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New Examples of Weakly Compact Approximation in Banach Spaces

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New Examples of Weakly Compact Approximation in Banach Spaces Annales Academiæ Scientiarum Fennicæ Mathematica Volumen 33, 2008, 429–438 NEW EXAMPLES OF WEAKLY COMPACT APPROXIMATION IN BANACH SPACES Eero Saksman and Hans-Olav Tylli University of Helsinki, Department of Mathematics and Statistics P.O. Box 68, FI-00014 University of Helsinki, Finland; eero.saksman@helsinki.fi University of Helsinki, Department of Mathematics and Statistics P.O. Box 68, FI-00014 University of Helsinki, Finland; [email protected].fi Abstract. The Banach space E has the weakly compact approximation property (W.A.P.) if there is C < 1 so that the identity map IE can be uniformly approximated on any weakly compact subset D ½ E by weakly compact operators V on E satisfying kV k · C. We show that the spaces N(`p; `q) of nuclear operators `p ! `q have the W.A.P. for 1 < q · p < 1, but that the Hardy space H1 does not have the W.A.P. 0. Introduction The Banach space E has the weakly compact approximation property (abbrevi- ated W.A.P.) if there is C < 1 so that for any weakly compact set D ½ E and " > 0 one finds a weakly compact operator V 2 W (E) satisfying (0:1) sup kx ¡ V xk < " and kV k · C: x2D Here V 2 W (E) if the image VBE of the closed unit ball BE of E is relatively weakly compact. This concept of weakly compact approximation is natural, but the resulting property differs completely from the classical bounded approximation properties defined in terms of finite rank or compact operators (for more about these properties see e.g. [C]). The W.A.P. was introduced in [AT], and some applications can be found in [AT] and [T2]. It was later more systematically studied in [OT] from the perspective of Banach space theory. The W.A.P. remains fairly rare and elusive for non-reflexive spaces (obviously any reflexive space has it). The following list reviews some of the known results. (0.2) If E is a L 1- or L 1-space, then E has the W.A.P. if and only if E has the 1 Schur property, see [AT, Cor. 3]. Thus ` has the W.A.P., while c0, C(0; 1) and L1(0; 1) fail to have it. (0.3) The direct sums `1(`p) and `p(`1) have the W.A.P. for 1 < p < 1, [OT, Prop. 5.3]. 2000 Mathematics Subject Classification: Primary 46B28; Secondary 46B20. Key words: Weakly compact approximation, nuclear operators, Hardy space. Supported by the Academy of Finland projects #113286 and #118765 (E.S.), #53893 and #210970 (H.-O.T.). 430 Eero Saksman and Hans-Olav Tylli (0.4) The quasi-reflexive James’ space J, as well as its dual J ¤, have the W.A.P., [OT, Thm. 2.2 and 3.3]. On the other hand, there is [ArT, Prop. 14.11] a quasi-reflexive hereditarily indecomposable space X that fails to have the W.A.P. Moreover, the related James’ tree space JT fails to have the W.A.P., [OT, Thm. 6.5]. As the first result of this note we show that the spaces N(`p; `q) consisting of nuclear operators have the W.A.P. for 1 < q · p < 1 (note that N(`p; `q) is reflex- ive if q > p). This result, which was motivated by timely questions of Zacharias and Defant, includes the Schatten trace class space C1 for p = q = 2. Secondly, we show that the Hardy space H1 does not have the W.A.P., which solves a question from [AT, p. 370] in the negative. In order to determine whether a given space E has the W.A.P. or not one is often forced to rely on very specific properties of E, and there still remains fairly concrete Banach spaces for which this property is not decided (see e.g. the Problems in Section 2 as well as [OT]). 