# A Note on Generalized Approximation Property

Authors: Antara Bhar; Manjul Gupta
Journal: Journal of Function Spaces and Applications
 (2013)

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## Abstract

We introduce a notion of generalized approximation property, which we refer to asLs-λ-AP possessed by a Banach space X, corresponding to an arbitrary Banach sequence space λ and a convex subset Ls(X) of ℒ(X), the class of bounded linear operators on X. This property includes approximation property studied by Grothendieck, p-approximation property considered by Sinha and Karn and Delgado et al., and also approximation property studied by Lissitsin et al. We characterize a Banach space having Ls-λ-AP with the help of λ-compact operators, λ-nuclear operators, and quasi-λ-nuclear operators. A particular case for λ=lp (1≤p≤∞) has also been characterized.


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## Body

## 1. Introduction

It is well known that the identity on an infinite dimensional Banach space is never compact, though it may be approximated by finite rank operators in the pointwise convergence topology, for instance, in the case whenX has a Schauder base. If the identity on X is approximated uniformly on compact sets by finite rank operators, it leads to the notion of approximation property of X, studied systematically by Grothendieck [1] in 1955. Now there are many reformulations of this property, all of them involve either subspaces or ideals of operators, for example, class of finite rank operators, compact operators, and so forth. One may refer to [2–5] and references given therein.In our recent work, using the duality theory of sequence spaces, we considered the notion ofλ-compact sets corresponding to a suitably restricted sequence space λ and studied λ-compact operators, specially their relationships with λ-summing, λ-nuclear, and quasi-λ-nuclear which were earlier considered by Ramanujan [6] in 1970. Replacing compact sets or p-compact sets by λ-compact sets, where λ is a Banach sequence space and the class of finite dimensional operators or compact operators, and so forth, by an arbitrary convex subset Ls(X) of the class ℒ(X) of bounded linear operators on a Banach space, we define and study Ls-λ-AP of a Banach space X in this paper. After giving preliminaries in the next section, we characterize this property in Section 3 and study its particular case for λ=lp in Section 4. This property includes, as particular cases, approximation property, p-approximation property, compact approximation property, and subspace approximation property (cf. [2–5]).


