A Note on Generalized Approximation Property

Abstract

We introduce a notion of generalized approximation property, which we refer to as 𝐿_𝑠-𝜆-AP possessed by a Banach space 𝑋, corresponding to an arbitrary Banach sequence space 𝜆 and a convex subset 𝐿_𝑠(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 𝐿_𝑠-𝜆-AP with the help of 𝜆-compact operators, 𝜆-nuclear operators, and quasi-𝜆-nuclear operators. A particular case for 𝜆=𝑙_𝑝 (1≤𝑝≤∞) has also been characterized.

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 when 𝑋 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 of 𝜆-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 𝐿_𝑠(𝑋) of the class ℒ(𝑋) of bounded linear operators on a Banach space, we define and study 𝐿_𝑠-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=𝑙_𝑝 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 𝑒^𝑛’s (𝑛≥1), where 𝑒^𝑛 is the 𝑛th unit vector in 𝜔; that is, 𝑒^𝑛={0,0,…,1,0,0,…}, where 1 is the 𝑛th coordinate of the sequence 𝑒^𝑛. A sequence space 𝜆 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,…,𝛼_𝑛,0,0,…}; that is, 𝛼^(𝑛)=∑^𝑛_𝑖=1𝛼_𝑖𝑒^𝑖.

A sequence space 𝜆 is called (i) symmetric if 𝛼_𝜎={𝛼_𝜎(𝑖)}∈𝜆 whenever 𝛼={𝛼_𝑖}∈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={𝛽_𝑖}∈𝜆 whenever |𝛽_𝑖|≤|𝛼_𝑖|, 𝑖≥1 for some 𝛼={𝛼_𝑖}∈𝜆.

The 𝛼-dual, cross-dual, or Köthe-dual 𝜆^𝛼 or 𝜆^× of 𝜆 is defined as 𝜆^×≡𝜆^𝛼=𝛽=𝛽_𝑖∈𝜔∶_𝑖≥1𝛼_𝑖𝛽_𝑖converges,∀𝛼∈𝜆. (1) 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 𝑃_𝑖∶𝜆→𝕂, 𝑃_𝑖(𝛼)=𝛼_𝑖 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: 𝛽_𝜆×=sup_𝑖≥1𝛼_𝑖𝛽_𝑖∶𝛼∈𝜆,‖𝛼‖_𝜆≤1. (2) The space (𝜆^×,  ‖⋅‖_𝜆^×) can easily be shown to be a BK-space provided that 0<sup_𝑛‖𝑒^𝑛‖_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 |𝛼_𝑖|≤|𝛽_𝑖|, for all 𝑖≥1.

An Orlicz function 𝑀 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 𝑀(𝑥)=^𝑥₀𝑝(𝑡)𝑑𝑡, (3) 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 𝑙_𝑀 (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 function 𝑀 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)=∫^𝑥₀𝑞(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 function 𝑀 is said to satisfy the Δ₂-condition for small 𝑥 or at 0, if, for each 𝑘>1, there exist 𝑅_𝑘>0 and 𝑥_𝑘>0 such that 𝑀(𝑘𝑥)≤𝑅_𝑘𝑀(𝑥),∀𝑥∈0,𝑥_𝑘. (4)

Corresponding to an Orlicz function 𝑀, the set 𝑙_𝑀̃ is defined by 𝑙_𝑀̃=𝛼∈𝜔∶_𝑖≥1𝑀𝛼_𝑖<∞. (5) If 𝑀 and 𝑁 are mutually complementary functions, the Orlicz sequence space 𝑙_𝑀 is defined as 𝑙_𝑀=𝛼∈𝜔∶_𝑖≥1𝛽_𝑖𝛼_𝑖converges,∀𝛽_𝑖∈𝑙_𝑁̃. (6) An equivalent way of defining 𝑙_𝑀 is 𝑙_𝑀=𝛼∈𝜔∶_𝑖≥1𝑀𝛼_𝑖𝜌<∞forsome𝜌>0. (7) Two equivalent norms on 𝑙_𝑀 are given by ‖𝛼‖_(𝑀)=sup_𝑖≥1𝛽_𝑖𝛼_𝑖∶_𝑖≥1𝑁𝛽_𝑖,‖≤1𝛼‖_𝑀=inf𝜌>0∶_𝑖≥1𝑀𝛼_𝑖𝜌≤1; (8) indeed, ‖𝛼‖_𝑀≤‖𝛼‖_(𝑀)≤2‖𝛼‖_𝑀,for𝛼∈𝑙_𝑀. (9)

If 𝑀 and 𝑁 are mutually complementary Orlicz functions and 𝑀 satisfies Δ₂-condition at 0, then (𝑙_𝑀)^×=𝑙_𝑁 (cf. [7]).

