Multiplication operators on L(L_{p}) and ℓ_{p}-strictly singular operators^{∗}

William B. Johnson^{†}^{‡} Gideon Schechtman^{§}

September 18, 2007

Abstract

A classification of weakly compact multiplication operators on L(L_{p}), 1 < p < ∞, is given. This answers a question raised by Saksman and Tylli in 1992. The classification involves the concept of ℓ_{p}-strictly singular operators, and we also investigate the structure of general ℓ_{p}-strictly singular operators on L_{p}. The main result is that if an operator T on L_{p}, 1 < p < 2, is ℓ_{p}-strictly singular and T_{|X} is an isomorphism for some subspace X of L_{p}, then X embeds into L_{r} for all r < 2, but X need not be isomorphic to a Hilbert space.

It is also shown that if T is convolution by a biased coin on L_{p} of the Cantor group, 1 ≤ p < 2, and T_{|X} is an isomorphism for some reflexive subspace X of L_{p}, then X is isomorphic to a Hilbert space. The case p = 1 answers a question asked by Rosenthal in 1976.

1 Introduction

Given (always bounded, linear) operators A, B on a Banach space X, define L_{A}, R_{B} on L(X) (the space of bounded linear operators on X) by L_{A}T = AT, R_{B}T = T B. Operators of the form L_{A}R_{B} on L(X) are called multiplication operators. The beginning point of this paper is a problem raised in 1992 by E. Saksman and H.-O. Tylli [ST1] (see also [ST2, Problem 2.8]):

Characterize the multiplication operators on L(L_{p}), 1 < p ≠ 2 < ∞, which are weakly compact.

∗AMS subject classification: 46B20,46E30. Key words: Elementary operators, multiplication operators, strictly singular operators, L_{p} spaces, biased coin

†Supported in part by NSF DMS-0503688 and U.S.-Israel Binational Science Foundation.

‡Corresponding author.

§Supported in part by Israel Science Foundation and U.S.-Israel Binational Science Foundation; participant, NSF Workshop in Analysis and Probability, Texas A&M University.

Here L_{p} is L_{p}(0, 1) or, equivalently, L_{p}(μ) for any purely non-atomic separable probability μ.

In Theorem 1 we answer the Saksman-Tylli question. The characterization is rather simple but gives rise to questions about operators on L_{p}, some of which were asked by Tylli. First we set some terminology. Given an operator T : X → Y and a Banach space Z, say that T is Z-strictly singular provided there is no subspace Z_{0} of X which is isomorphic to Z for which T_{|Z_{0}} is an isomorphism. An operator S : Z → W factors through an operator T : X → Y provided there are operators A : Z → X and B : Y → W so that S = BT A. If S factors through the identity operator on X, we say that S factors through X.

If T is an operator on L_{p}, 1 < p < ∞, then T is ℓ_{p}-strictly singular (respectively, ℓ_{2}-strictly singular) if and only if I_{ℓ_{p}} (respectively, I_{ℓ_{2}}) does not factor through T. This is true because every subspace of L_{p} which is isomorphic to ℓ_{p} (respectively, ℓ_{2}) has a subspace which is still isomorphic to ℓ_{p} (respectively, ℓ_{2}) and is complemented in L_{p}. Actually, a stronger fact is true: if {x_{n}}_{n=1}^{∞} is a sequence in L_{p} which is equivalent to the unit vector basis for either ℓ_{p} or ℓ_{2}, then {x_{n}}_{n=1}^{∞} has a subsequence which spans a complemented subspace of L_{p}. For p > 2, an even stronger theorem was proved by Kadec-Pełczyński [KP]. When 1 < p < 2 and {x_{n}}_{n=1}^{∞} is a sequence in L_{p} which is equivalent to the unit vector basis for ℓ_{2}, one takes \{y_{n}\}_{n=1}^{∞} in L_{p′} (where p′ = \frac{p}{p−1} is the conjugate index to p) which are uniformly bounded and biorthogonal to {x_{n}}_{n=1}^{∞}. By passing to a subsequence which is weakly convergent and subtracting the limit from each y_{n}, one may assume that y_{n} → 0 weakly and hence, by the Kadec–Pełczyński dichotomy [KP], has a subsequence that is equivalent to the unit vector basis of ℓ_{2} (since it is clearly impossible that {y_{n}}_{n=1}^{∞} has a subsequence equivalent to the unit vector basis of ℓ_{p′}) . This implies that that the corresponding subsequence of {x_{n}}_{n=1}^{∞} spans a complemented subspace of L_{p}. (Pełczyński showed this argument, or something similar, to one of the authors many years ago, and a closely related result was proved in [PR].) Finally, when 1 < p < 2 and {x_{n}}_{n=1}^{∞} is a sequence in L_{p} which is equivalent to the unit vector basis for ℓ_{p}, see the comments after the statement of Lemma 1.

Notice that the comments in the preceding paragraph yield that an operator on L_{p}, 1 < p < ∞, is ℓ_{p}-strictly singular (respectively, ℓ_{2}-strictly singular) if and only if T^{∗} is ℓ_{p′}-strictly singular (respectively, ℓ_{2}-strictly singular). Better known is that an operator on L_{p}, 1 < p < ∞, is strictly singular if it is both ℓ_{p}-strictly singular and ℓ_{2}-strictly singular (and hence T is strictly singular if and only if T^{∗} is strictly singular). For p > 2 this is immediate from [KP], and Lutz Weis [We] proved the case p < 2.

Although Saksman and Tylli did not know a complete characterization of the weakly compact multiplication operators on L(L_{p}), they realized that a classification must involve ℓ_{p} and ℓ_{2}-strictly singular operators on L_{p}. This led Tylli to ask us about possible classifications of the ℓ_{p} and ℓ_{2}-strictly singular operators on L_{p}. The ℓ_{2} case is known. It is enough to consider the case 2 < p < ∞. If T is an operator on L_{p}, 2 < p < ∞, and T is ℓ_{2}-strictly singular, then it is an easy consequence of the Kadec-Pełczyński dichotomy that I_{p,2}T is compact, where I_{p,r} is the identity mapping from L_{p} into L_{r}. But then by [Jo], T factors through ℓ_{p}. Tylli then asked whether the following conjecture is true:

Tylli Conjecture. If T is an ℓ_{p}-strictly singular operator on L_{p}, 1 < p < ∞, then T is in the closure (in the operator norm) of the operators on L_{p} that factor through ℓ_{2}. (It is clear that the closure is needed because not all compact operators on L_{p}, p ≠ 2, factor through ℓ_{2}.)