## 2. Preliminaries

Throughout this paper, we denote by (X, ∥·∥X) a Banach space equipped with the norm ∥·∥X and by X* its topological dual equipped with the dual operator norm topology ∥·∥X*.Let us begin with the basics of the sequence space theory, for which our reference is [7]. Let ω denote the family of all real or complex sequences, which is a vector space with usual pointwise addition and scalar multiplication, and let ϕ be the span of en’s (n≥1), where en is the nth unit vector in ω; that is, en={0,0,…,1,0,0,…}, where 1 is the nth coordinate of the sequence en. A sequence spaceλ is a subspace of ω containing ϕ. Members of λ are denoted by the symbols α-, β-, and so forth, where  α-={α1,α2,α3,…}  and β-={β1,β2,β3,…}. The nth section of α- for n∈ℕ is written as α-(n) and is defined as α-(n)={α1,α2,…,αn,0,0,…}; that is, α-(n)=∑i=1n‍αiei.A sequence spaceλ is called (i) symmetric if ασ-={ασ(i)}∈λ whenever α-={αi}∈λ and σ∈Π, where Π is collection of all permutations of the set of natural numbers ℕ, (ii) monotone if m0λ⊆λ, where m0=span{A}, and A is the collection of all sequences consisting of zero and one, and (iii) normal or solid if β-={βi}∈λ whenever |βi|≤|αi|, i≥1 for some α-={αi}∈λ.Theα-dual, cross-dual, or Köthe-dualλα or λ× of λ is defined as
(1)λ×≡λα={β-={βi}∈ω:∑i≥1‍|αiβi|converges,∀α-∈λ}.
A sequence space λ is said to be perfect if λ=λ××=(λ×)×. Every perfect sequence space is normal, and every normal sequence space is monotone.A Banach sequence space (λ, ∥·∥λ) is called a BK-space provided that each of the projection maps Pi:λ→𝕂, Pi(α-)=αi is continuous, for i≥1, where 𝕂 is the field of scalars and α-={α1,α2,…}. A BK-space (λ, ∥·∥λ) is called an AK-space if α-(n)→α-, for each α-∈λ.For a BK-space (λ, ∥·∥λ), we define the dual norm on λ× as follows:
(2)∥β-∥λ×=sup{∑i≥1‍|αi||βi|:α-∈λ,∥α-∥λ≤1}.
The space (λ×,  ∥·∥λ×) can easily be shown to be a BK-space provided that 0<supn∥en∥λ<∞.The norm∥·∥λ is said to be (i) k-symmetric if ∥α-∥λ=∥α-σ∥λ, for all σ∈Π and (ii) monotone if ∥α-∥λ≤∥β-∥λ for α-, β- in λ with |αi|≤|βi|, for all i≥1.AnOrlicz functionM is a continuous, convex function defined from [0,∞) to itself such that M(0)=0, M(x)>0 for x>0. Such function M always has the integral representation
(3)M(x)=∫0x‍p(t)dt,
where p(t), known as the kernel of M, is right continuous, nondecreasing function for t>0. Let us note that an Orlicz function M is always increasing, and M(x)→∞ as x→∞. Also tp(t)→∞ as t→∞, and tp(t)=0 for t=0 (cf. [4], page 139). However, p(t)>0 for t=0 is equivalent to the fact that the Orlicz sequence space lM (see the definition below) is isomorphic to l1 (cf. [7], page 309). Therefore, we assume here that the kernel p(t) has value 0 for t=0 and obviously p(t)→∞ as t→∞.Given an Orlicz functionM with kernel p, define q(s)=sup{t:p(t)≤s}, s≥0. Then q possesses the same properties as p, and the function N defined as N(x)=∫0x‍q(t)dt is an Orlicz function. The functions M and N are called mutually complementary Orlicz functions. For such functions M and N, we have Young’s inequality: xy≤M(x)+N(y), for x,y≥0 and also M(αx)≤αM(x) for 0<α<1 (cf. [7]).An Orlicz functionM is said to satisfy the Δ2-condition for smallxor at 0, if, for each k>1, there exist Rk>0 and xk>0 such that
