Banach Spaces of Linear Operators and Homogeneous Polynomials

Home , Approximation property

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This paper is based on part of the author’s doctoral thesis at the Universidade Estadual de Campinas. This research has been supported by CAPES and CNPq. The author is grateful to his thesis advisor, Professor Jorge Mujica, for his advice and help. He is also grateful to Professor Vinicius Fávaro for some helpful suggestions.

2 Banach spaces of linear operators without the approximation property

Let E and F denote Banach spaces over K, where K is R or C. Let E′ denote the dual of E. Let L(E; F ) (resp. LK(E; F )) denote the space of all bounded (resp. compact) linear operators from E into F . Let us recall that E is said to have the approximation property if given K ⊂ E compact and ǫ > 0, there is a ﬁnite rank operator T ∈L(E; E) such that kTx − xk <ǫ for every x ∈ K. If E has the approximation property, then every complemented subspace of E also has the approximation property. E is said to have the bounded approximation property if there exists λ ≥ 1 so that for every compact subset K ⊂ E and for every ǫ> 0 there is a ﬁnite rank operator T ∈ L(E; E) such that kT k ≤ λ and kTx − xk <ǫ for every x ∈ K. E is isomorphic to a complemented subspace of F if and only if there are A ∈L(E; F ) and B ∈L(F ; E) such that B ◦ A = I. These simple remarks will be repeatedly used throughout this paper.

Theorem 2.1. Let E and F be Banach spaces. If E and F contain complemented subspaces isomorphic to M and N, respectively, then L(E; F ) contains a complemented subspace isomorphic to L(M; N).

Proof. By hypothesis there are A1 ∈ L(M; E), B1 ∈ L(E; M), A2 ∈ L(N; F ), B2 ∈ L(F ; N) such that B1 ◦ A1 = I and B2 ◦ A2 = I. Consider the operators

C : S ∈L(M; N) → A2 ◦ S ◦ B1 ∈L(E; F ) and

D : T ∈L(E; F ) → B2 ◦ T ◦ A1 ∈L(M; N). Then D ◦ C = I and the desired conclusion follows.

The next theorem improves [11, Proposition 2.4].

Theorem 2.2. Let 1 < p ≤ q < ∞. If E and F contain complemented subspaces isomorphic to ℓp and ℓq, respectively, then L(E; F ) does not have the approximation property.

Proof. By Theorem 2.1 L(E; F ) contains a complemented subspace isomorphic to L(ℓp; ℓq). Then the conclusion follows from [11, Proposition 2.2 ].

The next result improves [11, Proposition 2.2].

Theorem 2.3. Let 1 < p ≤ q < ∞, and let E and F be closed inﬁnite dimensional subspaces of ℓp and ℓq, respectively. Then L(E; F ) does not have the approximation property.

Proof. By [21, Lemma 2 ] or [16, Proposition 2.a.2] E and F contain complemented subspaces isomorphic to ℓp and ℓq, respectively. Then the desired conclusion follows from Theorem 2.2.

2 The next result complements [11, Proposition 2.1].

Theorem 2.4. Let 2 < p ≤ q < ∞, and let E and F be closed inﬁnite dimensional subspaces of Lp[0, 1] and Lq[0, 1], respectively, with F not isomorphic to ℓ2. Then L(E; F ) does not have the approximation property.

Proof. (i) If E is not isomorphic to ℓ2, then it follows from [22, p. 206, 12.5] that E and F contain complemented subspaces isomorphic to ℓp and ℓq, respectively. Then the desired conclusion follows from Theorem 2.2.

(ii) If E is isomorphic to ℓ2, then the same argument shows that L(E; F ) contains a complemented subspace isomorphic to L(ℓ2; ℓq), and the desired conclusion follows as before.

The next result improves [13, Corollary 2.8].

Theorem 2.5. If 1

⊥ ′ Proof. We apply [13, Theorem 2.4]. By [15, Lemma 1] LK(ℓp; ℓq) is a complemented subspace of L(ℓp; ℓq) . By [18, Corollary 16.69] LK(ℓp; ℓq) has a Schauder basis. If we assume that L(ℓp; ℓq)/LK(ℓp; ℓq) has the approximation property, then [13, Theorem 2.4] would imply that L(ℓp; ℓq) has the approximation property, thus contradicting [11, Proposition 2.2].

Remark 2.6. If 1

Theorem 2.7. If E and F contain complemented subspaces isomorphic to M and N, respectively, then L(E; F )/LK (E; F ) contains a complemented subspace isomorphic to L(M; N)/LK (M; N).