We then formulated a weaker conjecture:

Weak Tylli Conjecture. If T is an ℓ_{p}-strictly singular operator on L_{p}, 1 < p < ∞, and J : L_{p} → ℓ_{∞} is an isometric embedding, then JT is in the closure of the operators from L_{p} into ℓ_{∞} that factor through ℓ_{2}.

It is of course evident that an operator on L_{p}, p ≠ 2, that satisfies the conclusion of the Weak Tylli Conjecture must be ℓ_{p}-strictly singular. There is a slight subtlety in these conjectures: while the Tylli Conjecture for p is equivalent to the Tylli Conjecture for p′, it is not at all clear and is even false that the Weak Tylli Conjecture for p is equivalent to the Weak Tylli Conjecture for p′. In fact, we observe in Lemma 2 (it is simple) that for p > 2 the Weak Tylli Conjecture is true, while the example in Section 4 yields that the Tylli Conjecture is false for all p ≠ 2 and the Weak Tylli Conjecture is false for p < 2.

There are however some interesting consequences of the Weak Tylli Conjecture that are true when p < 2. In Theorem 4 we prove that if T is an ℓ_{p}-strictly singular operator on L_{p}, 1 < p < 2, then T is ℓ_{r}-strictly singular for all p < r < 2. In view of theorems of Aldous [Al] (see also [KM]) and Rosenthal [Ro3], this proves that if such a T is an isomorphism on a subspace Z of L_{p}, then Z embeds into L_{r} for all r < 2. The Weak Tylli Conjecture would imply that Z is isomorphic to ℓ_{2}, but the example in Section 4 shows that this need not be true. When we discovered Theorem 4, we thought its proof bizarre and assumed that a more straightforward argument would yield a stronger theorem. The example suggests that in fact the proof may be “natural”.

In Section 5 we discuss convolution by a biased coin on L_{p} of the Cantor group, 1 ≤ p < 2. We do not know whether such an operator T on L_{p}, 1 < p < 2, must satisfy the Tylli Conjecture or the weak Tylli conjecture. We do prove, however, that if T_{|X} is an isomorphism for some reflexive subspace X of L_{p}, 1 ≤ p < 2, then X is isomorphic to a Hilbert space. This answers an old question of H. P. Rosenthal [Ro4].

The standard Banach space theory terminology and background we use can be found in [LT].

2 Weakly compact multiplication operators on L(L_{p})

We use freely the result [ST2, Proposition 2.5] that if A, B are in L(X) where X is a reflexive Banach space with the approximation property, then the multiplication operator L_{A}R_{B} on L(X) is weakly compact if and only if for every T in L(X), the operator AT B is compact. For completeness, in section 6 we give another proof of this under the weaker assumption that X is reflexive and has the compact approximation property. This theorem implies that for such an X, L_{A}R_{B} is weakly compact on L(X) if and only if L_{B^{∗}}R_{A^{∗}} is a weakly compact operator on L(X^{∗}). Consequently, to classify weakly compact multiplication operators on L(L_{p}), 1 < p < ∞, it is enough to consider the case p > 2. For p ≤ r we denote the identity operator from ℓ_{p} into ℓ_{r} by i_{p,r}. It is immediate from [KP] that an operator T on L_{p}, 2 < p < ∞, is compact if and only if i_{2,p} does not factor through T.

Theorem 1 Let 2 < p < ∞ and let A, B be bounded linear operators on L_{p}. Then the multiplication operator L_{A}R_{B} on L(L_{p}) is weakly compact if and only if one of the following (mutually exclusive) conditions hold.

(a) i_{2,p} does not factor through A (i.e., A is compact)

(b) i_{2,p} factors through A but i_{p,p} does not factor through A (i.e., A is ℓ_{p}-strictly singular) and i_{2,2} does not factor through B (i.e., B is ℓ_{2}-strictly singular)

(c) i_{p,p} factors through A but i_{2,p} does not factor through B (i.e., B is compact)

Proof: The proof is a straightforward application of the Kadec-Pełczyński dichotomy principle [KP]: if {x_{n}}_{n=1}^{∞} is a semi-normalized (i.e., bounded and bounded away from zero) weakly null sequence in L_{p}, 2 < p < ∞, then there is a subsequence which is equivalent to either the unit vector basis of ℓ_{p} or of ℓ_{2} and spans a complemented subspace of L_{p}. Notice that this immediately implies the “i.e.’s” in the statement of the theorem so that (a) and (c) imply that L_{A}R_{B} is weakly compact. Now assume that (b) holds and let T be in L(L_{P}). If AT B is not compact, then there is a normalized weakly null sequence {x_{n}}_{n=1}^{∞} in L_{p} so that AT Bx_{n} is bounded away from zero. By passing to a subsequence, we may assume that {x_{n}}_{n=1}^{∞} is equivalent to either the unit vector basis of ℓ_{p} or of ℓ_{2}. If {x_{n}}_{n=1}^{∞} is equivalent to the unit vector basis of ℓ_{p}, then since T Bx_{n} is bounded away from zero, we can assume by passing to another subsequence that also T Bx_{n} is equivalent to the unit vector basis of ℓ_{p} and similarly for AT Bx_{n}, which contradicts the assumption that A is ℓ_{p}-strictly singular. On the other hand, if {x_{n}}_{n=1}^{∞} is equivalent to the unit vector basis of ℓ_{2}, then since B is ℓ_{2}-strictly singular we can assume by passing to a subsequence that Bx_{n} is equivalent to the unit vector basis of ℓ_{p} and continue as in the previous case to get a contradiction.

Now suppose that (a), (b), and (c) are all false. If i_{p,p} factors through A and i_{2,p} factors through B then there is sequence {x_{n}}_{n=1}^{∞} equivalent to the unit vector basis of ℓ_{2} or of ℓ_{p} so that Bx_{n} is equivalent to the unit vector basis of ℓ_{2} or of ℓ_{p} (of course, only three of the four cases are possible) and Bx_{n} spans a complemented subspace of L_{p}. Moreover, there is a sequence {y_{n}}_{n=1}^{∞} in L_{p} so that both y_{n} and Ay_{n} are equivalent to the unit vector basis of ℓ_{p}. Since Bx_{n} spans a complemented subspace of L_{p}, the mapping Bx_{n} ↦ y_{n} extends to a bounded linear operator T on L_{p} and AT B is not compact. Finally, suppose that i_{2,p} factors through A but i_{p,p} does not factor through A and i_{2,2} factors through B. Then there is a sequence {x_{n}}_{n=1}^{∞} so that x_{n} and Bx_{n} are both equivalent to the unit vector basis of ℓ_{2} and Bx_{n} spans a complemented subspace of L_{p}. There is also a sequence {y_{n}}_{n=1}^{∞} equivalent to the unit vector basis of ℓ_{2} so that Ay_{n} is equivalent to the unit vector basis of ℓ_{2} or of ℓ_{p}. The mapping Bx_{n} ↦ y_{n} extends to a bounded linear operator T on L_{p} and AT B is not compact.