(4)M(kx)≤RkM(x),∀x∈(0,xk].Corresponding to an Orlicz functionM, the set l~M is defined by
(5)l~M={α-∈ω:∑i≥1‍M(|αi|)<∞}.
If M and N are mutually complementary functions, the Orlicz sequence spacelM is defined as
(6)lM={α-∈ω:∑i≥1‍βiαiconverges,∀{βi}∈l~N}.
An equivalent way of defining lM is
(7)lM={α-∈ω:∑i≥1‍M(|αi|ρ)<∞forsomeρ>0}.
Two equivalent norms on lM are given by
(8)∥α-∥(M)=sup{|∑i≥1‍βiαi|:∑i≥1‍N(|βi|)≤1},∥α-∥M=inf{ρ>0:∑i≥1‍M(|αi|ρ)≤1};
indeed,
(9)∥α-∥M≤∥α-∥(M)≤2∥α-∥M,forα-∈lM.IfM and N are mutually complementary Orlicz functions and M satisfies Δ2-condition at 0, then (lM)×=lN (cf. [7]).Corresponding to a sequence spaceλ and a Banach space X with its topological dual X* equipped with the operator norm topology generated by ∥·∥, the vector-valued sequence spaces λs(X) and λw(X) defined below, have been introduced and studied earlier in [6], under different notations. Indeed, we have
(10)λs(X)={x-={xn}⊂X:{∥xn∥X}∈λ},λw(X)={x-={xn}⊂X:{f(xn)}∈λ,∀f∈X*}.
In case ∥·∥λ is a monotone norm, the space λs(X) becomes a normed linear space with respect to the norm defined as
(11)∥x-∥λ=∥{∥xn∥X}∥λ,x-={xn}∈λs(X).
However, for x-∈λw(X), the norm on λw(X) is defined as
(12)∥x-∥λw=sup{∥{f(xn)}∥λ,f∈X*,∥f∥≤1},x-={xn}∈λw(X),which can be proved to be finite by applying closed graph theorem. For the norm∥y-∥λ×w on (λ×)w(X), we assume throughout that 0<supn∥en∥λ<∞ so that (λ×)w(X) equipped with this norm becomes a K-space.Corresponding to a Banach sequence spaceλ with 0<supn∥en∥λ<∞ and a Banach space X, we have introduced the concept of λ-compact sets and λ-compact operators in our work [8]. A subset K of X is said to be λ-compact if there exists x-={xn}∈λs(X) such that K⊆λ-co{xn}={∑n≥1‍αnxn:α-={αn}∈Bλ×}. For λ=lp, 1≤p≤∞, λ-compact sets are nothing but p-compact sets studied in [9].Concerning theλ-compact sets, we have the following [8].Theorem 1.
Letλ be a normal sequence space equipped with a monotone norm ∥·∥λ satisfying the condition 0<supn∥en∥λ<∞. Also let (λ, ∥·∥λ) be an AK-BK-reflexive sequence space. Then for x-={xn}∈λs(X), the set {∑n≥1‍αnxn:α-={αn}∈Bλ×} is norm closed in X.For Banach spacesX and Y, the symbol ℒ(X,Y) denotes the class of all bounded linear operators from X to Y, and ℒ denotes the collection of all bounded operators between any pair of Banach spaces. The notation 𝔉(X,Y), K(X,Y), and W(X,Y), respectively, stand for collection of all finite rank, compact, and weakly compact operators from X to Y. For Y=X, we write ℒ(X,Y)≡ℒ(X), 𝔉(X,Y)≡𝔉(X), and so forth.The spaceℒ(X,Y) endowed with the locally convex topology τp(1≤p≤∞) has been studied in [10], where τp is the topology of uniform convergence on p-compact sets in X. Concerning the dual space (ℒ(X,Y),τp)* for 1<p<∞, Choi and Kim [10] have proved the following.Theorem 2.
