An approximation property with respect to an operator ideal

STUDIA MATHEMATICA 214 (1) (2013) An approximation property with respect to an operator ideal by Juan Manuel Delgado (Sevilla) and C´ andido Pi˜ neiro (Huelva) Abstract. Given an operator ideal A, we say that a Banach space Xhas the approximation property with respect to Aif Tbelongs to {S◦T:S∈ F(X)}τcfor every Banach space Yand every T∈ A(Y, X), τcbeing the topology of uniform convergence on compact sets. We present several characterizations of this type of approximation property. It is shown that some of the existing approximation properties in the literature may be included in this setting. 1. Introduction. It is well known that a Banach space Xhas the approximation property (for short, AP) if and only if T∈ {S◦T:S∈ F(X)}τc for every Banach space Yand every T∈ L(Y, X), where F(X) denotes the space of all finite rank operators on Xand τcis the topology of uniform convergence on compact sets in X. Given an operator ideal A, we may ask the following: For which Banach spaces Xdo we have T∈ {S◦T:S∈ F(X)}τc for every Banach space Yand every T∈ A(Y, X)? This question leads to the notion of approximation property depending on the operator ideal A, which will be denoted by APA. This is a different approach compared with that by Reinov [17, 19], Grønbæk and Willis [9], Lissitsin, Mikkor and Oja [12] and, recently, Berrios and Botelho [2], which is already considered as classic. Obviously, if Xhas the AP, then Xhas the APAfor every operator ideal A. The main result of this paper is Theorem 2.3 in which several characterizations of the APAare established, some of them involving the notion of A-compactness introduced by Carl and Stephani [4]. As an application, it is proven that if p≥1 and A=Kp, the ideal of p-compact operators, then the APAis the p-approximation property in the sense of Sinha and Karn [21]. As another example, we show that the approximation property of order p(1/2≤p < 1) in the sense of Reinov [16] is included in this setting for a suitable choice of operator ideal. Finally, a characterization of the APA in terms of a trace condition is presented. 2010 Mathematics Subject Classification: Primary 46B28; Secondary 47L20, 46B50. Key words and phrases: approximation property, Banach ideal. DOI: 10.4064/sm214-1-4 [67] c Instytut Matematyczny PAN, 2013 68 J. M. Delgado and C. Pi˜neiro Our notation is standard. A Banach space Xwill be regarded as a subspace of its bidual X∗∗ under the canonical embedding iX:X→X∗∗. We denote the closed unit ball of Xby BX. If Ais a set in X, we write aco(A) for its absolutely convex hull. Given an absolutely convex and compact set A⊂X, span(A) is denoted by XA. This space is normed by the Minkowski functional of A, ρA(x) = inf{t > 0: x∈tA}. It is well known that (XA, ρA) is complete and Ais its closed unit ball. The canonical inclusion map from XAinto X, denoted by jA, is obviously compact. For Banach spaces Xand Y, the Banach space of all bounded linear operators from Xto Yis denoted by L(X, Y ). In this space, we denote by τc the topology of uniform convergence on compact sets in X. If Ais an operator ideal, then Addenotes its dual operator ideal and Asur the surjective hull of A. We denote by L,Kand Fthe operator ideals of bounded, compact and finite rank linear operators, respectively. We also need the following operator ideals: Np,p-nuclear operators; QNp, quasi p-nuclear operators; and Πp, p-summing operators. We refer to Lindenstrauss–Tzafriri’s book [11] for the classical approximation properties and to Pietsch’s book [15] for operator ideals (see also Diestel, Jarchow and Tonge [8] for common operator ideals like Npand Πp, and Persson and Pietsch [14] for QNp). 