Proof. By hypothesis there are A1 ∈ L(M; E), B1 ∈ L(E; M), A2 ∈ L(N; F ), B2 ∈ L(F ; N) such that B1 ◦A1 = I and B2 ◦A2 = I. Let C : L(M; N) →L(E; F ) and D : L(E; F ) →L(M; N) be the operators from the proof of Theorem 2.1. Since C(LK(M; N)) ⊂LK(E; F ) and D(LK (E; F )) ⊂LK(M; N), the operators

C˜ : [S] ∈L(M; N)/LK (M; N) → [A2 ◦ S ◦ B1] ∈L(E; F )/LK (E; F ) and

D˜ : [T ] ∈L(E; F )/LK (E; F ) → [B2 ◦ T ◦ A1] ∈L(M; N)/LK (M; N) are well deﬁned, and D˜ ◦ C˜ = I, thus completing the proof.

Theorem 2.8. Let 1 < p ≤ q < ∞. If E and F contain complemented subspaces isomorphic to ℓp and ℓq, respectively, then L(E; F )/LK (E; F ) does not have the approximation property.

Proof. By Theorem 2.7 L(E; F )/LK (E; F ) contains a complemented subspace isomorphic to L(ℓp; ℓq)/LK (ℓp; ℓq). Then the desired conclusion follows from Theorem 2.5.

By combining Theorem 2.8 and [21, Lemma 2] we obtain the following theorem.

Theorem 2.9. Let 1 < p ≤ q < ∞, and let E and F be closed inﬁnite dimensional subspaces of ℓp and ℓq, respectively. Then L(E; F )/LK (E; F ) does not have the approximation property.

3 3 Other examples of Banach spaces of linear operators without the approximation property

Example 3.1. Let U1 denote the universal space of Pelczynski (see, [16, Theorem 2.d.10]). U1 is a Banach space with an unconditional basis with the property that every Banach space with an unconditional basis is isomorphic to a complemented subspace of U1. Since every ℓp (1 ≤ p< ∞) has an unconditional basis, it follows that every ℓp (1 ≤ p< ∞) is isomorphic to a complemented subspace of U1. By Theorem 2.2 none of the spaces L(U1; U1), L(U1; ℓq) (1 0 . M n n=1 n nX=1

Then ℓM is a Banach space for the norm ∞ kxk = inf ρ> 0; M(|ξ /ρ) ≤ 1 . n nX=1

ℓM is called an Orlicz sequence space. p Example 3.3. Consider the Orlicz function Mp(t) = t (1 + | log t|), with 1

Theorem 2.2, L(ℓMp ; ℓMq ) does not have the approximation property. Deﬁnition 3.4. (see [16, p. 175]) ∞ Let 1 ≤ p < ∞ and let w = {wn} be a nonincreasing sequence of positive numbers such that w1 = 1, ∞ n=1

lim wn = 0 and wn = ∞. Let n→∞ nX=1 ∞ 1/p ∞ K p d(w,p)= x = (ξn)n=1 ⊂ : kxk = sup |ξπ(n)| wn < ∞ , π Xn=1 where π ranges over all permutations of N. Then d(w,p) is a Banach space, called a Lorentz sequence space. Example 3.5. It follows from [16, p. 177, Proposition 4.e.3] that every closed inﬁnite dimensional subspace of d(w,p) contains a complemented subspace isomorphic to ℓp. By Theorem 2.2, if 1 < p ≤ q < ∞, and E and F are closed inﬁnite dimensional subspaces of d(w,p) and d(w,q), respectively, then L(E; F ) does not have the approximation property.

4 Spaces of homogeneous polynomials without the approximation property

Let P(nE; F ) denote the Banach space of all continuous n-homogeneous polynomials from E into F . We omit F when F = K. A polynomial P ∈ P(nE; F ) is called compact if P takes bounded subsets in E into relatively n compact subsets in F . Let PK ( E; F ) denote the space of all compact n-homogeneous polynomials from E into F . We refer to [7] or [19] for background information on the theory of polynomials on Banach spaces.

An important tool in this section is a linearization theorem due to Ryan [24]. We will use the following version of Ryan’s linearization theorem, wich appeared in [20]. Here τc denotes the compact-open topology.

4 Theorem 4.1. For each Banach space E and each n ∈ N let

n n ′ Q( E) = (P( E),τc) , with the norm induced by P(nE), and let

n δn : x ∈ E → δx ∈ Q( E)

n n denote the evaluation mapping, that is, δx(P )= P (x) for every x ∈ E and P ∈ P( E). Then Q( E) is a Banach n n n space and δn ∈ P( E; Q( E)). The pair (Q( E), δn) has the following universal property: for each Banach n n space F and each P ∈ P( E; F ), there is a unique operator Tp ∈ L(Q( E); F ) such that Tp ◦ δn = P . The mapping n n P ∈ P( E; F ) → Tp ∈L(Q( E); F ) n n n is an isometric isomorphism. Moreover P ∈ PK( E; F ) if only if Tp ∈ LK(Q( E); F ). Furthermore Q( E) is isometrically isomorphic to ⊗ˆ n,s,πE, the completion of the space of n-symmetric tensors on E, with the projective topology.

Theorem 4.2. If E and F contain complemented subspaces isomorphic to M and N, respectively, then P(nE; F ) contains a complemented subspace isomorphic to P(nM; N).