Corresponding to a sequence space 𝜆 and a Banach space 𝑋 with its topological dual 𝑋^∗ equipped with the operator norm topology generated by ‖⋅‖, the vector-valued sequence spaces 𝜆^𝑠(𝑋) and 𝜆^𝑤(𝑋) defined below, have been introduced and studied earlier in [6], under different notations. Indeed, we have 𝜆^𝑠(𝑋)=𝑥=𝑥_𝑛⊂𝑋∶𝑥_𝑛𝑋,𝜆∈𝜆_𝑤(𝑋)=𝑥=𝑥_𝑛⊂𝑋∶𝑓𝑥_𝑛∈𝜆,∀𝑓∈𝑋^∗. (10) In case ‖⋅‖_𝜆 is a monotone norm, the space 𝜆^𝑠(𝑋) becomes a normed linear space with respect to the norm defined as ‖𝑥‖_𝜆=𝑥_𝑛𝑋𝜆,𝑥=𝑥_𝑛∈𝜆^𝑠(𝑋). (11) However, for 𝑥∈𝜆^𝑤(𝑋), the norm on 𝜆^𝑤(𝑋) is defined as ‖𝑥‖^𝑤_𝜆=sup𝑓𝑥_𝑛𝜆,𝑓∈𝑋^∗,𝑓≤1,𝑥=𝑥_𝑛∈𝜆^𝑤(𝑋), (12)

which can be proved to be finite by applying closed graph theorem. For the norm ‖𝑦‖^𝑤_𝜆^× on (𝜆^×)^𝑤(𝑋), we assume throughout that 0<sup_𝑛‖𝑒^𝑛‖_𝜆<∞ so that (𝜆^×)^𝑤(𝑋) equipped with this norm becomes a 𝐾-space.

Corresponding to a Banach sequence space 𝜆 with 0<sup_𝑛‖𝑒^𝑛‖_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={𝑥_𝑛}∈𝜆^𝑠(𝑋) such that 𝐾⊆𝜆−co{𝑥_𝑛}={∑_𝑛≥1𝛼_𝑛𝑥_𝑛∶𝛼={𝛼_𝑛}∈𝐵_𝜆^×}. For 𝜆=𝑙_𝑝, 1≤𝑝≤∞, 𝜆-compact sets are nothing but 𝑝-compact sets studied in [9].

Concerning the 𝜆-compact sets, we have the following [8].

Theorem 1. Let 𝜆 be a normal sequence space equipped with a monotone norm ‖⋅‖_𝜆 satisfying the condition 0<sup_𝑛‖𝑒^𝑛‖_𝜆<∞. Also let (𝜆, ‖⋅‖_𝜆) be an AK-BK-reflexive sequence space. Then for 𝑥={𝑥_𝑛}∈𝜆^𝑠(𝑋), the set {∑_𝑛≥1𝛼_𝑛𝑥_𝑛∶𝛼={𝛼_𝑛}∈𝐵_𝜆^×} is norm closed in 𝑋.

For Banach spaces 𝑋 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 space ℒ(𝑋,𝑌) endowed with the locally convex topology 𝜏_𝑝(1≤𝑝≤∞) has been studied in [10], where 𝜏_𝑝 is the topology of uniform convergence on 𝑝-compact sets in 𝑋. Concerning the dual space (ℒ(𝑋,𝑌),𝜏_𝑝)^∗ for 1<𝑝<∞, Choi and Kim [10] have proved the following.

Theorem 2. Let 1<𝑝, 𝑞<∞, and 1/𝑝+1/𝑞=1. Then (ℒ(𝑋,𝑌),𝜏_𝑝)^∗ consists of all linear functionals 𝜑 of the form 𝜑(𝑇)=_{𝑗≥1𝑛≥1}𝛼^𝑗_𝑛𝑔_𝑗𝑇𝑥_𝑛, (13) where 𝑥={𝑥_𝑛}∈𝑙^𝑠_𝑝(𝑋); 𝛼^𝑗={𝛼^𝑗_𝑛}_𝑛≥1 from 𝑙_𝑞, for each 𝑗∈ℕ and {𝑔_𝑗}⊆𝑌^∗ with ∑_𝑗≥1‖𝛼^𝑗‖_𝑞‖𝑔_𝑗‖<∞.