2. An approximation property with respect to an operator ideal Definition 2.1.We say that a Banach space Xhas the approximation property with respect to the operator ideal A(for short, APA) if for every Banach space Yand T∈ A(Y, X), we have T∈ {S◦T:S∈ F(X)}τc. It is well known that a Banach space Xhas the AP if and only if the identity map on Xcan be approximated by finite rank operators uniformly on compact subsets of X. In order to obtain an analogous result for the APA, we recall the notion of A-compactness introduced by Carl and Stephani [4]. If Ais an operator ideal and Xis a Banach space, a set A⊂Xis said to be A-compact if there exist a Banach space Z, an operator T∈ A(Z, X) and a compact set K⊂Zsuch that A⊂T(K). We denote by MA c(X) the family of all A-compact subsets of X. Relying on this concept, the notion of A-compact operator is defined in an obvious way: an operator T∈ L(X, Y ) is said to be A-compact if T(BX) is A-compact in Y. The set of all A- compact operators between Banach spaces is denoted by KA. For the sake of completeness, we list several properties of these concepts (see Sections 1 and 2 in [4] for more details): An approximation property 69 Proposition 2.2.Let Abe an operator ideal and let Xbe a Banach space. Then (i) MA c(X) = MAsur c(X), (ii) MA c(X) = MA◦K c(X), (iii) KA=Asur ◦ K. We are now ready to state our main result. Theorem 2.3.Let Abe an operator ideal and Xa Banach space. The following statements are equivalent: (i) Xhas the APA. (ii) For every space Yand T∈ A◦K(Y, X),T∈ {S◦T:S∈ F(X)}τc. (iii) For every space Yand T∈ A◦K(Y, X),T∈ {S◦T:S∈ F(X)}k·k. (iv) For every ε > 0and A∈MA c(X), there exists S∈ F(X)such that kSx −xk< ε for all x∈A. (v) For every space Y,F(Y, X)is τc-dense in Asur(Y, X). (vi) For every space Y,F(Y, X)is k·k-dense in KA(Y, X). Proof. (i)⇒(ii). This is obvious. (ii)⇒(iii). Consider a Banach space Yand T∈ A ◦ K(Y, X). Bearing in mind the well known fact that K=K ◦ K, we can write T=T1◦T2, where T1∈ A ◦ K(Z, X) and T2∈ K(Y, Z) for a certain Banach space Z. By hypothesis, given ε > 0, there exists an operator S∈ F(X) such that kS◦T1z−T1zk< ε for all z∈T2(BY) and this yields kS◦T−Tk< ε. (iii)⇒(iv). This is immediate in virtue of Proposition 2.2(ii). (iv)⇒(v). This is easy on taking a glance at Proposition 2.2(i). (v)⇒(vi). This follows easily using Proposition 2.2(iii). (vi)⇒(i). Let us consider a Banach space Y, an operator T∈ A(Y, X), a compact set K⊂Yand ε > 0. According to [20, Lemma 4.11], there exist two compact sets Dand Ein Ysatisfying K⊂D⊂Eand Dis compact in YE. If we put F=T(D) and G=T(E) then the operator jG:XG→Xbelongs to KA(XG, X) since jG(BXG) = Gis A-compact in X. By hypothesis, there exists R∈ F(XG, X) so that kR−jGk< ε/2. The operator Radmits a representation of the form R=Pn k=1 ψk⊗xkwith ψk∈(XG)∗and xk∈X. However, the map jGis injective so (jG)∗has dense image in (XG)∗for the topology of uniform convergence on compact sets in XG. Hence, for each k= 1, . . . , n, there exists x∗ k∈X∗such that sup x∈F |hψk−(jG)∗(x∗ k), xi| <ε 2Pn k=1 kxkk. Put S=Pn k=1 