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A Note on Generalized Approximation Property
We introduce a notion of generalized approximation property, which we refer to as --AP possessed by a Banach space , corresponding to an arbitrary Banach sequence space and a convex subset of , the class of bounded linear operators on . This property includes approximation property studied by Grothendieck, -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 --AP with the help of -compact operators, -nuclear operators, and quasi--nuclear operators. A particular case for () has also been characterized.
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  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 -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  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 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. [2–5]).
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 . 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 (), where is the th unit vector in ; that is, , 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 and . The th section of for is written as and is defined as ; that is, .
A sequence space is called (i) symmetric if whenever and , where is collection of all permutations of the set of natural numbers , (ii) monotone if , where , and is the collection of all sequences consisting of zero and one, and (iii) normal or solid if whenever , for some .
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 , is continuous, for , where is the field of scalars and . A BK-space (, ) is called an AK-space if , for each .
For a BK-space (, ), we define the dual norm on as follows: The space (, ) can easily be shown to be a BK-space provided that .
The norm is said to be (i) k-symmetric if , for all and (ii) monotone if for , in with , for all .
An Orlicz function is a continuous, convex function defined from to itself such that , for . Such function always has the integral representation where , known as the kernel of , is right continuous, nondecreasing function for . Let us note that an Orlicz function is always increasing, and as . Also as , and for (cf. , page 139). However, for is equivalent to the fact that the Orlicz sequence space (see the definition below) is isomorphic to (cf. , page 309). Therefore, we assume here that the kernel has value 0 for and obviously as .
Given an Orlicz function with kernel , define , . Then possesses the same properties as , and the function defined as is an Orlicz function. The functions and are called mutually complementary Orlicz functions. For such functions and , we have Young’s inequality: , for and also for (cf. ).
An Orlicz function is said to satisfy the -condition for small or at 0, if, for each , there exist and such that
Corresponding to an Orlicz function , the set 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 -condition at , then (cf. ).
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 , 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 so that equipped with this norm becomes a -space.
Corresponding to a Banach sequence space with and a Banach space , we have introduced the concept of -compact sets and -compact operators in our work . A subset of is said to be -compact if there exists such that . For , , -compact sets are nothing but -compact sets studied in .
Concerning the -compact sets, we have the following .
Theorem 1. Let be a normal sequence space equipped with a monotone norm satisfying the condition . Also let (, ) be an AK-BK-reflexive sequence space. Then for , the set is norm closed in .
For Banach spaces and , the symbol denotes the class of all bounded linear operators from to , and denotes the collection of all bounded operators between any pair of Banach spaces. The notation , , and , respectively, stand for collection of all finite rank, compact, and weakly compact operators from to . For , we write , , and so forth.
The space endowed with the locally convex topology has been studied in , where is the topology of uniform convergence on -compact sets in . Concerning the dual space for , Choi and Kim  have proved the following.
Theorem 2. Let , , and . Then consists of all linear functionals of the form where ; from , for each and with .
Let be a normal BK-sequence space equipped with the norm satisfying . An operator is said to be (i) -compact operator if is a -compact set in ; that is, maps bounded sets in to -compact sets in ,(ii) -nuclear if has the representation where with , for each ; with and ,(iii)quasi--nuclear if there exists such that and , for each . The symbols , , and , respectively, denote the collection of all -compact, -nuclear, and quasi--nuclear operators from to .
For the relationships of -compact, -nuclear, and quasi--nuclear operators, we have the following .
Theorem 3. Let (, ) be a normal, symmetric, BK-sequence space with . Consider the following.(i)For any pair of Banach spaces and , if is a -nuclear operator, then is a -compact operator;(ii)if is a -compact operator, then is a quasi--nuclear operator.
For and , we quote from .
(i) if and only if ; if and only if ;
(ii) if and only if there exists such that , for all .
We also make use of the following result from .
Theorem 5. Let be a symmetric normal AK-BK-sequence space equipped with a monotone norm satisfying . Then implies that if the range space has extension property; that is, for every injection and every operator , there exists an extension such that and .
For the salient features on operator ideals, one is referred to .
Writing , a subset of is said to be an operator ideal if it satisfies the following conditions: (i) contains all finite rank operators;(ii) for ;(iii)if and , then , and also if and , then .
The collection , for a given pair of Banach spaces and , is called a component of .
For an operator ideal , the dual operator ideal is defined as the one of which the component is given by
For an operator ideal , the subspace of the component is defined as
3. -Subset -Approximation Property
Throughout this section we denote by , a Banach space and by , a BK-sequence space equipped with a norm such that . Let be a convex subset of . 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 and any -compact set in ; there exists such that ; 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 . Assume that is also monotone. Then for a convex subset of , the following conditions are equivalent. (i) has --AP.(ii)For any Banach space and for any , , the closure being considered with respect to the norm in .
Proof. Let . Then is -compact set in . Since has --AP, for there exists such that . Thus .
For proving (i), consider a -compact set of the form for some . In view of Theorem 1, is norm closed and so it is complete. Hence the space is a Banach space with respect to the norm defined as for . As is the unit ball of , the inclusion map from to is -compact; that is, . Therefore by hypothesis, given any , we can find that such that This proves that has --AP.