It is perhaps worthwhile to restate Theorem 1 in a way that the cases where L_{A}R_{B} is weakly compact are not mutually exclusive.

Theorem 2 Let 2 < p < ∞ and let A, B be bounded linear operators on L_{p}. Then the multiplication operator L_{A}R_{B} on L(L_{p}) is weakly compact if and only if one of the following conditions hold.

(a) A is compact

(b) A is ℓ_{p}-strictly singular and B is ℓ_{2}-strictly singular

(c) B is compact

3 ℓ_{p}-strictly singular operators on L_{p}

We recall the well known

Lemma 1 Let W be a bounded convex symmetric subset of L_{p}, 1 ≤ p ≠ 2 < ∞. The following are equivalent:

1. No sequence in W equivalent to the unit vector basis for ℓ_{p} spans a complemented subspace of L_{p}.

2. For every C there exists n so that no length n sequence in W is C-equivalent to the unit vector basis of ℓ_{p}^{n}.

3. For each ε > 0 there is M_{ε} so that W ⊂ εB_{L_{p}} + M_{ε}B_{L_{∞}}.

4. |W |^{p} is uniformly integrable; i.e., lim_{t↓0} sup_{x∈W} sup_{μ(E)<t} ∥1_{E}x∥_{p} = 0.

When p = 1, the assumptions that W is convex and W symmetric are not needed, and the conditions in Lemma 1 are equivalent to the non weak compactness of the weak closure of W. This case is essentially proved in [KP] and proofs can also be found in books; see, e.g., [Wo, Theorem 3.C.12]). (Condition (3) does not appear in [Wo], but it is easy to check the equivalence of (3) and (4). Also, in the proof in [Wo, Theorem 3.C.12]) that not (4) implies not (1), Wojtaszczyk only constructs a basic sequence in W that is equivalent to the unit vector basis for ℓ_{1}; however, it is clear that the constructed basic sequence spans a complemented subspace.)

For p > 2, Lemma 1 and stronger versions of condition (1) can be deduced from [KP]. For 1 < p < 2, one needs to modify slightly the proof in [Wo] for the case p = 1. The only essential modification comes in the proof that not (4) implies not (1), and this is where it is needed that W is convex and symmetric. Just as in [Wo], one shows that not (4) implies that there is a sequence {x_{n}}_{n=1}^{∞} in W and a sequence {E_{n}}_{n=1}^{∞} of disjoint measurable sets so that inf ∥1_{E_{n}}x_{n}∥_{p} > 0. By passing to a subsequence, we can assume that {x_{n}}_{n=1}^{∞} converges weakly to, say, x. Suppose first that x = 0. Then by passing to a further subsequence, we may assume that {x_{n}}_{n=1}^{∞} is a small perturbation of a block basis of the Haar basis for L_{p} and hence is an unconditionally basic sequence. Since L_{p} has type p, this implies that there is a constant C so that for all sequences {a_{n}}_{n=1}^{∞} of scalars, ∥ ∑ a_{n}x_{n}∥_{p} ≤ C(∑ |a_{n}|^{p})^{1/p}. Let P be the norm one projection from L_{p} onto the closed linear span Y of the disjoint sequence {1_{E_{n}}x_{n}}_{n=1}^{∞}. Then P x_{n} is weakly null in a space isometric to ℓ_{p} and ∥P x_{n}∥_{p} is bounded away from zero, so there is a subsequence {P x_{n(k)}}_{k=1}^{∞} which is equivalent to the unit vector basis for ℓ_{p} and whose closed span is the range of a projection Q from Y. The projection QP from L_{p} onto the the closed span of {P x_{n(k)}}_{k=1}^{∞} maps x_{n(k)} to P x_{n(k)} and, because of the upper p estimate on {x_{n(k)}}_{k=1}^{∞}, maps the closed span of {x_{n(k)}}_{k=1}^{∞} isomorphically onto the closed span of {P x_{n(k)}}_{k=1}^{∞}. This yields that {x_{n(k)}}_{k=1}^{∞} is equivalent to the unit vector basis for ℓ_{p} and spans a complemented subspace. Suppose now that the weak limit x of {x_{n}}_{n=1}^{∞} is not zero. Choose a subsequence {x_{n(k)}}_{k=1}^{∞} so that inf ∥1_{E_{n(2k+1)}}(x_{n(2k)} − x_{n(2k+1)})∥_{p} > 0 and replace \{x_{n}\}_{n=1}^{∞} with \{\frac{x_{n(2k)}−x_{n(2k+1)}}{2} \}_{k=1}^{∞} in the argument above.

Notice that the argument outlined above gives that if {x_{n}}_{n=1}^{∞} is a sequence in L_{p}, 1 < p ≠ 2 < ∞, which is equivalent to the unit vector basis of ℓ_{p}, then there is a subsequence {y_{n}}_{n=1}^{∞} whose closed linear span in L_{p} is complemented. This is how one proves that the identity on ℓ_{p} factors through any operator on L_{p} which is not ℓ_{p}-strictly singular.

The Weak Tylli Conjecture for p > 2 is an easy consequence of the following lemma.

Lemma 2 Let T be an operator from a L_{1} space V into L_{p}, 1 < p < 2, so that W := T B_{V} satisfies condition (1) in Lemma 1. Then for each ε > 0 there is an operator S : V → L_{2} so that ∥T − I_{2,p}S∥ < ε.

Proof: Let ε > 0. By condition (3) in Lemma 1, for each norm one vector x in V there is a vector Ux in L_{2} with ∥Ux∥_{2} ≤ ∥Ux∥_{∞} ≤ M_{ε} and ∥T x − Ux∥_{p} ≤ ε. By the definition of L_{1} space, we can write V as a directed union ∪_{α}E_{α} of finite dimensional spaces that are uniformly isomorphic to ℓ_{1}^{n_{α}}, n_{α} = dim E_{α}, and let (x_{i}^{α})_{i=1}^{n_{α}} be norm one vectors in E_{α} which are, say, λ-equivalent to the unit vector basis for ℓ_{1}^{n_{α}} with λ independent of α. Let U_{α} be the linear extension to E_{α} of the mapping x_{i}^{α} ↦ Ux_{i}^{α}, considered as an operator into L_{2}. Then ∥T_{|E_{α}} − I_{2,p}U_{α}∥ ≤ λε and ∥U_{α}∥ ≤ λM_{ε}. A standard Lindenstrauss compactness argument produces an operator S : V → L_{2} so that ∥S∥ ≤ λM_{ε} and ∥T − I_{2,p}S∥ ≤ λε. Indeed, extend U_{α} to all of V by letting U_{α}x = 0 if x ∉ E_{α}. The net T_{α} has a subnet S_{β} so that for each x in V, S_{β}x converges weakly in L_{2}; call the limit Sx. It is easy to check that S has the properties claimed.