Let1<p, q<∞, and 1/p+1/q=1. Then (ℒ(X,Y),τp)* consists of all linear functionals φ of the form
(13)φ(T)=∑j≥1‍∑n≥1‍αnjgj(Txn),
where x-={xn}∈lps(X); α-j={αnj}n≥1 from lq, for each j∈ℕ and {gj}⊆Y* with ∑j≥1‍∥α-j∥q∥gj∥<∞.Letλ be a normal BK-sequence space equipped with the norm ∥·∥λ satisfying 0<infn∥en∥λ≤supn∥en∥λ<∞. An operator T∈ℒ(X,Y) is said to be (i)
λ-compact operator if T(BX) is a λ-compact set in Y; that is, T maps bounded sets in X to λ-compact sets in Y,(ii)
λ-nuclear if T has the representation
(14)Tx=∑n≥1‍αn<x,fn>yn,where{fn}⊆X* with ∥fn∥X*≤1, for each n; y-={yn}∈(λ×)w(Y) with ∥y-∥λ×w≤1 and α¯={αn}∈λ,(iii)
quasi-λ-nuclearif there exists {fn}⊆X* such that f-={fn}∈λs(X*) and ∥Tx∥≤∥{fn(x)}∥λ, for each x∈X.The symbols Kλ(X,Y), Nλ(X,Y), and QNλ(X,Y), respectively, denote the collection of all λ-compact, λ-nuclear, and quasi-λ-nuclear operators from X to Y.For the relationships ofλ-compact, λ-nuclear, and quasi-λ-nuclear operators, we have the following [8].Theorem 3.
Let (λ, ∥·∥λ) be a normal, symmetric, BK-sequence space with 0<infn∥en∥λ≤supn∥en∥λ<∞. Consider the following.(i)
For any pair of Banach spacesX and Y, if T∈ℒ(X,Y) is a λ-nuclear operator, then T*∈ℒ(Y*,X*) is a λ-compact operator;(ii)
ifT∈ℒ(X,Y) is a λ-compact operator, then T*∈ℒ(Y*,X*) is a quasi-λ-nuclear operator.Forλ=lp(1≤p<∞) and T∈ℒ(X,Y), we quote from [11].Proposition 4.
(i)T∈Kp(X,Y) if and only if T*∈QNp(Y*,X*); T∈QNp(X,Y) if and only if T*∈Kp(Y*,X*);
(ii)T*∈QNp(Y*,X*) if and only if there exists {yn}∈lps(Y) such that ∥T*g∥≤∥{g(yn)}∥p, for all g∈Y*.We also make use of the following result from [6].Theorem 5.
Letλ be a symmetric normal AK-BK-sequence space equipped with a monotone norm ∥·∥λ satisfying 0<supn∥en∥λ<∞. Then T∈QNλ(X,Y) implies that T∈Nλ(X,Y) if the range space Y has extension property; that is, for every injection J∈ℒ(X0,X) and every operator S0∈ℒ(X0,Y), there exists an extension S∈ℒ(X,Y) such that ∥S0∥=∥S∥ and S0=SJ.For the salient features on operator ideals, one is referred to [12].Writing𝒜(X,Y)=𝒜∩ℒ(X,Y), a subset 𝒜 of ℒ is said to be an operator ideal if it satisfies the following conditions: (i)
𝒜contains all finite rank operators;(ii)
T+S∈𝒜(X,Y) for S,T∈𝒜(X,Y);(iii)
ifT∈𝒜(X,Y) and S∈ℒ(Y,Z), then ST∈𝒜(X,Z), and also if  T∈ℒ(X,Y) and S∈𝒜(Y,Z), then ST∈𝒜(X,Z).The collection𝒜(X,Y), for a given pair of Banach spaces X and Y, is called a componentof 𝒜.It has been shown in [6, 8] that the collections of λ-compact, λ-nuclear, and quasi-λ-nuclear operators from X to Y are operator ideals for suitably chosen sequence space λ.For an operator ideal𝒜, the dual operator ideal𝒜d is defined as the one of which the component 𝒜d(X,Y) is given by
(15)𝒜d(X,Y)={T∈ℒ:T*∈𝒜(Y*,X*)}.For an operator ideal𝒜, the subspace 𝒜w*(X*,Y) of the component 𝒜(X*,Y) is defined as
(16)𝒜w*(X*,Y)={T∈𝒜(X*,Y):Tisσ(X*,X)-σ(Y,Y*)continuous}.