x∗ k⊗xkand notice that, for each x∈T(K)⊂F, kSx −xk=kS◦jG(x)−jG(x)k≤kS◦jG(x)−R(x)k+kR(x)−jG(x)k. This yields kS◦Ty −Tyk< ε for all y∈K. 70 J. M. Delgado and C. Pi˜neiro If A⊃K, then MA c(X) is the class Mc(X) of all relatively compact subsets of X. This is a consequence of Proposition 2.2(ii): Mc(X) = ML c(X) = ML◦K c(X) = MK c(X)⊂MA c(X). Thus, in view of Theorem 2.3(iv), we have: Corollary 2.4.If the operator ideal Acontains the ideal of all compact operators, then the APAis precisely the AP. In early 1980s, Reinov introduced the notion of the approximation property of order pwhen 0 < p ≤1 (see, e.g., [16]). A Banach space Xis said to have the approximation property of order p(for short, APp) if the restriction of j:Y∗ˆ ⊗X→ N(Y, X) to the subspace of Y∗ˆ ⊗Xconsisting of all tensors u=Pny∗ n⊗xn,y∗ n∈Y∗,xn∈X, with Pn(ky∗ nk kxnk)p<∞is injective for all Banach spaces Y. Setting q=p/(1 −p), it was observed in [3] and [18] that Xhas the APpif and only if the identity map on Xcan be approximated by finite rank operators uniformly on Bourgain–Reinov q-compact subsets, that is, the subsets Aof Xfor which there exists (xn)∈`q(X) such that A⊂ {Pnαnxn: (αn)∈B`1}= aco(xn). Bourgain–Reinov compact sets and corresponding operators were recently studied in [1]. If q≥1 and Ais the ideal consisting of all operators mapping bounded sets to Bourgain–Reinov q-compact sets, then MA c(X) is precisely the class of all Bourgain–Reinov q-compact sets in X. It is obvious that every subset belonging to MA c(X) is Bourgain–Reinov q-compact in X. For the converse, it suffices to bear in mind that, given a q-summable sequence (xn) in X, there exists a null sequence (δn)&0 such that (δ−1 nxn) remains q-summable; hence, aco(xn)⊂T(aco(δnxn)), where T∈ A(`1, X) is defined by T en= δ−1 nxn. According to Theorem 2.3(iv), we obtain: Corollary 2.5.Let p∈[1/2,1) and q=p/(1 −p). If Ais the ideal of all operators mapping bounded sets to Bourgain–Reinov q-compact sets, then the APAis precisely the APp. Let 1 ≤p < ∞and let p0be the conjugate index of p. A subset Aof X is said to be relatively p-compact if there exists a p-summable sequence (xn) in Xsuch that A⊂ {Pnαnxn: (αn)∈B`p0}. This notion appeared in [21] to introduce a sort of gradation of the approximation property: A Banach space Xhas the p-approximation property if the identity map on Xcan be approximated by finite rank operators uniformly on relatively p-compact subsets of X. Let us denote by Kpthe ideal consisting of all p-compact operators (i.e., operators mapping bounded sets to relatively p-compact sets). According to [7, Proposition 3.11], Kp= (Np)sur, where Np(X, Y ) = nT∈ L(X, Y ): T=X n x∗ n⊗yn,(x∗ n)∈`w p0(X∗),(yn)∈`p(Y)o. An approximation property 71 Thus, Kpis a surjective ideal as also is Πd p(being the dual of an injective ideal), so Theorem 2.3(v) and [6, Theorem 2.1] yield Corollary 2.6.Let A=Kp,Πd por Np. Then the APAis precisely the p-approximation property. In [21, Theorem 6.4], it is showed that every Banach space has the papproximation property for p≤2. The following result follows directly from Theorem 2.3(v) if we bear in mind the surjectivity of the ideal Γ2of 2- factorable operators and the fact that every Hilbert space enjoys the AP. Corollary 2.7.Every Banach space has the APAwhenever A ⊂ Γ2. Remark 2.8.A direct consequence of Proposition 2.2(i) and Theorem 2.3(iv) is that APA≡APAsur . From this, it