Theorem 3 Let T be an ℓ_{p}-strictly singular operator on L_{p}, 2 < p < ∞, and let J be an isometric embedding of L_{p} into an injective Z. Then for each ε > 0 there is an operator S : L_{p} → Z so that S factors through ℓ_{2} and ∥JT − S∥ < ε.

Proof: Lemma 2 gives the conclusion when J is the adjoint of a quotient mapping from ℓ_{1} or L_{1} onto L_{p′}. The general case then follows from the injectivity of Z.

The next proposition, when souped-up via “abstract nonsense” and known results, gives our main result about ℓ_{p}-strictly singular operators on L_{p}. Note that it shows that an ℓ_{p}-strictly singular operator on L_{p}, 1 < p < 2, cannot be the identity on the span of a sequence of r-stable independent random variables for any p < r < 2. We do not know another way of proving even this special case of our main result.

Proposition 1 Let T be an ℓ_{p}-strictly singular operator on L_{p}, 1 < p < 2. If X is a subspace of L_{p} and T_{|X} = aI_{X} with a ≠ 0, then X embeds into L_{s} for all s < 2.

Proof: By making a change of density, we can by [JJ] assume that T is also a bounded linear operator on L_{2}, so assume, without loss of generality, that ∥T ∥_{p} ∨ ∥T ∥_{2} = 1, so that, in particular, a ≤ 1. Lemma 1 gives for each ϵ > 0 a constant M_{ϵ} so that

T B_{L_{p}} ⊂ ϵB_{L_{p}} + M_{ϵ}B_{L_{2}}.  (1)

Indeed, otherwise condition (1) in Lemma 1 gives a bounded sequence {x_{n}}_{n=1}^{∞} in L_{p} so that {T x_{n}}_{n=1}^{∞} is equivalent to the unit vector basis of ℓ_{p}. By passing to a subsequence of differences of {x_{n}}_{n=1}^{∞}, we can assume, without loss of generality, that {x_{n}}_{n=1}^{∞} is a small perturbation of a block basis of the Haar basis for L_{p} and hence is an unconditionally basic sequence. Since L_{p} has type p, the sequence {x_{n}}_{n=1}^{∞} has an upper p estimate, which means that there is a constant C so that for all sequences {a_{n}}_{n=1}^{∞} of scalars, ∥ ∑ a_{n}x_{n}∥ ≤ C∥(∑ |a_{n}|^{p})^{1/p}∥. Since {T x_{n}}_{n=1}^{∞} is equivalent to the unit vector basis of ℓ_{p}, {x_{n}}_{n=1}^{∞} also has a lower p estimate and hence {x_{n}}_{n=1}^{∞} is equivalent to the unit vector basis of ℓ_{p}. This contradicts the ℓ_{p} strict singularity of T.

Iterating this we get for every n and 0 < ϵ < 1/2

a^{n}B_{X} ⊂ T^{n}B_{L_{p}} ⊂ ϵ^{n}B_{L_{p}} + 2M_{ϵ}B_{L_{2}}

or, setting A := 1/a,

B_{X} ⊂ A^{n}ϵ^{n}B_{L_{p}} + 2A^{n}M_{ϵ}B_{L_{2}}.

For f a unit vector in X write f = f_{n} + g_{n} with ∥f_{n}∥_{2} ≤ 2A^{n}M_{ϵ} and ∥g_{n}∥_{p} ≤ (Aϵ)^{n}. Then f_{n+1} − f_{n} = g_{n} − g_{n+1}, and since evidently f_{n} can be chosen to be of the form (f ∨ −k_{n}) ∧ k_{n} (with appropriate interpretation when the set [f_{n} = ±k_{n}] has positive measure), the choice of f_{n}, g_{n} can be made so that

∥f_{n+1} − f_{n}∥_{2} ≤ ∥f_{n+1}∥_{2} ≤ 2M_{ϵ}A^{n+1}
∥g_{n} − g_{n+1}∥_{p} ≤ ∥g_{n}∥_{p} ≤ (Aϵ)^{n}.

(Alternatively, to avoid thinking, just take any f = f_{n} + g_{n} so that ∥f_{n}∥_{2} ≤ 2A^{n}M_{ϵ} and ∥g_{n}∥_{p} ≤ (Aϵ)^{n}. Each left side of the two displayed inequalities is less than twice the corresponding right side as long as Aε ≤ 1.)

For p < s < 2 write \frac{1}{s} = \frac{θ}{2} + \frac{1−θ}{p}. Then

∥f_{n+1} − f_{n}∥_{s} ≤ ∥f_{n+1} − f_{n}∥_{2}^{θ}∥g_{n} − g_{n+1}∥_{p}^{1−θ} ≤ (2M_{ϵ}A)^{θ}(Aϵ^{1−θ})^{n}

which is summable if ϵ^{1−θ} < 1/A. But ∥f − f_{n}∥_{p} → 0 so f = f_{1} + ∑_{n=1}^{∞} f_{n+1} − f_{n} in L_{p} and hence also in L_{s} if ϵ^{1−θ} < 1/A. So for some constant C_{s} we get for all f ∈ X that ∥f∥_{p} ≤ ∥f∥_{s} ≤ C_{s}∥f∥_{p}.

We can now prove our main theorem. For background on ultrapowers of Banach spaces, see [DJT, Chapter 8].

Theorem 4 Let T be an ℓ_{p}-strictly singular operator on L_{p}, 1 < p < 2. If X is a subspace of L_{p} and T_{|X} is an isomorphism, then X embeds into L_{r} for all r < 2.