Proof. By hypothesis there are A1 ∈ L(M; E), B1 ∈ L(E; M), A2 ∈ L(N; F ), B2 ∈ L(F ; N) such that B1 ◦ A1 = I and B2 ◦ A2 = I. Consider the operators

n n C : P ∈ P( M; N) → A2 ◦ P ◦ B1 ∈ P( E; F ) and n n D : Q ∈ P( E; F ) → B2 ◦ Q ◦ A1 ∈ P( M; N). Then D ◦ C = I and the desired conclusion follows.

Corollary 4.3. If E contains a complemented subspace isomorphic to M, then P(nE) contains a complemented subspace isomorphic to P(nM).

Proof. Take F = N = K in Theorem 4.2.

The next theorem improves [11, Theorem 3.2].

n Theorem 4.4. Let 1

n n Proof. By Corollary 4.3 P( E) contains a complemented subspace isomorphic to P( ℓp). Then the conclusion follows from [11, Theorem 3.2].

Theorem 4.4 can be used to produce many additional counterexamples. For instance, by combining Theorem 4.4 and [21, Lemma 2] we obtain the following result.

n Theorem 4.5. Let 1

In a similar way we may obtain scalar-valued polynomial versions of Theorem 2.4 and Examples 3.1, 3.3 and 3.5. We leave the details to the reader.

5 Theorem 4.6. Let 1 < p ≤ q < ∞. If E and F contain complemented subspaces isomorphic to ℓp and ℓq, respectively, then P(nE; F ) does not have the approximation property for every n ≥ 1.

Proof. By [4, Proposition 5], L(E; F ) is isomorphic to a complemented subspace of P(nE; F ). Then the desired conclusion follows from Theorem 2.2.

Theorem 4.6 can be used to produce many additional counterexamples. For instance, by combining Theorem 4.6 and [21, Lemma 2] we obtain the following result.

Theorem 4.7. Let 1 < p ≤ q < ∞. and let E and F be closed inﬁnite dimensional subspaces of ℓp and ℓq, respectively. Then P(nE; F ) does not have the approximation property for every n ≥ 1.

In a similar way we may obtain vector-valued polynomial versions of Theorem 2.4 and Examples 3.1, 3.3 and 3.5. We leave the details to the reader.

n n Theorem 4.8. If n

Proof. By Theorem 4.1 we can write n n P( ℓp; ℓq)= L(Q( ℓp); ℓq) and n n PK( ℓp; ℓq)= LK(Q( ℓp); ℓq). n ⊥ n ′ We apply [13, Theorem 2.4]. By [15, Lemma 1] LK(Q( ℓp); ℓq) is a complemented subspace of L(Q( ℓp); ℓq) . n n By [11, Remark 3.3] P( ℓp) is a reﬂexive Banach space with a Schauder basis. Hence Q( ℓp) is also a reﬂexive n Banach space with a Schauder basis. Then by [18, Corollary 16.69] LK (Q( ℓp); ℓq) has a Schauder basis. If we n n assume that L(Q( ℓp); ℓq)/LK(Q( ℓp); ℓq) has the approximation property, then [13, Theorem 2.4] would imply n 1 that L(Q( ℓp); ℓq) has the approximation property. But this contradicts Theorem 2.8, since ℓp = Q( ℓp) is a n complemented subspace of Q( ℓp), by [4, Theorem 3]. This completes the proof.

n n Theorem 4.9. If E and F contain complemented subspaces isomorphic to M and N, respectively, then P( E; F )/PK ( E; F ) n n contains a complemented subspace isomorphic to P( M; N)/PK ( M; N).

Proof. By hypothesis there are A1 ∈ L(M; E), B1 ∈ L(E; M), A2 ∈ L(N; F ), B2 ∈ L(F ; N) such that B1 ◦ A1 = I and B2 ◦ A2 = I. Let C : P(nM; N) → P(nE; F ) and D : P(nE; F ) → P(nM; N)

n n n be the operators from the proof of Theorem 4.2. Since C(PK ( M; N)) ⊂ PK( E; F ) and D(PK ( E; F )) ⊂ n PK ( M; N), the operators

n n n n C˜ : [P ] ∈ P( M; N)/PK ( M; N) → [A2 ◦ P ◦ B1] ∈ P( E; F )/PK ( E; F ) and n n n n D˜ : [Q] ∈ P( E; F )/PK ( E; F ) → [B2 ◦ Q ◦ A1] ∈ P( M; N)/PK ( M; N) are well-deﬁned and D˜ ◦ C˜ = I, thus completing the proof.

Theorem 4.10. Let n

6 n n n n Proof. By Theorem 4.9 P( E; F )/PK ( E; F ) contains a complemented subspace isomorphic to P( ℓp; ℓq)/PK ( ℓp; ℓq). Thus the desired conclusion follows from Theorem 4.8.

Theorem 4.10 can be used to produce many additional counterexamples. For instance by combining Theorem 4.10 and [21, Lemma 2] we obtain the following theorem.

Theorem 4.11. Let n

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