is clear that (ii) and (iii) of Theorem 2.3 are also valid if A ◦ K is replaced with KA. Nevertheless, Asur cannot be replaced with Ain Theorem 2.3(v). For an example, consider A=Np. Then every Banach space Xhas the property that F(Y, X) is k · k-dense (thus, τc-dense) in Np(Y, X) for every space Y. But, if p > 2, there are Banach spaces Xfailing to have the p-approximation property [21, Theorem 6.7] so, according to Corollary 2.6, failing the APNp. Bearing in mind that Np=Np◦ K, this example also implies that KAcannot be replaced with A◦Kin Theorem 2.3(vi). Remark 2.9.Let (A,k·kA) be a Banach operator ideal. In [13], a Banach space Xis said to have the A-AP if F(Y, X) is k·kA-dense in A(Y, X) for all Banach spaces Y. The A-AP and the APAare different notions: as observed in [13], every Banach space has the Np-AP for all p; however, by Remark 2.8, if p > 2, there exist Banach spaces failing the APNp. Now, we present a specific characterization of the APAfor dual Banach spaces. The following lemma is needed. Lemma 2.10.Let Abe an operator ideal and Xand YBanach spaces. If T∈ KA(X, Y ), then T∗∗ ∈ KA(X∗∗, Y ). Proof. Consider T∈ KA(X, Y ) and choose a Banach space Z, an operator U∈ A(Z, Y ) and a compact set K⊂Zsuch that T(BX)⊂U(K). Then T∗∗(BX∗∗ ) = T∗∗(Bweak∗ X)⊂T∗∗(BX)weak∗ ⊂U(K)weak∗ . Since U(K) is compact, it follows that T∗∗(BX∗∗ )⊂U(K), and so we have T∗∗ ∈ KA(X∗∗, Y ). Theorem 2.11.Let Abe an operator ideal and Xa Banach space. The following statements are equivalent: (i) X∗has the APA. 72 J. M. Delgado and C. Pi˜neiro (ii) For every ε > 0and A∈MA c(X∗), there exists S∈ F(X)such that kS∗x∗−x∗k< ε for all x∗∈A. (iii) For every space Yand T∈(KA)d(X, Y ),T∈ {T◦S:S∈ F(X)}k·k . Proof. (i)⇒(ii). This is immediate in view of Theorem 2.3(iv) and bearing in mind the well known fact that the adjoints of finite rank operators are τc-dense in F(X∗). (ii)⇒(iii). Given ε > 0 and T∈(KA)d(X, Y ), there exists S∈ F(X) such that kS∗x∗−x∗k< ε for all x∗∈T∗(BY∗). Then kT◦S−Tk= kS∗◦T∗−T∗k< ε. (iii)⇒(i). By Theorem 2.3(iii) and Remark 2.8, we are going to show that, for every Banach space Yand T∈ KA(Y, X∗), T∈ {S◦T:S∈ F(X)}k·k. So, fix ε > 0 and T∈ KA(Y, X∗). By Lemma 2.10 and the hypothesis, there exists an operator S∈ F(X) so that kT∗◦iX−T∗◦iX◦Sk< ε. Thus, kT−S∗◦Tk≤ki∗ X◦T∗∗ −S∗◦i∗ X◦T∗∗k=k(T∗◦iX)∗−S∗◦(T∗◦iX)∗k< ε. Corollary 2.12.If p≤2, every Banach space has the APQNp. On the other hand, if p > 2, there exists a Banach space failing to have the APQN p. Proof. Since QNp⊂Γ2for every p≤2, the first assertion follows from Corollary 2.7. Now, if p > 2, suppose, for contradiction, that every Banach space has the APQNp. According to Theorem 2.11, we have (2.1) Kd QNp(X, Y )⊂F(X, Y )k·k for all Banach spaces Xand Y. On the other hand, from the equality QN p= QNp◦ K [14, p. 32], it is clear that (2.2) QNd p⊂(QNp◦ K)d⊂((QN p)sur ◦ K)d. Finally, recall that Kp=QN d p[7, Proposition 3.8]; so, in view of (2.1) and (2.2), we obtain Kp(X, Y )⊂ F(X, Y )k·k for all Banach spaces Xand Y, contradicting [21, Theorem 6.7]. Remark 2.13.Theorem 3.1 in [22] states that QNp=Πp◦ K. Hence, in virtue of Theorem 2.3(iv) and Proposition 2.2(ii), the above result is also valid if QNpis replaced with Πp. It is possible to characterize the APAin terms of a trace condition. Let us consider in L(X, Y ) the topology of uniform convergence on A-compact sets in X. We denote this locally