Proof: In view of Rosenthal’s theorem [Ro3], it is enough to prove that X has type s for all s < 2. By virtue of of the Krivine-Maurey-Pisier theorem, [Kr] and [MP] (or, alternatively, Aldous’ theorem, [Al] or [KM]), we only need to check that for p < s < 2, X does not contain almost isometric copies of ℓ_{s}^{n} for all n. (To apply the Krivine-Maurey-Pisier theorem we use that the second condition in Lemma 1, applied to the unit ball of X, yields that X has type s for some p < s ≤ 2). So suppose that for some p < s < 2, X contains almost isometric copies of ℓ_{s}^{n} for all n. By applying Krivine’s theorem [Kr] we get for each n a sequence (f_{i}^{n})_{i=1}^{n} of unit vectors in X which is 1 + ϵ-equivalent to the unit vector basis for ℓ_{s}^{n} and, for some constant C (which we can take independently of n), the sequence (CT f_{i}^{n})_{i=1}^{n} is also 1 + ϵ-equivalent to the unit vector basis for ℓ_{s}^{n}. By replacing T by CT, we might as well assume that C = 1. Now consider an ultrapower T_{U}, where U is a free ultrafilter on the natural numbers. The domain and codomain of T_{U} is the (abstract) L_{p} space (L_{p})_{U}, and T_{U} is defined by T_{U}(f_{1}, f_{2}, …) = (T f_{1}, T f_{2}, …) for any (equivalence class of a) bounded sequence (f_{1}, f_{2}, …). It is evident that T_{U} is an isometry on the ultraproduct of span (f_{i}^{n})_{i=1}^{n}; n = 1, 2, …, and hence T_{U} is an isometry on a subspace of (L_{p})_{U} which is isometric to ℓ_{s}. Since condition 2 in Lemma 1 is obviously preserved when taking an ultrapower of a set, we see that T_{U} is ℓ_{p}-strictly singular. Finally, by restricting T_{U} to a suitable subspace, we get an ℓ_{p}-strictly singular operator S on L_{p} and a subspace Y of L_{p} so that Y is isometric to ℓ_{s} and S_{|Y} is an isometry. By restricting the domain of S, we can assume that Y has full support and the functions in Y generate the Borel sets. It then follows from the Plotkin-Rudin theorem [Pl], [Ru] (see [KK, Theorem 1]) that S_{|Y} extends to an isometry W from L_{p} into L_{p}. Since any isometric copy of L_{p} in L_{p} is norm one complemented (see [La, §17]), there is a norm one operator V : L_{p} → L_{p} so that V W = I_{L_{p}}. Then V S_{|Y} = I_{Y} and V S is ℓ_{p}-strictly singular, which contradicts Proposition 1.

Remark 1 The ℓ_{1}-strictly singular operators on L_{1} also form an interesting class. They are the weakly compact operators on L_{1}. In terms of factorization, they are just the closure in the operator norm of the integral operators on L_{1} (see, e.g., the proof of Lemma 2).

4 The example

Rosenthal [Ro1] proved that if {x_{n}}_{n=1}^{∞} is a sequence of three valued, symmetric, independent random variables, then for all 1 < p < ∞, the closed span in L_{p} of {x_{n}}_{n=1}^{∞} is complemented by means of the orthogonal projection P, and ∥P ∥_{p} depends only on p, not on the specific sequence {x_{n}}_{n=1}^{∞}. Moreover, he showed that if p > 2, then for any sequence {x_{n}}_{n=1}^{∞} of symmetric, independent random variables in L_{p}, ∥ ∑ x_{n}∥_{p} is equivalent (with constant depending only on p) to (∑ ∥x_{n}∥_{p}^{p})^{1/p} ∨ (∑ ∥x_{n}∥_{2}^{2})^{1/2}. Thus if {x_{n}}_{n=1}^{∞} is normalized in L_{p}, p > 2, and w_{n} := ∥x_{n}∥_{2}, then ∥ ∑ a_{n}x_{n}∥_{p} is equivalent to ∥{a_{n}}_{n=1}^{∞}∥_{p,w} := (∑ |a_{n}|^{p})^{1/p} ∨ (∑ |a_{n}|^{2}w_{n}^{2})^{1/2}. The completion of the finitely non zero sequences of scalars under the norm ∥ · ∥_{p,w} is called X_{p,w}. It follows that if w = {w_{n}}_{n=1}^{∞} is any sequence of numbers in [0, 1], then X_{p,w} is isomorphic to a complemented subspace of L_{p}. Suppose now that w = {w_{n}}_{n=1}^{∞} and v = {v_{n}}_{n=1}^{∞} are two such sequences of weights and v_{n} ≥ w_{n}, then the diagonal operator D from X_{p,w} to X_{p,v} that sends the nth unit vector basis vector e_{n} to \frac{w_{n}}{v_{n}} e_{n} is contractive and it is more or less obvious that D is ℓ_{p}-strictly singular if \frac{w_{n}}{v_{n}} → 0 as n → ∞. Since X_{p,w} and X_{p,v} are isomorphic to complemented subspaces of L_{p}, the adjoint operator D^{∗} is ℓ_{p′} strictly singular and (identifying X_{p,w}^{∗} and X_{p,v}^{∗} with subspaces of L_{p′}) extends to a ℓ_{p′} strictly singular operator on L_{p′}. Our goal in this section is produce weights w and v so that D^{∗} is an isomorphism on a subspace of X_{p,v}^{∗} which is not isomorphic to a Hilbert space.

For all 0 < r < 2 there is a positive constant c_{r} such that

|t|^{r} = c_{r} ∫_{0}^{∞} \frac{1 − cos tx}{x^{r+1}}dx

for all t ∈ ℝ. It follows that for any closed interval [a, b] ⊂ (0, ∞) and for all ε > 0 there are 0 < x_{1} < x_{2} < ⋯ < x_{n+1} such that max_{1≤j≤n} |\frac{x_{j+1}−x_{j}}{x_{j}^{r+1}} | ≤ ε and

|c_{r} ∑_{j=1}^{n} \frac{x_{j+1} − x_{j}}{x_{j}^{r+1}} (1 − cos tx_{j}) − |t|^{r}| < ε  (2)

for all t with |t| ∈ [a, b].

Let 0 < q < r < 2 and define v_{j} and a_{j}, j = 1, …, n, by

vj^{\frac{2q}{2−q}} = c_{r} \frac{x_{j+1} − x_{j}}{x_{j}^{r+1}} , \frac{a_{j}}{vj^{\frac{2}{2−q}}} = x_{j}.