## 3.L-Subset λ-Approximation Property

Throughout this section we denote byX, a Banach space and by λ, a BK-sequence space equipped with a norm ∥·∥λ such that 0<supn∥en∥λ<∞. Let Ls(X) be a convex subset of ℒ(X). Recalling the definition of λ-compact sets in X from the previous section, we introduce the following.Definition 6.
A Banach spaceX is said to have L-subset λ-approximation property (Ls-λ-AP) if given ε>0 and any λ-compact set K in X; there exists R∈Ls(X) such that supx∈K∥Rx-x∥X<ε; that is, the identity map on X is approximated uniformly on a λ-compact set by a member of Ls(X).IfLs(X)=𝔉(X) and λ=c0, it is the approximation property studied by Grothendieck [1]. If Ls(X)=𝔉(X) and λ=lp, it is the p-approximation property (cf. [3, 9]), and if Ls(X)=K(X) and λ=c0, it is the compact approximation property (cf. [4]).Theorem 7.
Let(λ,∥·∥λ) be an AK-BK-reflexive sequence space such that the norm ∥·∥λ is monotone satisfying the condition 0<supn∥en∥λ<∞. Assume that λ is also monotone. Then for a convex subset Ls(X) of ℒ(X), the following conditions are equivalent. (i)
X has Ls-λ-AP.(ii)
For any Banach spaceY and for any T∈Kλ(Y,X), T∈{RT:R∈Ls(X)}-∥·∥, the closure being considered with respect to the norm ∥·∥ in ℒ(Y,X).Proof.
(
i
)
⇒
(
ii
) Let T∈Kλ(Y,X). Then T(BY) is λ-compact set in X. Since X has Ls-λ-AP, for ε>0 there exists R0∈Ls(X) such that supx∈T(BY)∥x-R0x∥<ε⇒∥T-R0T∥<ε. Thus T∈{RT:R∈Ls(X)}-∥·∥.
(
ii
)
⇒
(
i
) For proving (i), consider a λ-compact set K of the form K=λ-co{xn} for some x¯={xn}∈λs(X). In view of Theorem 1, K is norm closed and so it is complete. Hence the space Y=⋃n≥1‍nK is a Banach space with respect to the norm |||·||| defined as |||y|||=inf{ρ>0:y∈ρK} for y∈Y. As K is the unit ball of Y, the inclusion map I from Y to X is λ-compact; that is, I∈Kλ(Y,X). Therefore by hypothesis, given any ε>0, we can find that R0∈Ls(X) such that
(17)∥I-R0∥=supx∈K∥x-R0x∥<ε.
This proves that X has Ls-λ-AP.The above result leads to the following.Proposition 8.
Let (λ, ∥·∥λ) be a sequence space as in Theorem 7, and let Ls(X) be a convex subset of ℒ(X). Then X has Ls-λ-AP if T∈{TR*:R∈Ls(X)}-∥·∥, for any Banach space Y and any T∈QNλ(X*,Y).Proof.
LetT∈Kλ(Y,X) where Y is an arbitrary Banach space. Then T*∈QNλ(X*,Y*) by Theorem 3(ii). Hence for any ε>0, there exists R0∈Ls(X) such that ∥T*-T*R0*∥<ε, by hypothesis. Thus ∥T-R0T∥<ε; that is, T∈{RT:R∈Ls(X)}-∥·∥. Now apply Theorem 7 to get that X has Ls-λ-AP.If the Banach spacesX and Y are restricted further, we get the converse of Proposition 8 in the following form.Proposition 9.
Let (λ, ∥·∥λ) be as in Proposition 8. Assume that X is a reflexive Banach space and Y is a Banach space having extension property. If X has Ls-λ-AP, then T∈{TR*:R∈Ls(X)}-∥·∥, for any T∈QNλ(X*,Y).Proof.
LetT∈QNλ(X*,Y). Then T∈Nλ(X*,Y) by Theorem 5, and so T*∈Kλ(Y*,X) by Theorem 3(i). By hypothesis and Theorem 7, for ε>0, there exists R0∈Ls(X) such that ∥T-TR0*∥=∥T*-R0T*∥<ε; that is, T∈{TR*:R∈Ls(X)}-∥·∥.For the dualX* of X, we introduce the following.Definition 10.
LetX be a Banach space with its topological dual X* equipped with the operator norm topology, and let Ls(X) be a convex subset of ℒ(X). Then X* is said to haveLs-λapproximation property with conjugate operator (Ls*-λ-AP) if, for anyλ-compact set K* of X* and ε>0, there exists R0∈Ls(X) such that supf∈K*∥R0*f-f∥<ε; that is, the identity operator on X* is approximation by a conjugate operator of a member in Ls(X), uniformly on λ-compact subset of X*.ConcerningLs*-λ-AP of X*, we prove the following.Theorem 11.
Let (λ, ∥·∥λ) and Ls(X) be as in Theorem 7.(i)
IfY is a Banach space with extension property and X* has Ls*-λ-AP, then T∈{TR:R∈Ls(X)}-∥·∥ for any T∈QNλ(X,Y).(ii)
LetX be a reflexive Banach space. If, for any Banach space Y and T∈QNλ(X,Y), T∈{TR:R∈Ls(X)}¯∥·∥, then X* has Ls*-λ-AP.Proof.
Omitted as these are analogous to the proofs of Propositions8 and 9.Remark 12.
Let us note here that the restriction onλ is such that we may include the Orlicz sequence spaces lM, where M is an Orlicz function satisfying Δ2-condition at zero. One may call such approximation property as Ls-lM-AP and may be worth investigating further. However, for Ls(X)=𝔉(X), we illustrate lM-AP by the following.Example 13.
We consider Orlicz functionsM1(u)=u2/(log(e+u)) and M2(u)=u(eu-1), for u≥0. Then lM1=l2=lM2 (cf. [13]). Therefore lN1=l2=lN2, where Ni are the complementary Orlicz functions of Mi, respectively, for i=1,2. Let K be an lN1-compact set in X. Then there exists x¯={xn}∈lN1s(X)=l2s(X) such that K⊆{∑n≥1‍αnxn:α¯={αn}∈BlM1}. As ∥α¯∥M1≤∥α¯∥2≤C∥α¯∥M1, we have K⊆{∑n≥1‍αn(Cxn):α¯={αn}∈Bl2}. Thus every lN1-compact set is l2-compact set. Hence every Banach space has lN1-approximation property (cf. [9], page 27). Similarly we can show that every Banach space has lN2-approximation property.