convex topology by τc(A). It is defined by the family of seminorms pA(T) = sup x∈A kTxk, where Ais running over all the sets in MA c(X). The following result is immediate. An approximation property 73 Proposition 2.14.Let Abe an operator ideal and Xa Banach space. The following statements are equivalent: (i) Xhas the APA. (ii) For every space Yand T∈ L(X, Y ),T∈ {T◦S:S∈ F(X)}τc(A). (iii) For every space Y,F(X, Y )is τc(A)-dense in L(X, Y ). (iv) F(X)is τc(A)-dense in L(X). (v) For every space Yand T∈ L(Y, X),T∈ {S◦T:S∈ F(X)}τc(A). (vi) For every space Y,F(Y, X)is τc(A)-dense in L(Y, X). According to [4, Theorem 1.1], if A⊂Xis A-compact, then A⊂ aco (xn), where the sequence (xn) in Xis A-convergent to zero (i.e., there exist an operator S∈ A(Z, X) and a null sequence (zn) in Zfor which xn=Szn). Arguing as in the classical case, this property leads to the representation of every element Φin (L(X, Y ), τc(A))∗by Φ(T) = X n hy∗ n, Txni for all T∈ L(X, Y ), where the sequence (xn) in Xis A-convergent to zero and (y∗ n)∈`1(Y∗). Thus, we obtain Proposition 2.15.Let Abe an operator ideal and Xa Banach space. The following statements are equivalent: (i) Xhas the APA. (ii) For every sequence (xn)in Xthat is A-convergent to zero, and every (x∗ n)∈`1(X∗)satisfying Pnhx∗ n, xixn= 0 for all x∈X, we have Pnhx∗ n, xni= 0. Corollary 2.16.Let Abe an operator ideal and Xa Banach space. If X∗∗ has the APA, then so does X. By Proposition 2.14, if Xhas the APAthen F(Y, X) is τc(A)-dense in K(Y, X) for every Banach space Y. In [5, Theorem 4.1] it is proved that the converse statement is true for the p-approximation property. The key to proving this result is the equality Kp=K ◦ Kp[5, Theorem 3.1]. Proposition 2.17.Let Abe an operator ideal satisfying A=K ◦ A. Then Xhas the APAif and only if F(Y, X)is τc(A)-dense in K(Y, X)for every Banach space Y. Proof. First we suppose Ais a surjective ideal. To prove that Xhas the APA, we will show that F(Y, X) is τc-dense in A(Y, X) for every Banach space Y(Theorem 2.3(v)). Given ε > 0, T∈ A(Y, X) and a compact set K⊂Y, there exist a Banach space Zand operators T1∈ K(Z, X) and T2∈ A(Y, Z) so that T=T1◦T2. By hypothesis, there exists S∈ F(Z, X) 74 J. M. Delgado and C. Pi˜neiro such that kSz −T1zk< ε for all z∈T2(K)∈MA c(Z). Then we have kS◦T2y−Tyk< ε for all y∈K. Now we consider an arbitrary operator ideal A. 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Reinov, A survey of some results in connection with Grothendieck approximation property, Math. Nachr. 119 (1984), 257–264. [19] O. I. Reinov, How bad can a Banach space with approximation property be? II, J. Math. Sci. (New York) 112 (2002), 4065–4072. [20] R. Ryan, Introduction to Tensor Products of Banach Spaces, Springer, London, 2002. [21] D. P. Sinha and A. K. Karn, Compact operators whose adjoints factor through subspaces of `p, Studia Math. 150 (2002), 17–33. [22] D. P. Sinha and A. K. Karn, Compact operators which factor through subspaces of `p, Math. Nachr. 281 (2008), 412–423. Juan Manuel Delgado Departamento de Matem´atica Aplicada I Escuela T´ecnica Superior de Arquitectura Avenida Reina Mercedes, 2 41012 Sevilla, Spain E-mail: [email protected] C´andido Pi˜neiro Departamento de Matem´aticas Facultad de Ciencias Experimentales Campus Universitario del Carmen 21071 Huelva, Spain E-mail: [email protected] Received May 28, 2012 Revised version March 10, 2013 (7525)