Let 𝜆 be a normal BK-sequence space equipped with the norm ‖⋅‖_𝜆 satisfying 0<inf_𝑛‖𝑒^𝑛‖_𝜆≤sup_𝑛‖𝑒^𝑛‖_𝜆<∞. An operator 𝑇∈ℒ(𝑋,𝑌) is said to be (i) 𝜆-compact operator if 𝑇(𝐵_𝑋) 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 𝑇𝑥=_𝑛≥1𝛼_𝑛<𝑥,𝑓_𝑛>𝑦_𝑛, (14) where {𝑓_𝑛}⊆𝑋^∗ with ‖𝑓_𝑛‖_𝑋∗≤1, for each 𝑛; 𝑦={𝑦_𝑛}∈(𝜆^×)^𝑤(𝑌) with ‖𝑦‖^𝑤_𝜆^×≤1 and 𝛼={𝛼_𝑛}∈𝜆,(iii)quasi-𝜆-nuclear if there exists {𝑓_𝑛}⊆𝑋^∗ such that 𝑓={𝑓_𝑛}∈𝜆^𝑠(𝑋^∗) and ‖𝑇𝑥‖≤‖{𝑓_𝑛(𝑥)}‖_𝜆, for each 𝑥∈𝑋. The symbols 𝐾_𝜆(𝑋,𝑌), 𝑁_𝜆(𝑋,𝑌), and 𝑄𝑁_𝜆(𝑋,𝑌), respectively, denote the collection of all 𝜆-compact, 𝜆-nuclear, and quasi-𝜆-nuclear operators from 𝑋 to 𝑌.

For the relationships of 𝜆-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<inf_𝑛‖𝑒^𝑛‖_𝜆≤sup_𝑛‖𝑒^𝑛‖_𝜆<∞. Consider the following.(i)For any pair of Banach spaces 𝑋 and 𝑌, if 𝑇∈ℒ(𝑋,𝑌) is a 𝜆-nuclear operator, then 𝑇^∗∈ℒ(𝑌^∗,𝑋^∗) is a 𝜆-compact operator;(ii)if 𝑇∈ℒ(𝑋,𝑌) is a 𝜆-compact operator, then 𝑇^∗∈ℒ(𝑌^∗,𝑋^∗) is a quasi-𝜆-nuclear operator.

For 𝜆=𝑙_𝑝(1≤𝑝<∞) and 𝑇∈ℒ(𝑋,𝑌), we quote from [11].

Proposition 4. (i)  𝑇∈𝐾_𝑝(𝑋,𝑌) if and only if 𝑇^∗∈𝑄𝑁_𝑝(𝑌^∗,𝑋^∗); 𝑇∈𝑄𝑁_𝑝(𝑋,𝑌) if and only if 𝑇^∗∈𝐾_𝑝(Y^∗,𝑋^∗);
(ii)  𝑇^∗∈𝑄𝑁_𝑝(𝑌^∗,𝑋^∗) if and only if there exists {𝑦_𝑛}∈𝑙^𝑠_𝑝(𝑌) such that ‖𝑇^∗𝑔‖≤‖{𝑔(𝑦_𝑛)}‖_𝑝, for all 𝑔∈𝑌^∗.

We also make use of the following result from [6].

Theorem 5. Let 𝜆 be a symmetric normal AK-BK-sequence space equipped with a monotone norm ‖⋅‖_𝜆 satisfying 0<sup_𝑛‖𝑒^𝑛‖_𝜆<∞. Then 𝑇∈𝑄𝑁_𝜆(𝑋,𝑌) implies that 𝑇∈𝑁_𝜆(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].

Writing 𝒜(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)𝒜 contains all finite rank operators;(ii)𝑇+𝑆∈𝒜(𝑋,𝑌) for 𝑆,𝑇∈𝒜(𝑋,𝑌);(iii)if 𝑇∈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 collection 𝒜(𝑋,𝑌), for a given pair of Banach spaces 𝑋 and 𝑌, is called a component of 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 ideal 𝒜, the dual operator ideal 𝒜^𝑑 is defined as the one of which the component 𝒜^𝑑(𝑋,𝑌) is given by 𝒜^𝑑(𝑋,𝑌)=𝑇∈ℒ∶𝑇^∗∈𝒜𝑌^∗,𝑋^∗. (15)

For an operator ideal 𝒜, the subspace 𝒜^𝑤∗(𝑋^∗,𝑌) of the component 𝒜(𝑋^∗,𝑌) is defined as 𝒜^𝑤∗𝑋^∗,𝑌=𝑇∈𝒜𝑋^∗∶𝑋,𝑌𝑇is𝜎^∗,𝑋−𝜎𝑌,𝑌^∗continuous. (16)

3. 𝐿-Subset 𝜆-Approximation Property

Throughout this section we denote by 𝑋, a Banach space and by 𝜆, a BK-sequence space equipped with a norm ‖⋅‖_𝜆 such that 0<sup_𝑛‖𝑒^𝑛‖_𝜆<∞. Let 𝐿_𝑠(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 space 𝑋 is said to have 𝐿-subset 𝜆-approximation property (𝐿_𝑠-𝜆-AP) if given 𝜀>0 and any 𝜆-compact set 𝐾 in 𝑋; there exists 𝑅∈𝐿_𝑠(𝑋) such that sup_𝑥∈𝐾‖𝑅𝑥−𝑥‖_𝑋<𝜀; that is, the identity map on 𝑋 is approximated uniformly on a 𝜆-compact set by a member of 𝐿_𝑠(𝑋).