Let Y_{j}, j = 1, …, n, be independent, symmetric, three valued random variables such

that |Y_{j}| = vj^{\frac{−2}{2−q}} 1_{B_{j}} with Prob(B_{j}) = vj^{\frac{2q}{2−q}}, so that in particular ∥Y_{j}∥_{q} = 1 and v_{j} = ∥Y_{j}∥_{q}/∥Y_{j}∥_{2}. Then the characteristic function of Y_{j} is

φ_{Y_{j}}(t) = 1 − vj^{\frac{2q}{2−q}} + vj^{\frac{2q}{2−q}} cos(tvj^{\frac{−2}{2−q}}) = 1 − vj^{\frac{2q}{2−q}} (1 − cos(tvj^{\frac{−2}{2−q}}))

and

\begin{matrix}φ_{∑ a_{j}Y_{j}}(t) = & ∏_{j=1}^{n}(1 − vj^{\frac{2q}{2−q}} (1 − cos(ta_{j}vj^{\frac{−2}{2−q}}))) \\ = & ∏_{j=1}^{n}(1 − c_{r} \frac{x_{j+1}−x_{j}}{x_{j}^{r+1}} (1 − cos(tx_{j}))) \end{matrix}  (3)

To evaluate this product we use the estimates on \frac{x_{j+1}−x_{j}}{x_{j}^{r+1}} to deduce that, for each j

| log(1 − c_{r} \frac{x_{j+1}−x_{j}}{x_{j}^{r+1}} (1 − cos(tx_{j})))+ c_{r} \frac{x_{j+1}−x_{j}}{x_{j}^{r+1}} (1 − cos(tx_{j}))|
≤ Cεc_{r}^{2} \frac{x_{j+1}−x_{j}}{x_{j}^{r+1}} (1 − cos(tx_{j}))

for some absolute C < ∞. Then, by (2),

| ∑_{j=1}^{n} log(1 − c_{r} \frac{x_{j+1}−x_{j}}{x_{j}^{r+1}} (1 − cos(tx_{j})))+ c_{r} ∑_{j=1}^{n} \frac{x_{j+1}−x_{j}}{x_{j}^{r+1}} (1 − cos(tx_{j}))|
≤ Cεc_{r}(ε + b^{r}).

Using (2) again we get

| ∑_{j=1}^{n} log(1 − c_{r} \frac{x_{j+1} − x_{j}}{x_{j}^{r+1}} (1 − cos(tx_{j}))) + |t|^{r}| ≤ (C + 1)ε(ε + b^{r})

(assuming as we may that b ≥ 1), and from (3) we get

φ_{∑ a_{j}Y_{j}}(t) = (1 + O(ε)) exp(−|t|^{r})

for all |t| ∈ [a, b], where the function hiding under the O notation depends on r and b but on nothing else. It follows that, given any η > 0, one can find a, b and ε, such that for the corresponding {a_{j}, Y_{j}} there is a symmetric r-stable Y (with characteristic function e^{−|t|^{r}}) satisfying

∥Y − ∑_{j=1}^{n} a_{j}Y_{j}∥_{q} ≤ η.

This follows from classical translation of various convergence notions; see e.g. [Ro2, p. 154].

Let now 0 < δ < 1. Put w_{j} = δv_{j}, j = 1, …, n, and let Z_{j}, j = 1, …, n, be

independent, symmetric, three valued random variables such that |Z_{j}| = wj^{\frac{−2}{2−q}} 1_{C_{j}} with

Prob(C_{j}) = wj^{\frac{2q}{2−q}}, so that in particular ∥Z_{j}∥_{q} = 1 and w_{j} = ∥Z_{j}∥_{q}/∥Z_{j}∥_{2}. In a similar manner to the argument above we get that,

φ_{∑ δa_{j}Z_{j}}(t) = ∏_{j=1}^{n}(1 − wj^{\frac{2q}{2−q}} (1 − cos(tδa_{j}wj^{\frac{−2}{2−q}})))
= ∏_{j=1}^{n}(1 − δ^{\frac{2q}{2−q}} vj^{\frac{2q}{2−q}} (1 − cos(tδ^{\frac{−q}{2−q}} a_{j}vj^{\frac{−2}{2−q}})))
= (1 + O(ε)) exp(−δ^{\frac{q(2−r)}{2−q}} |t|^{r})

for all |t| ∈ [δ^{\frac{q}{2−q}} a, δ^{\frac{q}{2−q}} b], where the O now depends also on δ.

Assuming δ^{\frac{q(2−r)}{2−q}} > 1/2 and for a choice of a, b and ε, depending on δ, r, q and η we get that there is a symmetric r-stable random variable Z (with characteristic function

e^{−δ^{\frac{q(2−r)}{2−q}} |t|^{r}}) such that

∥Z − ∑_{j=1}^{n} δa_{j}Z_{j}∥_{q} ≤ η.

Note that the ratio between the L_{q} norms of Y and Z are bounded away from zero and infinity by universal constants and each of these norms is also universally bounded away from zero. Consequently, if ε is small enough the ratio between the L_{q} norms of ∑_{j=1}^{n} a_{j}Y_{j} and ∑_{j=1}^{n} δa_{j}Z_{j} are bounded away from zero and infinity by universal constants.

Let now δ_{i} be any sequence decreasing to zero and r_{i} any sequence such that q < r_{i} ↑ 2

and satisfying δi^{\frac{q(2−r_{i})}{2−q}} > 1/2. Then, for any sequence ε_{i} ↓ 0 we can find two sequences of symmetric, independent, three valued random variables {Y_{i}} and {W_{i}}, all normalized in L_{q}, with the following additional properties:

• put v_{j} = ∥Y_{j}∥_{q}/∥Y_{j}∥_{2} and w_{j} = ∥Z_{j}∥_{q}/∥Z_{j}∥_{2}. Then there are disjoint finite subsets of the integers σ_{i}, i = 1, 2, …, such that w_{j} = δ_{i}v_{j} for j ∈ σ_{i}.

• There are independent random variables {Ȳ_{i}} and {Z̄_{i}} with Ȳ_{i} and Z̄_{i} r_{i} stable with bounded, from zero and infinity, ratio of L_{q} norms and there are coefficients {a_{j}} such that

∥Ȳ_{i} − ∑_{j∈σ_{i}} a_{j}Y_{j}∥_{q} < ε_{i} and ∥Z̄_{i} − ∑_{j∈σ_{i}} δ_{i}a_{j}Z_{j}∥_{q} < ε_{i}.

From [Ro1] we know that the spans of {Y_{j}} and {Z_{j}} are complemented in L_{q}, 1 < q < 2, and the dual spaces are naturally isomorphic to the spaces X_{p,{v_{j}}} and X_{p,{w_{j}}} respectively; both the isomorphism constants and the complementation constants depend only on q. Here p = q/(q − 1) and

∥{α_{j}}∥_{X_{p,{u_{j} }}} = max{(∑ |α_{j}|^{p})^{1/p}, (∑ u_{j}^{2}α_{j}^{2})^{1/2}}.