1. The spaces N(`p; `q) of nuclear operators have the W.A.P. Let E and F be Banach spaces. Recall that T : E ! F is a nuclear operator, ¤ ¤ denoted T 2 N(E; F ), if there are sequences (xj ) ½ E and (yj) ½ F so that P1 ¤ P1 ¤ ¤ j=1 kxj k kyjk < 1 and T = j=1 xj ­ yj. Here xj ­ yj denotes the rank-1 ¤ operator x 7! xj (x)yj. Then (N(E; F ); k ¢ kN ) is a Banach space, where the nuclear norm of T 2 N(E; F ) is ½ X1 X1 ¾ ¤ ¤ kT kN = inf kxj k kyjk : T = xj ­ yj : j=1 j=1 Recall that kASBkN · kAk kBk kSkN whenever S 2 N(E; F ) and A; B are com- 2 patible bounded operators. One may isometrically identify N(` ) = C1, where C1 is the Schatten trace class space, see e.g. [P, Sect. 0.b] or [Pi, Sect. 2.11]. Theorem 1 below is the main result of this section. Observe that in its statement the spaces N(`p; `q) are actually reflexive for 1 < p < q < 1, so that only the cases 1 < q · p < 1 contain non-trivial information. In fact, N(`p; `q) = K(`q; `p)¤ in the trace-duality hU; V i = tr(VU);U 2 N(`p; `q);V 2 K(`q; `p); where the space K(`q; `p) of compact operators `q ! `p is reflexive once 1 < p < q < 1, see e.g. [R, Cor. 2.6] or [K, Sect. 2, Cor. 2]. Theorem 1 can also be rephrased p q in the terms of the projective tensor products ` ­b ¼` , see the Remarks following Lemma 2. Theorem 1. N(`p; `q) has the W.A.P. whenever 1 < p; q < 1. The proof of Theorem 1 is based on Lemma 2 below, which contains a basic characterization of the relatively weakly compact subsets of the non-reflexive spaces p q p N(` ; ` ) for 1 < q · p < 1. Let (ej) be the unit vector basis of ` for 1 < p < 1. New examples of weakly compact approximation in Banach spaces 431 p We denote the natural basis projection of ` onto [e1; : : : ; en] by Pn, and set Qn = I ¡Pn for n 2 N. For m < n we also put P(m;n] = Pn ¡Pm = PnQm = QmPn, which p is the natural projection of ` onto [em+1; : : : ; en]. We denote the corresponding basis q e e e projections on ` by Pn; Qn and P(m;n], respectively. We will frequently use the facts e e p q that kS ¡ PnSPnkN ! 0 and kQnSQnkN ! 0 as n ! 1 for any S 2 N(` ; ` ). Lemma 2. Suppose that 1 < q · p < 1 and let D ½ N(`p; `q) be a bounded subset. Then D is relatively weakly compact in N(`p; `q) if and only if e (1:1) lim sup kQnSQnkN = 0: n!1 S2D We first complete the proof of Theorem 1 with the help of (1.1) before estab- lishing the more technical Lemma 2. Proof of Theorem 1. We may assume that 1 < q · p < 1 since N(`p; `q) is reflexive for 1 < p < q < 1, see the comment preceding Theorem 1. Suppose that D ½ N(`p; `q) is a weakly compact subset and let " > 0 be given. Write ¡ ¢ e e e e e (1:2) S = PnSPn + PnSQn + QnSPn + QnSQn ´ Ãn(S) + QnSQn p q p q p q for S 2 N(` ; ` ) and n 2 N. Here Ãn : N(` ; ` ) ! N(` ; ` ) for n 2 N are the e e e p q bounded linear maps defined by Ãn(S) = PnSPn+PnSQn+QnSPn for S 2 N(` ; ` ). Clearly kÃnk · 3 for any n. It follows from (1.1) and (1.2) that e sup kS ¡ Ãn(S)kN = sup kQnSQnkN ! 0 as n ! 1; S2D S2D so that supS2D kS ¡ Ãn(S)kN < " once n is large enough. Consequently it will be enough to verify that p q (1:3) Ãn 2 W (N(` ; ` )); n 2 N: This fact can be deduced from a suitable combination of general results, see the proofs of [LS, Prop. 2.2 and 2.3] or the survey [ST, p. 262], but we sketch a direct argument for completeness. Note first that the maps ¤ ¤ ¤ ¤ ¤ 'y;y¤ (S) = (y ­ y)S = S y ­ y; Áx;x¤ (S) = S(x ­ x) = x ­ Sx are weakly compact on N(`p; `q) for any y¤ 2 `q0 , y 2 `q, x 2 `p and x¤ 2 `p0 , where p0 0 ¤ and q are the respective dual exponents. In fact, 'y;y¤ (BN(`p;`q)) ½ ky k kyk(B`p0 ­ ¤ ¤ ¤ y), since kS y ­ ykN · ky k kyk for S 2 BN(`p;`q). Clearly the set B`p0 ­ y is relatively weakly compact in N(`p; `q), since z¤ 7! z¤ ­ y embeds `p0 isomorphically p q into N(` ; ` ) for y 6= 0. The case of Áx;x¤ is analogous. Finally, (1.3) follows since the individual operators defining Ãn, such as S 7! e PnSQn, are sums of weakly compact ones composed with bounded ones. The proof of Theorem 1 will be complete once Lemma 2 has been established. ¤ Proof of Lemma 2. Suppose first that (1.1) holds. According to (1.2) we get that (1:4) D ½ Ãn(D) + ±nBN(`p;`q); n 2 N; 432 Eero Saksman and Hans-Olav Tylli e where ±n ´ supS2D kQnSQnkN ! 0 as n ! 1. Here Ãn(D) is a relatively weakly compact subset of N(`p; `q) for all n by (1.3). It is a standard fact that (1.4) then implies that D is a relatively weakly compact subset of N(`p; `q).
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