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Ideals of Operators, Approximability in the Strong Operator Topology, and the Approximation Property

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IDEALS OF OPERATORS, APPROXIMABILITY IN THE STRONG OPERATOR TOPOLOGY, AND THE APPROXIMATION PROPERTY ASVALD˚ LIMA AND EVE OJA Abstract. Let F, K, and W denote the operator ideals of finite rank operators, compact operators, and weakly compact operators. Let X be a closed subspace of a Banach space Y . We prove that, in the case when A = F or A = K and B = K or B = W, the space A(Z; X) is an ideal in B(Z; Y ) for all Banach spaces Z if and only if there exists a Hahn- Banach extension operator Á : X¤ ! Y ¤ such that, for every Banach space Z and every T 2 B(Z; Y ) there exists a net (T®) in A(Z; X) with ¤ ¤ sup® kT®k · kT k such that T® ! T ± Á in the strong operator topology. Using this result, we show that X (resp. X¤) has the approxima- tion property if and only if, for every Banach space Z and every T 2 ¤¤ W(Z; X) (resp. T 2 W(Z; X )), there exists a net (T®) in F(Z; X) with ¤ ¤ sup® kT®k · kT k such that T® ! T jX¤ in the strong operator topology. It follows that conditions considered by Grothendieck as necessary for the approximation property are in fact equivalent to it. 1. Introduction Let us recall that a linear subspace F of a Banach space E is an ideal in E if F ? is the kernel of a norm one projection in E¤. The notion of an ideal was introduced and studied by Godefroy, Kalton, and Saphar in [3]. A linear operator Á: F ¤ ! E¤ is called a Hahn-Banach extension operator if (Áx¤)(x) = x¤(x) and kÁx¤k = kx¤k for all x 2 F and all x¤ 2 F ¤. Let us denote the set of all Hahn-Banach extension operators Á : F ¤ ! E¤ by HB(F; E). It is well known (and straightforward to verify) that HB(F; E) 6= ; if and only if F is an ideal in E. Let X be a Banach space. In [10, Theorem 3.1] Lima, Nygaard, and Oja proved that, for all Banach spaces Z, the space of compact operators K(Z; X) from Z to X, is an ideal in W(Z; X), the space of weakly compact operators from Z to X, if and only if K(Z; X) is an ideal in W(Z; X) for all reflexive Banach spaces Z. They also showed (see [10, Theorem 4.1]) that these condi- tions can be equivalently characterized through the approximability of weakly Date: November 9, 2001. 2000 Mathematics Subject Classification. Primary: 46B20, 46B28, 47L05. Key words and phrases. Spaces of operators, ideals, strong operator topology, approxi- mation property. The research of the second-named author was partially supported by Estonian Science Foun- dation Grant 4400. Filename: LO4.tex. AmsLaTeX. 1 2 ASVALD˚ LIMA AND EVE OJA compact operators in the strong operator topology as follows: for every Ba- nach space Z and for every T 2 W(Z; X), there exists a net (T®) in K(Z; X) with sup® kT®k · kT k such that T® ! T in the strong operator topology. Let X now be a closed subspace of a Banach space Y . In [12, Theorem 4.4] the authors proved that K(Z; X) is an ideal in W(Z; Y ) for all Banach spaces Z if and only if K(Z; X) is an ideal in W(Z; Y ) for all reflexive Banach spaces Z, thus extending the first-mentioned result by Lima, Nygaard, and Oja. The main purpose of this article is to characterize also this more general situation by means of the approximability of weakly compact operators in the strong operator topology. The key result that enables us to achieve this purpose comes from the main theorem of the article [13] by the authors (see Theorem 2.4 and its proof in [13]) and reads as follows. Lemma 1.1. Let X be a closed subspace of a Banach space Y . Let A and B be closed operator ideals satisfying A ½ B. If A(Z; X) is an ideal in B(Z; Y ) for all reflexive Banach spaces Z, then there exists Á 2 HB(X; Y ) such that, for every reflexive Banach space Z, there exists Φ 2 HB(A(Z; X); B(Z; Y )) such that Φ(x¤ ­ z) = (Áx¤) ­ z; 8x¤ 2 X¤; 8z 2 Z: By F, F, K, W, and L, we shall denote the operator ideals (in the sense of Pietsch [18]) of finite rank operators, approximable operators (i.e. norm limits of finite rank operators), compact