Under this duality the adjoint D^{∗} to the operator D that sends Y_{j} to δ_{i}Z_{j} for j ∈ σ_{i} is formally the same diagonal operator between X_{p,{w_{i}}} and X_{p,{v_{i}}}. The relation w_{j} = δ_{i}v_{j} for j ∈ σ_{i} easily implies that this is a bounded operator. δ_{i} → 0 implies that this operator is ℓ_{q} strictly singular. If ε_{i} → 0 fast enough, D^{∗} preserves a copy of span{Ȳ_{i}}. Finally, if r_{i} tend to 2 not too fast this span is not isomorphic to a Hilbert space. Indeed, let 1 ≤ s_{j} ↑ 2 be arbitrary and let {n_{j}}_{j=1}^{∞} be a sequence of positive integers with

n_{j}^{\frac{1}{s_{j}} − \frac{1}{2}} ≥ j, j = 1, 2, …, say. For 1 ≤ k ≤ n_{j}, put r_{n_{1}+n_{2}+···+n_{j−1}+k} = s_{j}. Then the span of {Y_{i}}_{i=n_{1}+···+n_{j−1}+1}^{n_{1}+···+n_{j}} is isomorphic, with constant independent of j, to ℓ_{s_{j}}^{n_{j}} and this last space is of distance at least j from a Euclidean space.

It follows that if J : L_{q} → ℓ_{∞} is an isometric embedding, then JD^{∗} cannot be arbitrarily approximated by an operator which factors through a Hilbert space, and hence the Weak Tylli Conjecture is false in the range 1 < q < 2.

5 Convolution by a biased coin

In this section we regard L_{p} as L_{p}(Δ), where Δ = {−1, 1}^{ℕ} is the Cantor group and the measure is the Haar measure μ on Δ; i.e., μ = ∏_{n=1}^{∞} μ_{n}, where μ_{n}(−1) = μ_{n}(1) = 1/2. For 0 < ε < 1, let ν_{ε} be the ε- biased coin tossing measure; i.e., ν_{ε} = ∏_{n=1}^{∞} ν_{ε,n}, where ν_{ε,n}(1) = \frac{1+ε}{2} and ν_{ε,n}(−1) = \frac{1−ε}{2}. Let T_{ε} be convolution by ν_{ε}, so that for a μ-integrable function f on Δ, (T_{ε}f)(x) = (f ∗ ν_{ε})(x) = ∫_{Δ} f(xy) dν_{ε}(y). The operator T_{ε} is a contraction on L_{p} for all 1 ≤ p ≤ ∞. Let us recall how T_{ε} acts on the characters on Δ. For t = {t_{n}}_{n=1}^{∞} ∈ Δ, let r_{n}(t) = t_{n}. The characters on Δ are finite products of these Rademacher functions r_{n} (where the void product is the constant one function). For A a finite subset of ℕ, set w_{A} = ∏_{n∈A} r_{n} and let W_{n} be the linear span of {w_{A} : |A| = n}. Then T_{ε}w_{A} = ε^{|A|}w_{A}.

We are interested in studying T_{ε} on L_{p}, 1 ≤ p < 2. The background we mention below is all contained in Bonami’s paper [Bo] (or see [Ro4]). On L_{p}, 1 < p < 2, T_{ε} is ℓ_{p}-strictly singular; in fact, T_{ε} even maps L_{p} into L_{r} for some r = r(p, ε) > p. Indeed, by interpolation it is sufficient to check that T_{ε} maps L_{s} into L_{2} for some s = s(ε) < 2. But there is a constant C_{s} which tends to 1 as s ↑ 2 so that for all f ∈ W_{n}, ∥f∥_{2} ≤ C_{s}^{n}∥f∥_{s} and the orthogonal projection P_{n} onto (the closure of) W_{n} satisfies ∥P_{n}∥_{p} ≤ C_{s}^{n}. From this it is easy to check that if εC_{s}^{2} < 1, then T_{ε} maps L_{s} into L_{2}. We remark in passing that Bonami [Bo] found for each p (including p ≥ 2) and ε the largest value of r = r(p, ε) such that T_{ε} maps L_{p} into L_{r}.

Thus Theorem 4 yields that if X is a subspace of L_{p}, 1 < p < 2, and T_{ε} (considered as an operator from L_{p} to L_{p}) is an isomorphism on X, then X embeds into L_{s} for all s < 2. Since, as we mentioned above, T_{ε} maps L_{s} into L_{2} for some s < 2, it then follows from an argument in [Ro4] that X must be isomorphic to a Hilbert space. (Actually, as we show after the proof, Lemma 3 is strong enough that we can prove Theorem 5 without using Theorem 4.) Since [Ro4] is not generally available, we repeat Rosenthal’s argument in Lemma 3 below.

Now T_{ε} is not ℓ_{1} strictly singular on L_{1}. Nevertheless, we still get that if X is a reflexive subspace of L_{1} and T_{ε} (considered as an operator from L_{1} to L_{1}) is an isomorphism on X, then X is isomorphic to a Hilbert space. Indeed, Rosenthal showed (see Lemma 3) that then there is another subspace X_{0} of L_{1} which is isomorphic to X so that X_{0} is contained in L_{p} for some 1 < p < 2, the L_{p} and L_{1} norms are equivalent on X_{0}, and T_{ε} is an isomorphism on X_{0}. This implies that as an operator on L_{p}, T_{ε} is an isomorphism on X_{0} and hence X_{0} is isomorphic to a Hilbert space. (To apply Lemma 3, use the fact [Ro3] that if X is a relexive subspace of L_{1}, then X embeds into L_{p} for some 1 < p < 2.)

We summarize this discussion in the first sentence of Theorem 5. The case p = 1 solves Problem B from Rosenthal’s 1976 paper [Ro4].

Theorem 5 Let 1 ≤ p < 2, let 0 < ε < 1, and let T_{ε} be considered as an operator on L_{p}. If X is a reflexive subspace of L_{p} and the restriction of T_{ε} to X is an isomorphism, then X is isomorphic to a Hilbert space. Moreover, if p > 1, then X is complemented in L_{p}.

We now prove Rosenthal’s lemma [Ro4, proof of Theorem 5] and defer the proof of the “moreover” statement in Theorem 5 until after the proof of the lemma. .