If 𝐿_𝑠(𝑋)=𝔉(𝑋) and 𝜆=𝑐₀, it is the approximation property studied by Grothendieck [1]. If 𝐿_𝑠(𝑋)=𝔉(𝑋) and 𝜆=𝑙_𝑝, it is the 𝑝-approximation property (cf. [3, 9]), and if 𝐿_𝑠(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<sup_𝑛‖𝑒^𝑛‖_𝜆<∞. Assume that 𝜆 is also monotone. Then for a convex subset 𝐿_𝑠(𝑋) of ℒ(𝑋), the following conditions are equivalent. (i)𝑋 has 𝐿_𝑠-𝜆-AP.(ii)For any Banach space 𝑌 and for any 𝑇∈𝐾_𝜆(𝑌,𝑋), 𝑇∈{𝑅𝑇∶𝑅∈𝐿_𝑠(𝑋)}^‖⋅‖, the closure being considered with respect to the norm ‖⋅‖ in ℒ(𝑌,𝑋).

Proof. (i)⇒(ii) Let 𝑇∈𝐾_𝜆(𝑌,𝑋). Then 𝑇(𝐵_𝑌) is 𝜆-compact set in 𝑋. Since 𝑋 has 𝐿_𝑠-𝜆-AP, for 𝜀>0 there exists 𝑅₀∈𝐿_𝑠(𝑋) such that sup_{𝑥∈𝑇(𝐵𝑌)}‖𝑥−𝑅₀𝑥‖<𝜀⇒‖𝑇−𝑅₀𝑇‖<𝜀. Thus 𝑇∈{𝑅𝑇∶𝑅∈𝐿_𝑠(𝑋)}^‖⋅‖.
(ii)⇒(i) For proving (i), consider a 𝜆-compact set 𝐾 of the form 𝐾=𝜆−co{𝑥_𝑛} for some 𝑥={𝑥_𝑛}∈𝜆^𝑠(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∈𝐾_𝜆(𝑌,𝑋). Therefore by hypothesis, given any 𝜀>0, we can find that 𝑅₀∈𝐿_𝑠(𝑋) such that 𝐼−𝑅₀=sup_𝑥∈𝐾𝑥−𝑅₀𝑥<𝜀. (17) This proves that 𝑋 has 𝐿_𝑠-𝜆-AP.

The above result leads to the following.

Proposition 8. Let (𝜆, ‖⋅‖_𝜆) be a sequence space as in Theorem 7, and let 𝐿_𝑠(𝑋) be a convex subset of ℒ(𝑋). Then 𝑋 has 𝐿_𝑠-𝜆-AP if 𝑇∈{𝑇𝑅^∗∶𝑅∈𝐿_𝑠(𝑋)}^‖⋅‖, for any Banach space 𝑌 and any 𝑇∈𝑄𝑁_𝜆(𝑋^∗,𝑌).

Proof. Let 𝑇∈𝐾_𝜆(𝑌,𝑋) where 𝑌 is an arbitrary Banach space. Then 𝑇^∗∈𝑄𝑁_𝜆(𝑋^∗,𝑌^∗) by Theorem 3(ii). Hence for any 𝜀>0, there exists 𝑅₀∈𝐿_𝑠(𝑋) such that ‖𝑇^∗−𝑇^∗𝑅^∗₀‖<𝜀, by hypothesis. Thus ‖𝑇−𝑅₀𝑇‖<𝜀; that is, 𝑇∈{𝑅𝑇∶𝑅∈𝐿_𝑠(𝑋)}^‖⋅‖. Now apply Theorem 7 to get that 𝑋 has 𝐿_𝑠-𝜆-AP.

If the Banach spaces 𝑋 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 𝐿_𝑠-𝜆-AP, then 𝑇∈{𝑇𝑅^∗∶𝑅∈𝐿_𝑠(𝑋)}^‖⋅‖, for any 𝑇∈𝑄𝑁_𝜆(𝑋^∗,𝑌).

Proof. Let 𝑇∈𝑄𝑁_𝜆(𝑋^∗,𝑌). Then 𝑇∈𝑁_𝜆(𝑋^∗,𝑌) by Theorem 5, and so 𝑇^∗∈𝐾_𝜆(𝑌^∗,𝑋) by Theorem 3(i). By hypothesis and Theorem 7, for 𝜀>0, there exists 𝑅₀∈𝐿_𝑠(𝑋) such that ‖𝑇−𝑇𝑅^∗₀‖=‖𝑇^∗−𝑅₀𝑇^∗‖<𝜀; that is, 𝑇∈{𝑇𝑅^∗∶𝑅∈𝐿_𝑠(𝑋)}^‖⋅‖.