## 4.L-Subset p-Approximation Property

In this section we studyLs-p-approximation property which is a particular case of L-subset λ approximation property for λ=lp(1≤p≤∞). Throughout this section, we denote by Ls(X) a convex subset of ℒ(X), where X is a Banach space. Let us begin with the following.Definition 14.
LetX be a Banach space with X* as its topological dual equipped with the operator norm topology. Then (i) X is said to have Ls(X)-p-approximation property(Ls-p-AP) if, for ε>0 and a p-compact subset K of X, there exists R0∈Ls(X) such that supx∈K∥R0x-x∥X<ε, and (ii) X* is said to have Ls(X)-p-approximation property with conjugate operator(Ls*-p-AP) if, for any ε>0 and a p-compact set K* in X*, there exists R0∈Ls(X) such that supf∈K*∥R0*f-f∥<ε.Characterizing Banach spaces havingLs-p-AP, we have the following.Theorem 15.
The following conditions are equivalent.(i)
X has Ls-p-AP.(ii)
For any Banach spaceY and any T∈Kp(Y,X), T∈{RT:R∈Ls(X)}¯∥·∥.(iii)
For any Banach spaceY and any T∈Kp(Y,X), T∈{RT:R∈Ls(X)}¯τ, where τ is the topology of uniform convergence on compact subsets of Y.Proof.
(
i
)
⇔
(
ii
) follows from Theorem 7.
(
ii
)
⇒
(
iii
) is obvious as norm topology is finer than τ.
(
iii
)
⇒
(
i
) The proof of this implication is almost the same as that of Theorem 1 (c⇒a) proved in [3]. However, for the sake of completeness, we sketch its proof. Let K be a p-compact subset of X of the type K={∑n≥1‍αnxn:α¯={αn}∈Blq}, where x¯={xn}∈lps(X) and 1/p+1/q=1. We can find that β¯={βn}∈c0 such that {βn-1xn}∈lps(X). Now let us consider the diagonal map D:lq→lq defined as D({αn})={αnβn} and ψ:lq→X such that ψ({αn})=∑n≥1‍αnβn-1xn. Clearly, D is a compact operator, and ψ is a p-compact operator. Write Y=lq/kerψ and Q for the quotient map from lq to Y. Define a p-compact operator ψ^:Y→X as ψ^(Q(α¯))=ψ(α¯). Now, Q∘D(Blq) is a compact set in Y. By hypothesis, for this compact set and any ε>0, there exists R0∈Ls(X) such that
(18)supx∈K∥R0x-x∥=supα¯^∈Q∘D(Blq)∥R0ψ^(α¯^)-ψ^(α¯^)∥<ε.
As K is an arbitrary p-compact set, the Banach space X has Ls-p-AP.To prove the next result we make use of the following.Lemma 16.
The Banach spaceX has Ls-p-AP for 1<p<∞ if and only if
(19)infR∈Ls(X)|∑j≥1‍∑n≥1‍αnjfj(Rxn-xn)|=0,
where x¯={xn}∈lps(X); α¯j={αnj}n≥1 are in lq for each j∈ℕ and {fj} in X* satisfying the condition ∑j≥1‍∥α¯j∥q∥fj∥X*<∞.Proof.
X has Ls-p-AP⇔I∈Ls(X)¯τp⇔infR∈Ls(X)|φ(I-R)|=0, for φ in (ℒ(X),τp)*. Now the result is immediate from Theorem 2.Making use of Proposition4(i) and the above lemma, we further characterize Ls-p-AP in the following.Theorem 17.
For1<p, q<∞, and 1/p+1/q=1, the following conditions are equivalent. (i)
X has Ls-p-AP.(ii)
For any Banach spaceY and for any T∈QNpw*(X*,Y), T∈{TR*:R∈Ls(X)}¯∥·∥.(iii)
For any reflexive separable Banach spaceY and any T∈QNpw*(X*,Y), there exists a net {Ri}i∈I from Ls(X) such that TRi*→T in weak operator topology on ℒ(X*,Y); that is, TRi*f→Tf in weak topology σ(Y,Y*) of Y for each f∈X*.Proof.
(
i
)
⇒
(
ii
