A Note on Generalized Approximation Property

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) ofL(X), the class of bounded linear operators onX.This property includes approximation property studied by Grothendieck, p-approximation property considered by Sinha and Karn andDelgado 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.


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  has a Schauder base.If the identity on  is approximated uniformly on compact sets by finite rank operators, it leads to the notion of approximation property of , 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][3][4][5] and references given therein.
In our recent work, using the duality theory of sequence spaces, we considered the notion of -compact sets corresponding to a suitably restricted sequence space  and studied -compact operators, specially their relationships with summing, -nuclear, and quasi--nuclear which were earlier considered by Ramanujan [6] in 1970.Replacing compact sets or -compact sets by -compact sets, where  is a Banach sequence space and the class of finite dimensional operators or compact operators, and so forth, by an arbitrary convex subset   () of the class L() of bounded linear operators on a Banach space, we define and study   --AP of a Banach space  in this paper.After giving preliminaries in the next section, we characterize this property in Section 3 and study its particular case for  =   in Section 4. This property includes, as particular cases, approximation property, approximation property, compact approximation property, and subspace approximation property (cf.[2][3][4][5]).

Preliminaries
Throughout this paper, we denote by (, ‖ ⋅ ‖  ) a Banach space equipped with the norm ‖ ⋅ ‖  and by  * its topological dual equipped with the dual operator norm topology ‖ ⋅ ‖  * .
Let us begin with the basics of the sequence space theory, for which our reference is [7].Let  denote the family of all real or complex sequences, which is a vector space with usual pointwise addition and scalar multiplication, and let  be the span of   's ( ≥ 1), where   is the th unit vector in ; that is,   = {0, 0, . . ., 1, 0, 0, . ..},where 1 is the th coordinate of the sequence   .A sequence space  is a subspace of  containing .Members of  are denoted by the symbols , , and so forth, where  = { 1 ,  2 ,  3 , . ..} and  = { 1 ,  2 ,  3 , . ..}.The th section of  for  ∈ N is written as  () and is defined as  () = { 1 ,  2 , . . .,   , 0, 0, . ..}; that is,  () = ∑  =1     .A sequence space  is called (i) symmetric if   = { () } ∈  whenever  = {  } ∈  and  ∈ Π, where Π is collection of all permutations of the set of natural numbers N, (ii) monotone if  0  ⊆ , where  0 = span{}, and  is the collection of all sequences consisting of zero and one, and (iii) The -dual, cross-dual, or Köthe-dual   or  × of  is defined as A sequence space  is said to be perfect if  =  ×× = ( × ) × .Every perfect sequence space is normal, and every normal sequence space is monotone.A Banach sequence space (, ‖ ⋅ ‖  ) is called a BK-space provided that each of the projection maps   :  → K,   () =   is continuous, for  ≥ 1, where K is the field of scalars and  = { 1 ,  2 , . ..}.A BK-space (, ‖ ⋅ ‖  ) is called an AK-space if  () → , for each  ∈ .
An Orlicz function  is said to satisfy the Δ 2 -condition for small  or at 0, if, for each  > 1, there exist   > 0 and   > 0 such that Corresponding to an Orlicz function , the set l is defined by If  and  are mutually complementary functions, the Orlicz sequence space   is defined as An equivalent way of defining   is Two equivalent norms on   are given by indeed, If  and  are mutually complementary Orlicz functions and  satisfies Δ 2 -condition at 0, then (  ) × =   (cf.[7]).
Corresponding to a sequence space  and a Banach space  with its topological dual  * equipped with the operator norm topology generated by ‖ ⋅ ‖, the vector-valued sequence spaces   () and   () defined below, have been introduced and studied earlier in [6], under different notations.Indeed, we have In case ‖ ⋅ ‖  is a monotone norm, the space   () becomes a normed linear space with respect to the norm defined as However, for  ∈   (), the norm on   () is defined as which can be proved to be finite by applying closed graph theorem.For the norm ‖‖   × on ( × )  (), we assume throughout that 0 < sup  ‖  ‖  < ∞ so that ( × )  () equipped with this norm becomes a -space.
Concerning the -compact sets, we have the following [8].
Theorem 1.Let  be a normal sequence space equipped with a monotone norm ‖ ⋅ ‖  satisfying the condition 0 < sup  ‖  ‖  < ∞.Also let (, ‖ ⋅ ‖  ) be an AK-BK-reflexive sequence space.Then for  = {  } ∈   (), the set For Banach spaces  and , the symbol L(, ) denotes the class of all bounded linear operators from  to , and L denotes the collection of all bounded operators between any pair of Banach spaces.The notation F(, ), (, ), and (, ), respectively, stand for collection of all finite rank, compact, and weakly compact operators from  to .For  = , we write L(, ) ≡ L(), F(, ) ≡ F(), and so forth.
For the relationships of -compact, -nuclear, and quasi--nuclear operators, we have the following [8].We also make use of the following result from [6].
For the salient features on operator ideals, one is referred to [12].
The collection A(, ), for a given pair of Banach spaces  and , is called a component of A.
It has been shown in [6,8] that the collections of compact, -nuclear, and quasi--nuclear operators from  to  are operator ideals for suitably chosen sequence space .
For an operator ideal A, the dual operator ideal A  is defined as the one of which the component A  (, ) is given by For an operator ideal A, the subspace A  * ( * , ) of the component A( * , ) is defined as

𝐿-Subset 𝜆-Approximation Property
Throughout this section we denote by , a Banach space and by , a BK-sequence space equipped with a norm ‖ ⋅ ‖  such that 0 < sup  ‖  ‖  < ∞.Let   () be a convex subset of L().Recalling the definition of -compact sets in  from the previous section, we introduce the following.Definition 6.A Banach space  is said to have -subset approximation property (  --AP) if given  > 0 and any -compact set  in ; there exists  ∈   () such that sup ∈ ‖ − ‖  < ; that is, the identity map on  is approximated uniformly on a -compact set by a member of   ().
Theorem 7. Let (, ‖ ⋅ ‖  ) be an AK-BK-reflexive sequence space such that the norm ‖ ⋅ ‖  is monotone satisfying the condition 0 < sup  ‖  ‖  < ∞.Assume that  is also monotone.Then for a convex subset   () of L(), the following conditions are equivalent.(ii) ⇒ (i) For proving (i), consider a -compact set  of the form  =  − co{  } for some  = {  } ∈   ().In view of Theorem 1,  is norm closed and so it is complete.Hence the space  = ⋃ ≥1  is a Banach space with respect to the norm ||| ⋅ ||| defined as |||||| = inf{ > 0 :  ∈ } for  ∈ .As  is the unit ball of , the inclusion map  from  to  is -compact; that is,  ∈   (, ).Therefore by hypothesis, given any  > 0, we can find that  0 ∈   () such that      −  0     = sup This proves that  has   --AP.
The above result leads to the following.