For the dual 𝑋^∗ of 𝑋, we introduce the following.

Definition 10. Let 𝑋 be a Banach space with its topological dual 𝑋^∗ equipped with the operator norm topology, and let 𝐿_𝑠(𝑋) be a convex subset of ℒ(𝑋). Then 𝑋^∗ is said to have 𝐿_𝑠-𝜆 approximation property with conjugate operator (𝐿^∗_𝑠-𝜆-AP) if, for any𝜆-compact set 𝐾^∗ of 𝑋^∗ and 𝜀>0, there exists 𝑅₀∈𝐿_𝑠(𝑋) such that sup_{𝑓∈𝐾∗}‖𝑅^∗₀𝑓−𝑓‖<𝜀; that is, the identity operator on 𝑋^∗ is approximation by a conjugate operator of a member in 𝐿_𝑠(𝑋), uniformly on 𝜆-compact subset of 𝑋^∗.

Concerning 𝐿^∗_𝑠-𝜆-AP of 𝑋^∗, we prove the following.

Theorem 11. Let (𝜆, ‖⋅‖_𝜆) and 𝐿_𝑠(𝑋) be as in Theorem 7.(i)If 𝑌 is a Banach space with extension property and 𝑋^∗ has 𝐿^∗_𝑠-𝜆-AP, then 𝑇∈{𝑇𝑅∶𝑅∈𝐿_𝑠(𝑋)}^‖⋅‖ for any 𝑇∈𝑄𝑁_𝜆(𝑋,𝑌).(ii)Let 𝑋 be a reflexive Banach space. If, for any Banach space 𝑌 and 𝑇∈𝑄𝑁_𝜆(𝑋,𝑌), 𝑇∈{𝑇𝑅∶𝑅∈𝐿_𝑠(𝑋)}^‖⋅‖, then 𝑋^∗ has 𝐿^∗_𝑠-𝜆-AP.

Proof. Omitted as these are analogous to the proofs of Propositions 8 and 9.

Remark 12. Let us note here that the restriction on 𝜆 is such that we may include the Orlicz sequence spaces 𝑙_𝑀, where 𝑀 is an Orlicz function satisfying Δ₂-condition at zero. One may call such approximation property as 𝐿_𝑠-𝑙_𝑀-AP and may be worth investigating further. However, for 𝐿_𝑠(𝑋)=𝔉(𝑋), we illustrate 𝑙_𝑀-AP by the following.

Example 13. We consider Orlicz functions 𝑀₁(𝑢)=𝑢²/(log(𝑒+𝑢)) and 𝑀₂(𝑢)=𝑢(𝑒^𝑢−1), for 𝑢≥0. Then 𝑙_𝑀1=𝑙₂=𝑙_𝑀2 (cf. [13]). Therefore 𝑙_𝑁1=𝑙₂=𝑙_𝑁2, where 𝑁_𝑖 are the complementary Orlicz functions of 𝑀_𝑖, respectively, for 𝑖=1,2. Let 𝐾 be an 𝑙_𝑁1-compact set in 𝑋. Then there exists 𝑥={𝑥_𝑛}∈𝑙^𝑠_𝑁1(𝑋)=𝑙^𝑠₂(𝑋) such that 𝐾⊆{∑_𝑛≥1𝛼_𝑛𝑥_𝑛∶𝛼={𝛼_𝑛}∈𝐵_𝑙𝑀1}. As ‖𝛼‖_𝑀1≤‖𝛼‖₂≤𝐶‖𝛼‖_𝑀1, we have 𝐾⊆{∑_𝑛≥1𝛼_𝑛(𝐶𝑥_𝑛)∶𝛼={𝛼_𝑛}∈𝐵_𝑙2}. Thus every 𝑙_𝑁1-compact set is 𝑙₂-compact set. Hence every Banach space has 𝑙_𝑁1-approximation property (cf. [9], page 27). Similarly we can show that every Banach space has 𝑙_𝑁2-approximation property.

4. 𝐿-Subset 𝑝-Approximation Property

In this section we study 𝐿_𝑠-𝑝-approximation property which is a particular case of 𝐿-subset 𝜆 approximation property for 𝜆=𝑙_𝑝(1≤𝑝≤∞). Throughout this section, we denote by 𝐿_𝑠(𝑋) a convex subset of ℒ(𝑋), where 𝑋 is a Banach space. Let us begin with the following.

