr(K
0
) and cr(K
0
) ≤ dr(K).
In addition , if K is a compact operator then cr(K
0
Lemma 4. [8] Let K ∈ B. Then xr(K) = yr(K 0 ) and yr(K 0 )= xr(K) .
The dual of an operator ideal = is denoted by =0 and it is defined as [6] =0(X, Y) =nK ∈ B(X, Y) : K0 ∈ =Y0, X0o .
An operator ideal = is called symmetric if = ⊂ =0 and is called completely symmetric if == =0 [6]. Let E = (En) be a partition of finite subsets of the positive integers which satisfies
max En< min En+1
for n ∈ N+. In [21] Foroutannia defined the sequence space lp(E) by
lp(E)= x= (xn) ∈ω : ∞ X n=1 X j∈En xj p < ∞ , (1 ≤ p < ∞) with the seminorm k|·|kp,E, which defined as:
k|x|kp,E = ∞ X n=1 X j∈En xj p 1 p .
For example, if En = {3n − 2, 3n − 1, 3n} for all n, then x = (xn) ∈ lp(E) if and only if ∞
P
n=1
|x3n−2+ x3n−1+ x3n|
p < ∞. It is obvious that k|·|k
p,E is not a norm, since we have k|x|kp,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)= lpand k|x|kp,E= kxkp.
For more information about block sequence spaces, we refer the reader to [17, 22–25]. 2. Results
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: Zu, v; lp(E) = x ∈ω : ∞ X n=1 un n X k=1 X j∈Ek vjxj p < ∞ . An operator K ∈ B (X, Y) is in the class of s-type Zu, v; lp(E)
if ∞ X n=1 un n X k=1 X j∈Ek vjsj(K) p < ∞, (1 < p < ∞) . The class of all s-type Zu, 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 .
Theorem 1. Fix 1 < p < ∞. If
∞
P
n=1
(un)
p< ∞ and M > 0 is such that v
2k−1+ v2k ≤ Mvk, M > 0 for all
k ∈ N, then Lz,p,E is an operator ideal.
Proof. Let x∗∈ X0
and y ∈ Y. Since the rank of the operator x∗⊗ y is one, s
n(x∗⊗ y)= 0 for n ≥ 2. By
using this fact
∞ X n=1 un n X k=1 X j∈Ek vjsj(x ∗ ⊗ y) p = ∞ X n=1 (un) p (v1s1(x ∗ ⊗ y))p = ∞ X n=1 (un)p(v1)pkx∗⊗ yk p = ∞ X n=1 (un)p(v1)pkx∗k p kykp < ∞. Therefore x∗⊗ y ∈ L z,p,E(X, Y).
Let L, K ∈ Lz,p,E(X, Y). Then
∞ X n=1 un n X k=1 X j∈Ek vjsj(L) p < ∞, ∞ X n=1 un n X k=1 X j∈Ek vjsj(K) p < ∞. To show that L+ K ∈ Lz,p,E(X, Y), let us begin with
n X k=1 X j∈Ek vjsj(L+ K) ≤ n X k=1 X j∈Ek v2 j−1s2 j−1(L+ K) + X j∈Ek v2 js2 j(L+ K) ≤ n X k=1 X j∈Ek v2 j−1+ v2 j s2 j−1(L+ K) ≤ M n X k=1 X j∈Ek vj sj(L)+ sj(K) ≤ M n X k=1 X j∈Ek vjsj(L)+ X j∈Ek vjsj(K) . By using Minkowski inequality we get;
∞ X n=1 un n X k=1 X j∈Ek vjsj(L+ K) p 1 p ≤ M ∞ X n=1 un n X k=1 X j∈Ek vjsj(L)+ X j∈Ek vjsj(K) p 1 p ≤ M ∞ X n=1 un n X k=1 X j∈Ek vjsj(L) p 1 p + ∞ X n=1 un n X k=1 X j∈Ek vjsj(K) p 1 p < ∞.