Lemma 3 Suppose that T is an operator on L_{p}, 1 ≤ p < r < s < 2, X is a subspace of L_{p} which is isomorphic to a subspace of L_{s}, and T_{|X} is an isomorphism. Then there is another subspace X_{0} of L_{p} which is isomorphic to X so that X_{0} is contained in L_{r}, the L_{r} and L_{p} norms are equivalent on X_{0}, and T is an isomorphism on X_{0}.

Proof: We want to find a measurable set E so that

(1) X_{0} := {1_{E}x : x ∈ X} is isomorphic to X,

(2) X_{0} ⊂ L_{r},

(3) T_{|X_{0}} is an isomorphism.

(We did not say that ∥ · ∥_{p} and ∥ · ∥_{r}, are equivalent on X_{0} since that follows formally from the closed graph theorem. The isomorphism X → X_{0} guaranteed by (a) is of course the mapping x ↦ 1_{E}x.)

Assume, without loss of generality, that ∥T ∥ = 1. Take a > 0 so that ∥T x∥_{p} ≥ a∥x∥_{p} for all x in X. Since ℓ_{p} does not embed into L_{s} we get from (4) in Lemma 1 that there is η > 0 so that if E has measure larger than 1 − η, then ∥1_{∼E}x∥_{p} ≤ \frac{a}{2}∥x∥_{p} for all x in x. Obviously (1) and (3) are satisfied for any such E. It is proved in [Ro3] that there is strictly positive g with ∥g∥_{1} = 1 so that \frac{x}{g} is in L_{r} for all x in X. Now simply choose t < ∞ so that E := [g < t] has measure at least 1 − η; then E satisfies (1), (2), and (3).

Next we remark how to avoid using Theorem 4 in proving Theorem 5. Suppose that T_{ε} is an isomorphism on a reflexive subspace X of L_{p}, 1 ≤ p < 2. Let s be the supremum of those r ≤ 2 such that X is isomorphic to a subspace of L_{r}, so 1 < s ≤ 2. It is sufficient to show that s = 2. But if s < 2, we get from the interpolation formula that if r < s is sufficiently close to s, then T_{ε} maps L_{r} into L_{t} for some t > s and hence, by Lemma 3, X embeds into L_{t}.

Finally we prove the “moreover” statement in Theorem 5. We now know that X is isomorphic to a Hilbert space. In the proof of Lemma 3, instead of using Rosenthal’s result from [Ro3], use Grothendieck’s theorem [DJT, Theorem 3.5], which implies that there is strictly positive g with ∥g∥_{1} = 1 so that \frac{x}{g} is in L_{2} for all x in X. Choosing E the same way as in the proof of Lemma 3 with T := T_{ε}, we get that (1), (2), and (3) are true with r = 2. Now the L_{2} and L_{p} norms are equivalent on both X_{0} and on T_{ε}X_{0}. But it is clear that the only way that T_{ε} can be an isomorphism on a subspace X_{0} of L_{2} is for the orthogonal projection P_{n} onto the closed span of W_{k}, 0 ≤ k ≤ n, to be an isomorphism on X_{0} for some finite n. But then also in the L_{p} norm the restriction of P_{n} to X_{0} is an isomorphism because the L_{p} norm and the L_{2} norm are equivalent on the span of W_{k}, 0 ≤ k ≤ n and P_{n} is bounded on L_{p} (since p > 1). It follows that the operator S := P_{n} ∘ 1_{E} on L_{p} maps X_{0} isomorphically onto a complemented subspace of L_{p}, which implies that X_{0} is also complemented in L_{p}.

Here is the problem that started us thinking about ℓ_{p}-strictly singular operators:

Problem 1 Let 1 < p < 2 and 0 < ε < 1. On L_{p}(Δ), does T_{ε} satisfy the conclusion of the Tylli Conjecture?

After we submitted this paper, G. Pisier [Pi] answered problem 1 in the affirmative. Although the example in section 4 shows that the Tylli Conjecture is false, something

close to it may be true:

Problem 2 Let 1 < p < r < 2. Is every ℓ_{p}-strictly singular operator on L_{p} in the closure of the operators on L_{p} that factor through L_{r}?

6 Appendix

In this appendix we prove a theorem that is essentially due to Saksman and Tylli. The only novelty is that we assume the compact approximation property rather than the approximation property.

Theorem 6 Let X be a reflexive Banach space and let A, B be in L(X). Then

(a) If AT B is a compact operator on X for every T in L(X), then L_{A}R_{B} is a weakly compact operator on L(X).

(b) If X has the compact approximation property and L_{A}R_{B} is a weakly compact operator on L(X), then AT B is a compact operator on X for every T in L(X).

Proof: To prove (a), recall [Kal] that for a reflexive space X, on bounded subsets of K(X) the weak topology is the same as the weak operator topology (the operator T ↦ f_{T} ∈ C((B_{X}, weak) × (B_{X^{∗}}, weak)), where f_{T} (x, x^{∗}) := ⟨x^{∗}, T x⟩, is an isometric isomorphism from K(X) into a space of continuous functions on a compact Hausdorff space). Now if (T_{α}) is a bounded net in L(X), then since X is reflexive there is a subnet (which we still denote by (T_{α})) which converges in the weak operator topology to, say T ∈ L(X). Then AT_{α}B converges in the the weak operator topology to AT B. But since all these operators are in K(X), AT_{α}B converges weakly to AT B by Kalton’s theorem. This shows that L_{A}R_{B} is a weakly compact operator on L(X).

To prove (b), suppose that we have a T ∈ L(X) with AT B not compact. Then there is a weakly null normalized sequence {x_{n}}_{n=1}^{∞} in X and δ > 0 so that for all n, ∥AT Bx_{n}∥ > δ. Since a reflexive space with the compact approximation property also has the compact metric approximation property [CJ], there are C_{n} ∈ K(X) with ∥C_{n}∥ < 1 + 1/n, C_{n}Bx_{i} = Bx_{i} for i ≤ n. Since the C_{n} are compact, for each n, ∥C_{n}Bx_{m}∥ → 0 as m → ∞. Thus A(T C_{n})Bx_{i} = AT Bx_{i} for i ≤ n and ∥A(T C_{n})Bx_{m}∥ → 0 as m → ∞. This implies that no convex combination of {A(T C_{n})B}_{n=1}^{∞} can converge in the norm of L(X) and hence {A(T C_{n})B}_{n=1}^{∞} has no weakly convergent subsequence. This contradicts the weak compactness of L_{A}R_{B} and completes the proof.

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William B. Johnson Department Mathematics Texas A&M University College Station, TX, USA E-mail: johnson@math.tamu.edu

Gideon Schechtman Department of Mathematics Weizmann Institute of Science Rehovot, Israel E-mail: gideon@weizmann.ac.il