Definition 14. Let 𝑋 be a Banach space with 𝑋^∗ as its topological dual equipped with the operator norm topology. Then (i) 𝑋 is said to have 𝐿_𝑠(𝑋)-p-approximation property (𝐿_𝑠-𝑝-AP) if, for 𝜀>0 and a 𝑝-compact subset 𝐾 of 𝑋, there exists 𝑅₀∈𝐿_𝑠(𝑋) such that sup_𝑥∈𝐾‖𝑅₀𝑥−𝑥‖_𝑋<𝜀, and (ii) 𝑋^∗ is said to have 𝐿_𝑠(𝑋)-𝑝-approximation property with conjugate operator (𝐿^∗_𝑠-𝑝-AP) if, for any 𝜀>0 and a 𝑝-compact set 𝐾^∗ in 𝑋^∗, there exists 𝑅₀∈𝐿_𝑠(𝑋) such that sup_{𝑓∈𝐾∗}‖𝑅^∗₀𝑓−𝑓‖<𝜀.

Characterizing Banach spaces having 𝐿_𝑠-𝑝-AP, we have the following.

Theorem 15. The following conditions are equivalent. (i)𝑋 has 𝐿_𝑠-𝑝-AP. (ii)For any Banach space 𝑌 and any 𝑇∈𝐾_𝑝(𝑌,𝑋), 𝑇∈{𝑅𝑇∶𝑅∈𝐿_𝑠(𝑋)}^‖⋅‖.(iii)For any Banach space 𝑌 and any 𝑇∈𝐾_𝑝(𝑌,𝑋), 𝑇∈{𝑅𝑇∶𝑅∈𝐿_𝑠(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≥1𝛼_𝑛𝑥_𝑛∶𝛼={𝛼_𝑛}∈𝐵_𝑙𝑞}, where 𝑥={𝑥_𝑛}∈𝑙^𝑠_𝑝(𝑋) and 1/𝑝+1/𝑞=1. We can find that 𝛽={𝛽_𝑛}∈𝑐₀ such that {𝛽_𝑛⁻¹𝑥_𝑛}∈𝑙^𝑠_𝑝(𝑋). Now let us consider the diagonal map 𝐷∶𝑙_𝑞→𝑙_𝑞 defined as 𝐷({𝛼_𝑛})={𝛼_𝑛𝛽_𝑛} and 𝜓∶𝑙_𝑞→𝑋 such that 𝜓({𝛼_𝑛})=∑_𝑛≥1𝛼_𝑛𝛽_𝑛⁻¹𝑥_𝑛. Clearly, 𝐷 is a compact operator, and 𝜓 is a 𝑝-compact operator. Write 𝑌=𝑙_𝑞/ker𝜓 and 𝑄 for the quotient map from 𝑙_𝑞 to 𝑌. Define a 𝑝-compact operator 𝜓∶𝑌→𝑋 as 𝜓(𝑄(𝛼))=𝜓(𝛼). Now, 𝑄∘𝐷(𝐵_𝑙𝑞) is a compact set in 𝑌. By hypothesis, for this compact set and any 𝜀>0, there exists 𝑅₀∈𝐿_𝑠(𝑋) such that sup_𝑥∈𝐾𝑅₀𝑥−𝑥=sup𝛼_{∈𝑄∘𝐷(𝐵𝑙𝑞)}𝑅₀𝜓𝛼−𝜓𝛼<𝜀. (18) As 𝐾 is an arbitrary 𝑝-compact set, the Banach space 𝑋 has 𝐿_𝑠-𝑝-AP.

To prove the next result we make use of the following.

Lemma 16. The Banach space 𝑋 has 𝐿_𝑠-𝑝-AP for 1<𝑝<∞ if and only if inf_{𝑅∈𝐿𝑠(𝑋)𝑗≥1𝑛≥1}𝛼^𝑗_𝑛𝑓_𝑗𝑅𝑥_𝑛−𝑥_𝑛=0, (19) where 𝑥={𝑥_𝑛}∈𝑙^𝑠_𝑝(𝑋); 𝛼^𝑗={𝛼^𝑗_𝑛}_𝑛≥1 are in 𝑙_𝑞 for each 𝑗∈ℕ and {𝑓_𝑗} in 𝑋^∗ satisfying the condition ∑_𝑗≥1‖𝛼^𝑗‖_𝑞‖𝑓_𝑗‖_𝑋∗<∞.

Proof. 𝑋 has 𝐿_𝑠-𝑝-AP⇔𝐼∈𝐿_𝑠(𝑋)^𝜏𝑝⇔inf_{𝑅∈𝐿𝑠(𝑋)}|𝜑(𝐼−𝑅)|=0, for 𝜑 in (ℒ(𝑋),𝜏_𝑝)^∗. Now the result is immediate from Theorem 2.

Making use of Proposition 4(i) and the above lemma, we further characterize 𝐿_𝑠-𝑝-AP in the following.