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, ∞ X n=1 un n X k=1 X j∈Ek vjsj(MLK) p ≤ ∞ X n=1 un n X k=1 X j∈Ek kMk kKk vjsj(L) p ≤ kMkpkKkp ∞ X n=1 un n X k=1 X j∈Ek vjsj(L) p < ∞. So MLK ∈ Lz,p,E(X0,Y0).
Therefore Lz,p,E(X, Y) is an operator ideal.
Theorem 2. kKkz,p,E = ∞ P n=1 un n P k=1 P j∈Ek vjsj(K) !p!1p ∞ P n=1 (un)p !1p v1
is a quasi-norm on the operator ideal Lz,p,E.
Proof. Let x∗ ∈ X0 and y ∈ Y. Since the rank of the operator x∗⊗ y is one , sn(x∗⊗ y) = 0 for n ≥ 2.
Then ∞ P n=1 un n P k=1 P j∈Ek vjsj(x∗⊗ y) !p!1p ∞ P n=1 (un)p !1p v1 = ∞ P n=1 (un) p ! v1pkx∗⊗ ykp !1p ∞ P n=1 (un)p !1p v1 = kx∗ ⊗ yk= kx∗k kyk . Therefore kx∗⊗ yk z,p,E = kx∗k kyk.
Let L, K ∈ Lz,p,E(X, Y). Then
n X k=1 X j∈Ek vjsj(L+ K) ≤ n X k=1 X j∈Ek v2 j−1s2 j−1(L+ K) + X j∈En v2 js2 j(L+ K) ≤ n X k=1 X j∈Ek v2 j−1+ v2 j s2 j−1(L+ K) ≤ M n X k=1 X j∈Ek vj sj(L)+ sj(K) .
By using Minkowski inequality we get; ∞ X n=1 un n X k=1 X j∈Ek vjsj(L+ K) p 1 p ≤ ∞ X n=1 Mun n X k=1 X j∈Ek vj sj(L)+ sj(K) p 1 p
≤ M ∞ P n=1 un n P k=1 P j∈Ek vjsj(L) !p!1p + P∞ n=1 un n P k=1 P j∈Ek vjsj(K) !p!1p . Hence
kL+ Kkz,p,E ≤ MkS kz,p,E + kKkz,p,E . Let M ∈ B(Y, Y0), L ∈ Lz,p,E(X, Y) and K ∈ B(X0, X)
∞ X n=1 un n X k=1 X j∈Ek vjsj(MLK) p 1 p ≤ ∞ X n=1 un n X k=1 X j∈Ek kMk kKk vjsj(L) p 1 p ≤ kMk kKk ∞ X n=1 un n X k=1 X j∈Ek vjsj(L) p 1 p < ∞ kMLKkz,p,E ≤ kMk kKk kLkz,p,E.
Therefore kKkz,p,E is a quasi-norm on Lz,p,E.
Theorem 3. Let 1 < p < ∞.hLz,p,E(X, Y), kKkz,p,E
i
is a quasi-Banach operator ideal. Proof. Let X, Y be any two Banach spaces and 1 ≤ p < ∞. The following inequality holds
kKkz,p,E = ∞ P n=1 un n P k=1 P j∈Ek vjsj(K) !p!1p ∞ P n=1 (un)p !1p v1 ≥ kKk
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
kKm− Klkz,p,E < ε (2.1)
for all m, l ≥ n0. It follows that
kKm− Klk ≤ kKm− Klkz,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 kKm− Kk → 0 as m → ∞ for some K ∈ B (X, Y) . Now we show that kKm− Kkz,p,E → 0 as
m → ∞for K ∈ Lz,p,E(X, Y) .
The operators Kl− Km, K − Kmare in the class B (X, Y) for Km, Kl, K ∈ B (X, Y) .