operators, weakly compact operators, and bounded operators. In Section 2, relying on Lemma 1.1 and the uniform isometric version of the Davis-Figiel-Johnson-PeÃlczy´nskifactorization theorem due to Lima, Nygaard, and Oja [10], we prove, in the case when A = F or A = K and B = K or B = W, that A(Z; X) is an ideal in B(Z; Y ) for all (reflexive) Banach spaces Z if and only if there exists a Hahn-Banach extension operator Á : X¤ ! Y ¤ such that, for every Banach space Z and every T 2 B(Z; Y ), there exists a net ¤ ¤ (T®) in A(Z; X) with sup® kT®k · kT k such that T® ! T ± Á in the strong operator topology (see Theorem 2.3). Results of Section 2 are applied in Sections 3 and 4 to characterize the ap- proximation property of X and X¤ in terms of the approximability of weakly compact operators in the strong operator topology, a problem that goes back to the fundamental work [6] of Grothendieck. Here the main result is The- orem 3.2 that lists four conditions equivalent to the approximation property of X¤, expressed through the approximability of weakly compact or compact operators by finite rank operators in the strong operator topology. The final application, Corollary 4.3, asserts that if X is an M-ideal in its bidual, then X¤ has the approximation property if and only if F(Z; X) is an M-ideal in K(Z; X¤¤) for all Banach spaces Z or, equivalently, F(Z; X) is an ideal in K(Z; X¤¤) for all reflexive Banach spaces Z. Let us fix some more notation. In a linear normed space X, we denote the closed unit ball by BX and the open ball with center x and radius r by B(x; r). The closure of a set A ½ X is denoted by A and its linear span by span A. The identity operator on X is denoted by IX . We shall always regard X as a subspace of X¤¤. IDEALS OF OPERATORS 3 2. Ideals of operators and approximability in the strong operator topology To prove the main result of this article, Theorem 2.3, which together with its background was described in the Introduction, we shall also use the next two Lemmas. Lemma 2.1. Let X be a closed subspace of a Banach space Y and let Z be a Banach space. Let A ½ L(Z; X) and B ½ L(Z; Y ) be subspaces such that F(Z; X) ½ A, F(Z; Y ) ½ B, and A ½ B. Assume that Á 2 HB(X; Y ), Φ 2 HB(A; B), and that ¤ ¤ ¤ ¤ ZΦÁ = fz 2 Z : Φ(x ­ z) = (Áx ) ­ z; 8x 2 X g is reflexive. Then, for every operator T 2 B, there exists a net (T®) ½ A with ¤ ¤ sup® kT®k · kT k such that (T®jZΦÁ ) ! (T jZΦÁ ) ± Á in the strong operator ¤ ¤ topology of L(X ;ZΦÁ). Proof. We follow the idea of the proof of [10, Lemma 1.4]. Let P be the norm one projection on B¤ with ker P = A? defined by Φ, i.e. ¤ P f = Φ(fjA); f 2 B : ¤ ?? ¤¤ ¤ Since P (T ) 2 A ½ B and kP (T )k · kT k, there exists a net (T®) ½ A ¤ ¤ ¤¤ with sup® kT®k · kT k such that T® ! P (T ) weak in B . In particular, for ¤ ¤ all y 2 Y and z 2 ZΦÁ, we have ¤ ¤ ¤ ¤ ¤ y (T®z) = (y ­ z)(T®) ! (y ­ z)(P (T )) = (P (y ­ z))(T ) ¤ ¤ = (Φ(y jX ­ z))(T ) = (Á(y jX ))(T z): This means that ¤ ¤ ¤ ¤ x (T®z) ! (Áx )(T z); 8x 2 X ; 8z 2 ZΦÁ; ¤ ¤ or, equivalently, that (T®jZΦÁ ) ! (T jZΦÁ ) ± Á in the weak operator topol- ¤ ¤ ogy of L(X ;ZΦÁ). Since the weak and strong operator topologies yield the same dual space (cf. e.g. [1, Theorem VI.1.4]), after passing to convex com- binations, we may assume the desired convergence in the strong operator topology. ¤ The following local formulation of ideals is well known. A proof can be found in [9]. Lemma 2.2. Let F be a closed subspace of a Banach space E. The following statements are equivalent. (a) F is an ideal in E. (b) F is locally 1-complemented in E, i.e. for every finite dimensional subspace G of E and for all " > 0, there is an operator U : G ! F such that kUk · 1 + " and Ux = x for all x 2 G \ F . Remark 2.1. It is straightforward to verify that the condition Ux = x for all x 2 G \ F in Lemma 2.2 can be replaced by kUx ¡ xk · " for all x 2 BG\F . Let us recall that, for a linear subspace F of a Banach space E (as it is clear from the definition of the ideal), F is an ideal in E if and only if F is an ideal in E.
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