) Let T∈QNpw*(X*,Y). As T is σ(X*,X)-σ(Y,Y*) continuous, range(T*)⊂X and so T*∈Kp(Y*,X). Now for given ε>0, there exists R0∈Ls(X) such that ∥R0T*-T*∥<ε by Theorem 15. Hence ∥TR0*-T∥<ε, and (ii) follows.
(
ii
)
⇒
(
iii
) Immediate.
(
iii
)
⇒
(
i
) For proving that X has Ls-p-AP, we make use of Lemma 16. Hence we may assume without loss of generality that x¯={xn}∈lps(X), α¯j={αnj}n≥1∈Blq, for each j≥1, and {fj}⊂X* satisfies ∑j≥1‍∥fj∥X*<∞. Consider the map ψ:lq→X defined as ψ({αn})=∑n≥1‍αnxn. Then ψ is a p-compact map. Hence ψ*:X*→lp is a quasi-p-nuclear map by Proposition 4(i); that is, ψ*∈QNp(X*,lp). As lp is a reflexive separable Banach space and ψ**:lq→X**, we get range(ψ**)⊂X. Consequently, ψ*∈QNpw*(X*,lp). Now by hypothesis, there exists a net {Ri}i∈I from Ls(X) such that ψ*Ri*→ψ* in weak operator topology on ℒ(X*,lp). Hence by uniform boundedness principle, there exists M≥0 such that ∥ψ*Ri*∥≤M, for all i∈I. Also we get that ψ*Ri*(fj)→ψ*(fj), for each j≥1. Choose Ri0 in {Ri}i∈I such that
(20)|(ψ*Ri0*(fj)-ψ*(fj))(α¯j)|<ε2n0,forj=1,2,…,n0,
where n0 is the integer satisfying
(21)∑j≥n0+1‍∥fj∥<ε2(M+∥x¯∥ps).
Now consider
(22)|∑j≥1‍∑n≥1‍αnjfj(Ri0xn-xn)|=|∑j≥1‍(ψ*Ri0*fj-ψ*fj)(α¯j)|≤∑j=1n0‍|(ψ*Ri0*fj-ψ*fj)(α¯j)|+∑j≥n0+1‍|(ψ*Ri0*fj-ψ*fj)(α¯j)|<ε2+ε2,
since (∥ψ*Ri0*∥+∥ψ*∥)≤M+∥x¯∥ps and ∥α¯j∥q≤1, for all j≥1.
HenceinfR∈Ls(X)|∑j≥1‍∑n≥1‍αnjfj(Rxn-xn)|=0. Therefore X hasLs-p-AP by Lemma 16.A characterization ofLs*-p-AP of a Banach space X* is contained in the following.Theorem 18.
X
* has Ls*-p-AP if and only if, for every Banach space Y and every T∈QNp(X,Y), T∈{TR:R∈Ls(X)}¯∥·∥.Proof.
The proof of necessity part is omitted as it is analogous to that of(i)⇒(ii), Theorem 7.
For converse, let us consider thatT∈Kp(Y,X*), where Y is an arbitrary Banach space. Then T*∈QNp(X**,Y*) by Proposition 4(i). Using Proposition 4(ii), we get T*∈QNp(X,Y*). Now, by hypothesis, for given ε>0, we can find that R0∈Ls(X) such that ∥T*R0-T*∥<ε. Hence ∥R0*T-T∥<ε. Now apply Theorem 15 to get that X* has Ls*-p-AP.Finally, we consider a result involving operator ideals andp-compact operators, which is analogous to the one proved in [5] for compact operators.Proposition 19.
Let𝒜 be an operator ideal and Y an arbitrary Banach space. If, for R∈Ls(X) and any T∈Ad(Y,X), RT is a p-compact operator from Y to X, then for R∈Ls(X) and any T∈𝒜w*(X*,Y) the operator RT* is also p-compact operator from Y* to X. Converse follows when 𝒜⊆QNp.Proof.
IfR∈Ls(X) and T∈𝒜w*(X*,Y), then range(T*)⊂X. Then T*=JXTas where Tas:Y*→X is the astriction map of T*, and JX is the canonical embedding from X to X**. Then Tas∈𝒜d(Y*,X) as Tas*=JYT∈𝒜(X*,Y**). By hypothesis, RT*=RTas∈Kp(Y*,X).
For converse, consider thatR∈Ls(X) and T∈𝒜d(Y,X). Then T*∈QNp(X*,Y*) by hypothesis. By Proposition 4(i) T∈Kp(Y,X)⊂K(Y,X)⊂W(Y,X). Consequently, range(T**)⊂X (cf. [14], page 185) (see also [15], page 341). Hence T*∈𝒜w*(X*,Y*). Consequently, RT**∈Kp(Y**,X) by hypothesis. As RT=RT**|Y, the result follows.



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*Source: 325141-2013-10-02.xml*