Theorem 17. For 1<𝑝, 𝑞<∞, and 1/𝑝+1/𝑞=1, the following conditions are equivalent. (i)𝑋 has 𝐿_𝑠-𝑝-AP. (ii)For any Banach space 𝑌 and for any 𝑇∈𝑄𝑁^𝑤∗_𝑝(𝑋^∗,𝑌), 𝑇∈{𝑇𝑅^∗∶𝑅∈𝐿_𝑠(𝑋)}^‖⋅‖.(iii)For any reflexive separable Banach space 𝑌 and any 𝑇∈𝑄𝑁^𝑤∗_𝑝(𝑋^∗,𝑌), there exists a net {𝑅_𝑖}_𝑖∈𝐼 from 𝐿_𝑠(𝑋) such that 𝑇𝑅^∗_𝑖→𝑇 in weak operator topology on ℒ(𝑋^∗,𝑌); that is, 𝑇𝑅^∗_𝑖𝑓→𝑇𝑓 in weak topology 𝜎(𝑌,𝑌^∗) of 𝑌 for each 𝑓∈𝑋^∗.

Proof. (i)⇒(ii) Let 𝑇∈𝑄𝑁^𝑤∗_𝑝(𝑋^∗,𝑌). As 𝑇 is 𝜎(𝑋^∗,𝑋)−𝜎(Y,𝑌^∗) continuous, range(𝑇^∗)⊂𝑋 and so 𝑇^∗∈𝐾_𝑝(𝑌^∗,𝑋). Now for given 𝜀>0, there exists 𝑅₀∈𝐿_𝑠(𝑋) such that ‖𝑅₀𝑇^∗−𝑇^∗‖<𝜀 by Theorem 15. Hence ‖𝑇𝑅^∗₀−𝑇‖<𝜀, and (ii) follows.
(ii)⇒(iii) Immediate.
(iii)⇒(i) For proving that 𝑋 has 𝐿_𝑠-𝑝-AP, we make use of Lemma 16. Hence we may assume without loss of generality that 𝑥={𝑥_𝑛}∈𝑙^𝑠_𝑝(𝑋), 𝛼^𝑗={𝛼^𝑗_𝑛}_𝑛≥1∈𝐵_𝑙𝑞, for each 𝑗≥1, and {𝑓_𝑗}⊂𝑋^∗ satisfies ∑_𝑗≥1‖𝑓_𝑗‖_𝑋∗<∞. Consider the map 𝜓∶𝑙_𝑞→𝑋 defined as 𝜓({𝛼_𝑛})=∑_𝑛≥1𝛼_𝑛𝑥_𝑛. Then 𝜓 is a 𝑝-compact map. Hence 𝜓^∗∶𝑋^∗→𝑙_𝑝 is a quasi-𝑝-nuclear map by Proposition 4(i); that is, 𝜓^∗∈𝑄𝑁_𝑝(𝑋^∗,𝑙_𝑝). As 𝑙_𝑝 is a reflexive separable Banach space and 𝜓^∗∗∶𝑙_𝑞→𝑋^∗∗, we get range(𝜓^∗∗)⊂𝑋. Consequently, 𝜓^∗∈𝑄𝑁^𝑤∗_𝑝(𝑋^∗,𝑙_𝑝). Now by hypothesis, there exists a net {𝑅_𝑖}_𝑖∈𝐼 from 𝐿_𝑠(𝑋) such that 𝜓^∗𝑅^∗_𝑖→𝜓^∗ in weak operator topology on ℒ(𝑋^∗,𝑙_𝑝). Hence by uniform boundedness principle, there exists 𝑀≥0 such that ‖𝜓^∗𝑅^∗_𝑖‖≤𝑀, for all 𝑖∈𝐼. Also we get that 𝜓^∗𝑅^∗_𝑖(𝑓_𝑗)→𝜓^∗(𝑓_𝑗), for each 𝑗≥1. Choose 𝑅_𝑖0 in {𝑅_𝑖}_𝑖∈𝐼 such that 𝜓^∗𝑅^∗_𝑖0𝑓_𝑗−𝜓^∗𝑓_𝑗𝛼^𝑗<𝜀2𝑛₀,for𝑗=1,2,…,𝑛₀, (20) where 𝑛₀ is the integer satisfying _{𝑗≥𝑛0+1}𝑓_𝑗<𝜀2𝑀+‖𝑥‖^𝑠_𝑝. (21) Now consider _{𝑗≥1𝑛≥1}𝛼^𝑗_𝑛𝑓_𝑗𝑅_𝑖0𝑥_𝑛−𝑥_𝑛=_𝑗≥1𝜓^∗𝑅^∗_𝑖0𝑓_𝑗−𝜓^∗𝑓_𝑗𝛼^𝑗≤^𝑛0_𝑗=1𝜓^∗𝑅^∗_𝑖0𝑓_𝑗−𝜓^∗𝑓_𝑗𝛼^𝑗+_{𝑗≥𝑛0+1}𝜓^∗𝑅^∗_𝑖0𝑓_𝑗−𝜓^∗𝑓_𝑗𝛼^𝑗<𝜀2+𝜀2, (22) since (‖𝜓^∗𝑅^∗_𝑖0‖+‖𝜓^∗‖)≤𝑀+‖𝑥‖^𝑠_p and ‖𝛼^𝑗‖_𝑞≤1, for all 𝑗≥1.
Hence inf_{𝑅∈𝐿𝑠(𝑋)}|∑_𝑗≥1∑_𝑛≥1𝛼^𝑗_𝑛𝑓_𝑗(𝑅𝑥_𝑛−𝑥_𝑛)|=0. Therefore 𝑋 has𝐿_𝑠-𝑝-AP by Lemma 16.