Since Kl → K as l → ∞ we obtain
sn(Kl− Km) → sn(K − Km) (2.2)
It follows from (2.1) that the statement
kKm− Klkz,p,E = ∞ P n=1 un n P k=1 P j∈Ek vjsj(Km− Kl) !p!1p ∞ P n=1 (un)p !1p v1 < ε
holds for all m, l ≥ n0.We obtain from (2.2) that
∞ P n=1 un n P k=1 P j∈Ek vjsj(Km− K) !p!1p ∞ P n=1 (un)p !1p v1 ≤ε. Hence we have
kKm− Kkz,p,E < ε for all m ≥ n0.
Finally we show that K ∈ Lz,p,E(X, Y) . ∞ X n=1 un n X k=1 X j∈Ek vjsj(K) p ≤ ∞ X n=1 un n X k=1 X j∈Ek v2 j−1s2 j−1(K)+ un n X k=1 X j∈Ek v2 js2 j(K) p ≤ ∞ X n=1 un n X k=1 X j∈Ek v2 j−1+ v2 j s2 j−1(K − Km+ Km) p ≤ M ∞ X n=1 un n X k=1 X j∈Ek vj sj(K − Km)+ sj(Km) p
By using Minkowski inequality; since Km ∈ Lz,p,E(X, Y) for all m and kKm− Kkz,p,E → 0 as m → ∞,
we have M ∞ X n=1 un n X k=1 X j∈Ek vj sj(K − Km)+ sj(Km) p 1 p ≤ M ∞ P n=1 un n P k=1 P j∈Ek vjsj(K − Km) !p!1p + P∞ n=1 un n P k=1 P j∈Ek vjsj(Km) !p!1p < ∞
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 viaµ = (µi(K)) is defined
as L(µ)z,p,E(X, Y)= K ∈ B(X, Y) : ∞ X n=1 un n X k=1 X j∈Ek vjµj(K) p < ∞, (1 < p < ∞) . And the corresponding norm kKk(µ)z,p,E for each class is defined as
kKk(µ)z,p,E = ∞ P n=1 un n P k=1 P j∈Ek vjµj(K) !p!1p ∞ P n=1 (un)p !1p v1 .
Proposition 4. The inclusion L(a)z,p,E ⊆ L(a)z,q,E holds for1 < p ≤ q < ∞.
Proof. Since lp ⊆ lqfor 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 hL(s)z,p,E, kKk(s)z,p,Ei 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
J K ∈ L(s)z,p,E(X, Y0). Then ∞ X n=1 un n X k=1 X j∈Ek vjsj(J K) p < ∞ Since s= (sn) is injective, we have
sn(K) = sn(J K) for all K ∈ B (X, Y) , n= 1, 2, . . . . (2.3) Hence we get ∞ X n=1 un n X k=1 X j∈Ek vjsj(K) p = ∞ X n=1 un n X k=1 X j∈Ek vjsj(J K) p < ∞ Thus K ∈ L(s)z,p,E(X, Y) and we have from (2.3)
kJ Kk(s)z,p,E = ∞ P n=1 un n P k=1 P j∈Ek vjsj(J K) !p!1p ∞ P n=1 (un)p !1p v1 = ∞ P n=1 un n P k=1 P j∈Ek vjsj(K) !p!1p ∞ P n=1 (un)p !1p v1 = kKk(s) z,p,E
Hence the operator idealhL(s)z,p,E, kKk(s)z,p,Eiis injective. Conclusion 1. [8, p.90–94] Since the number sequences (cn(K)) and (xn(K)) are injective , the
quasi-Banach operator idealshL(c)z,p,E, kKk(c)z,p,EiandhL(x)z,p,E, kKk(x)z,p,Eiare injective.