The above result leads to the following.
Proposition 8. Let (, ) be a sequence space as in Theorem 7, and let be a convex subset of . Then has --AP if , for any Banach space and any .
If the Banach spaces and are restricted further, we get the converse of Proposition 8 in the following form.
Proposition 9. Let (, ) be as in Proposition 8. Assume that is a reflexive Banach space and is a Banach space having extension property. If has --AP, then , for any .
For the dual of , we introduce the following.
Definition 10. Let be a Banach space with its topological dual equipped with the operator norm topology, and let be a convex subset of . Then is said to have - approximation property with conjugate operator (--AP) if, for any-compact set of and , there exists such that ; that is, the identity operator on is approximation by a conjugate operator of a member in , uniformly on -compact subset of .
Concerning --AP of , we prove the following.
Theorem 11. Let (, ) and be as in Theorem 7.(i)If is a Banach space with extension property and has --AP, then for any .(ii)Let be a reflexive Banach space. If, for any Banach space and , , then has --AP.
Remark 12. Let us note here that the restriction on is such that we may include the Orlicz sequence spaces , where is an Orlicz function satisfying -condition at zero. One may call such approximation property as --AP and may be worth investigating further. However, for , we illustrate -AP by the following.
Example 13. We consider Orlicz functions and , for . Then (cf. ). Therefore , where are the complementary Orlicz functions of , respectively, for . Let be an -compact set in . Then there exists such that . As , we have . Thus every -compact set is -compact set. Hence every Banach space has -approximation property (cf. , page 27). Similarly we can show that every Banach space has -approximation property.
4. -Subset -Approximation Property
In this section we study --approximation property which is a particular case of -subset approximation property for . Throughout this section, we denote by a convex subset of , where is a Banach space. Let us begin with the following.
Definition 14. Let be a Banach space with as its topological dual equipped with the operator norm topology. Then (i) is said to have -p-approximation property --AP) if, for and a -compact subset of , there exists such that , and (ii) is said to have --approximation property with conjugate operator --AP) if, for any and a -compact set in , there exists such that .
Characterizing Banach spaces having --AP, we have the following.
Theorem 15. The following conditions are equivalent. (i) has --AP. (ii)For any Banach space and any , .(iii)For any Banach space and any , , where is the topology of uniform convergence on compact subsets of .
Proof. follows from Theorem 7.
is obvious as norm topology is finer than .
The proof of this implication is almost the same as that of Theorem 1 () proved in . However, for the sake of completeness, we sketch its proof. Let be a -compact subset of of the type , where and . We can find that such that . Now let us consider the diagonal map defined as and such that . Clearly, is a compact operator, and is a -compact operator. Write and for the quotient map from to . Define a -compact operator as . Now, is a compact set in . By hypothesis, for this compact set and any , there exists such that As is an arbitrary -compact set, the Banach space has --AP.
To prove the next result we make use of the following.
Lemma 16. The Banach space has --AP for if and only if where ; are in for each and in satisfying the condition .
Proof. has , for in . Now the result is immediate from Theorem 2.
Making use of Proposition 4(i) and the above lemma, we further characterize --AP in the following.
Theorem 17. For , , and , the following conditions are equivalent. (i) has --AP. (ii)For any Banach space and for any , .(iii)For any reflexive separable Banach space and any , there exists a net from such that in weak operator topology on ; that is, in weak topology of for each .
Proof. Let . As is continuous, and so . Now for given , there exists such that by Theorem 15. Hence , and (ii) follows.
For proving that has --AP, we make use of Lemma 16. Hence we may assume without loss of generality that , , for each , and satisfies . Consider the map defined as . Then is a -compact map. Hence is a quasi--nuclear map by Proposition 4(i); that is, . As is a reflexive separable Banach space and , we get . Consequently, . Now by hypothesis, there exists a net from such that in weak operator topology on . Hence by uniform boundedness principle, there exists such that , for all . Also we get that , for each . Choose in such that where is the integer satisfying Now consider since and , for all .
Hence . Therefore has--AP by Lemma 16.
A characterization of --AP of a Banach space is contained in the following.
Theorem 18. has --AP if and only if, for every Banach space and every , .
Proof. The proof of necessity part is omitted as it is analogous to that of , Theorem 7.
For converse, let us consider that , where is an arbitrary Banach space. Then by Proposition 4(i). Using Proposition 4(ii), we get . Now, by hypothesis, for given , we can find that such that . Hence . Now apply Theorem 15 to get that has --AP.
Finally, we consider a result involving operator ideals and -compact operators, which is analogous to the one proved in  for compact operators.
Proposition 19. Let be an operator ideal and an arbitrary Banach space. If, for and any , is a -compact operator from to , then for and any the operator is also -compact operator from to . Converse follows when .
Proof. If and , then . Then where is the astriction map of , and is the canonical embedding from to . Then as . By hypothesis, .
For converse, consider that and . Then by hypothesis. By Proposition 4(i) . Consequently, (cf. , page 185) (see also , page 341). Hence . Consequently, by hypothesis. As , the result follows.
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