A characterization of 𝐿^∗_𝑠-𝑝-AP of a Banach space 𝑋^∗ is contained in the following.

Theorem 18. 𝑋^∗ has 𝐿^∗_𝑠-𝑝-AP if and only if, for every Banach space 𝑌 and every 𝑇∈𝑄𝑁_𝑝(𝑋,𝑌), 𝑇∈{𝑇𝑅∶𝑅∈𝐿_𝑠(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 that 𝑇∈𝐾_𝑝(𝑌,𝑋^∗), where 𝑌 is an arbitrary Banach space. Then 𝑇^∗∈𝑄𝑁_𝑝(𝑋^∗∗,𝑌^∗) by Proposition 4(i). Using Proposition 4(ii), we get 𝑇^∗∈𝑄𝑁_𝑝(𝑋,𝑌^∗). Now, by hypothesis, for given 𝜀>0, we can find that 𝑅₀∈𝐿_𝑠(𝑋) such that ‖𝑇^∗𝑅₀−𝑇^∗‖<𝜀. Hence ‖𝑅^∗₀𝑇−𝑇‖<𝜀. Now apply Theorem 15 to get that 𝑋^∗ has 𝐿^∗_𝑠-𝑝-AP.

Finally, we consider a result involving operator ideals and 𝑝-compact operators, which is analogous to the one proved in [5] for compact operators.

Proposition 19. Let 𝒜 be an operator ideal and 𝑌 an arbitrary Banach space. If, for 𝑅∈𝐿_𝑠(𝑋) and any 𝑇∈𝐴^𝑑(𝑌,𝑋), 𝑅𝑇 is a 𝑝-compact operator from 𝑌 to 𝑋, then for 𝑅∈𝐿_𝑠(𝑋) and any 𝑇∈𝒜^𝑤∗(𝑋^∗,𝑌) the operator 𝑅𝑇^∗ is also 𝑝-compact operator from 𝑌^∗ to 𝑋. Converse follows when 𝒜⊆𝑄𝑁_𝑝.

Proof. If 𝑅∈𝐿_𝑠(𝑋) and 𝑇∈𝒜^𝑤∗(𝑋^∗,𝑌), then range(𝑇^∗)⊂𝑋. Then 𝑇^∗=𝐽_𝑋𝑇_𝑎𝑠 where 𝑇_𝑎𝑠∶𝑌^∗→𝑋 is the astriction map of 𝑇^∗, and 𝐽_𝑋 is the canonical embedding from 𝑋 to 𝑋^∗∗. Then 𝑇_𝑎𝑠∈𝒜^𝑑(𝑌^∗,𝑋) as 𝑇^∗_𝑎𝑠=𝐽_𝑌𝑇∈𝒜(𝑋^∗,𝑌^∗∗). By hypothesis, 𝑅𝑇^∗=𝑅𝑇_𝑎𝑠∈𝐾_𝑝(𝑌^∗,𝑋).
For converse, consider that 𝑅∈𝐿_𝑠(𝑋) and 𝑇∈𝒜^𝑑(𝑌,𝑋). Then 𝑇^∗∈𝑄𝑁_𝑝(𝑋^∗,𝑌^∗) by hypothesis. By Proposition 4(i) 𝑇∈𝐾_𝑝(𝑌,𝑋)⊂𝐾(𝑌,𝑋)⊂𝑊(𝑌,𝑋). Consequently, range(𝑇^∗∗)⊂𝑋 (cf. [14], page 185) (see also [15], page 341). Hence 𝑇^∗∈𝒜^𝑤∗(𝑋^∗,𝑌^∗). Consequently, 𝑅𝑇^∗∗∈𝐾_𝑝(𝑌^∗∗,𝑋) by hypothesis. As 𝑅𝑇=𝑅𝑇^∗∗|_𝑌, the result follows.