Theorem 5. Let 1 < p < ∞. The quasi-Banach operator ideal hL(s)z,p,E, kKk(s)z,p,Ei 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
∞ X n=1 un n X k=1 X j∈Ek vjsj(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 ∞ X n=1 un n X k=1 X j∈Ek vjsj(K) p = ∞ X n=1 un n X k=1 X j∈Ek vjsj(KS) p < ∞. Thus K ∈ L(s)z,p,E(X, Y) and we have from (2.4)
kKSk(s)z,p,E = ∞ P n=1 un n P k=1 P j∈Ek vjsj(KS) !p!1p ∞ P n=1 (un)p !1p v1 = ∞ P n=1 un n P k=1 P j∈Ek vjsj(K) !p!1p ∞ P n=1 (un)p !1p v1 = kKk(s) z,p,E.
Hence the operator idealhL(s)z,p,E, kKk(s)z,p,Eiis surjective.
Conclusion 2. [8, p.95] Since the number sequences (dn(K)) and (yn(K)) are surjective , the
quasi-Banach operator idealshL(d)z,p,E, kKk(d)z,p,EiandhL(y)z,p,E, kKk(y)z,p,Eiare 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
∞ X n=1 un n X k=1 X j∈Ek vjsj(K) p < ∞
where 1 < p < ∞. And from Proposition 3, we have; ∞ X n=1 un n X k=1 X j∈Ek vjhj(K) p ≤ ∞ X n=1 un n X k=1 X j∈Ek vjxj(K) p ≤ ∞ X n=1 un n X k=1 X j∈Ek vjcj(K) p ≤ ∞ X n=1 un n X k=1 X j∈Ek vjaj(K) p < ∞ and ∞ X n=1 un n X k=1 X j∈Ek vjhj(K) p ≤ ∞ X n=1 un n X k=1 X j∈Ek vjyj(K) p ≤ ∞ X n=1 un n X k=1 X j∈Ek vjdj(K) p ≤ ∞ X n=1 un n X k=1 X j∈Ek vjaj(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 Lz,p,E(h) 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 ⊆ Lz,p,E(a)
0
holds. Let K ∈ L(a)z,p,E. Then
∞ X n=1 un n X k=1 X j∈Ek vjaj(K) p < ∞. It follows from [6, p.152] an
K0 ≤ an(K) for K ∈ B. Hence we get
∞ X n=1 un n X k=1 X j∈Ek vjaj T0 p ≤ ∞ X n=1 un n X k=1 X j∈Ek vjaj(K) p < ∞. Therefore K ∈L(a)z,p,E
0
. Thus L(a)z,p,E is symmetric. Now we prove that the equation L(h)z,p,E = L(h)z,p,E
0
holds. It follows from [8, p.97] that hn
K0 = hn(K) for K ∈ B. Then we can write
∞ X n=1 un n X k=1 X j∈Ek vjhj K0 p = ∞ X n=1 un n X k=1 X j∈Ek vjhj(K) p .
∞ X n=1 un n X k=1 X j∈Ek vjhj K 0 p = ∞ X n=1 un n X k=1 X j∈Ek vjhj(K) p .
Hence Lz,p,E(h) is completely symmetric.
Theorem 8. Let 1 < p < ∞. The equation L(c)z,p,E = L(d)z,p,E
0
and the inclusion relation L(d)z,p,E ⊆ L(c)z,p,E
0
holds. Also, for a compact operator K, K ∈ L(d)z,p,E if and only if K0 ∈L(c)z,p,E.
Proof. Let 1 < p < ∞. For K ∈ B it is known from [8] that cn(K) = dn
K0 and cn
K0 ≤ dn(K).
Also, when K is a compact operator, the equality cn
K0 = dn(K) holds. Thus the proof is clear.
Theorem 9. L(x)z,p,E = L(y)z,p,E
0
and L(y)z,p,E =L(x)z,p,E
0
hold for1 < p < ∞. Proof. Let 1 < p < ∞. For K ∈ B we have from [8] that xn(K) = yn
K0and yn(K) = xn
K0 . Thus
the proof is clear.
Acknowledgments
The authors would like to thank anonymous referees for their careful corrections and valuable comments on the original version of this paper.
Conflict of interest
The authors declare no conflict of interest. References
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