UNIVERSAL NON-COMPLETELY-CONTINUOUS OPERATORS

Maria Girardi*^{‡} and William B. Johnson^{∗}

Israel Journal of Mathematics 99 (1997) 207–219

Abstract. A bounded linear operator between Banach spaces is called completely continuous if it carries weakly convergent sequences into norm convergent sequences. Isolated is a universal operator for the class of non-completely-continuous operators from L_{1} into an arbitrary Banach space, namely, the operator from L_{1} into ℓ_{∞} defined by

T_{0}(f) = (∫ r_{n}f dμ)_{n≥0} ,

where r_{n} is the n^{th} Rademacher function. It is also shown that there does not exist a universal operator for the class of non-completely-continuous operators between two arbitrary Banach space. The proof uses the factorization theorem for weakly compact operators and a Tsirelson-like space.

Suppose that ℭ is a class of (always bounded, linear, between Banach spaces) operators so that an operator S is in ℭ whenever the domain of S is the domain of some operator in ℭ and there exist operators A, B so that BSA is in ℭ; the natural examples of such classes are all the operators that do not belong to a given operator ideal. A subset 𝔖 of such a class ℭ is said to be universal for ℭ provided for each U in ℭ, some member of 𝔖 factors through U; that is, there exist operators A and B so that BUA is in 𝔖. In case 𝔖 is singleton; say, 𝔖 = {S}; we say that S is universal for ℭ.

In order to study a class ℭ of operators, it is natural to try to find a universal subclass of ℭ consisting of specific, simple operators. For certain classes, such a subclass is known to exist. For example, Lindenstrauss and Pełczyński, who introduced the concept of universal operator, proved [LP] that the “summing operator” from ℓ_{1} to ℓ_{∞}, defined by {a_{n}}_{n=1}^{∞} ↦ {∑_{k=1}^{n} a_{k}}_{n=1}^{∞}, is universal for the class of non-weakly-compact operators; while in [J] it was pointed out that the formal identity from ℓ_{1} to ℓ_{∞} is universal for the class of non-compact operators.

Typeset by AMS-TEX

1991 Mathematics Subject Classification. 47B99, 47A68, 47D50, 47B38, 46B20, 46B28, 46B07, 46B22.

*Supported in part by NSF grants DMS-9306460 and DMS-9003550.

‡Participant, NSF Workshop in Linear Analysis & Probability, Texas A&M University.

An operator between Banach spaces is called completely continuous if it carries weakly convergent sequences into norm convergent sequences. The operator from L_{1} into ℓ_{∞} given by

T_{0}(f) = {∫ r_{n}f dμ}_{n=0}^{∞} ,

where r_{n} is the n^{th} Rademacher function, is not completely continuous. We prove in Corollary 4 that T_{0} is universal for the class of non-completely-continuous operators from an L_{1}-space; however, in Theorem 5 we show that there does not exist a universal non-completely-continuous operator.

Throughout this paper, 𝔛 denotes an arbitrary Banach space, 𝔛^{∗} the dual space of 𝔛, and S(𝔛) the unit sphere of 𝔛. The triple (Ω, Σ, μ) refers to the Lebesgue measure space on [0, 1], Σ^{+} to the sets in Σ with positive measure, and L_{1} to L_{1}(Ω, Σ, μ). All notation and terminology, not otherwise explained, are as in [DU] or [LT].

To crystalize the ideas in Theorem 1, we introduce some terminology. A system A = {A_{k}^{n} ∈ Σ: n = 0, 1, 2, … and k = 1, …, 2^{n}} is a dyadic splitting of A_{1}^{0} ∈ Σ^{+} if each A_{k}^{n} is partitioned into the two sets A_{2k−1}^{n+1} and A_{2k}^{n+1} of equal measure for each admissible n and k . Thus the collection π_{n} = {A_{k}^{n}: k = 1, …, 2^{n}} of sets along the n^{th}-level partition A_{1}^{0} with π_{n+1} refining π_{n} and μ(A_{k}^{n}) = 2^{−n}μ(A_{1}^{0}). To a dyadic splitting corresponds a (normalized) Haar system {h_{j}}_{j≥1} along with its natural blocking {H_{n}}_{n≥0} where

h_{1} = \frac{1}{μ(A_{1}^{0})} 1_{A_{1}^{0}}  and  h_{2^{n}+k} = \frac{2^{n}}{μ(A_{1}^{0})} (1_{A_{2k−1}^{n+1}} − 1_{A_{2k}^{n+1}})

for n = 0, 1, 2, …, k = 1, …, 2^{n}, and H_{n} = {h_{j}: 2^{n−1} < j ≤ 2^{n}}. The usual Haar system \{h̃_{j}\} corresponds to the usual dyadic splitting \{[\frac{k−1}{2^{n}}, \frac{k}{2^{n}} )\}_{n,k}. Let L_{1}(A) be the closed subspace of L_{1} with basis {h_{j}}_{j≥1}.

A set N in the unit sphere of the dual of a Banach space 𝔛 is said to norm a subspace 𝔛_{0} within τ > 1 if for each x ∈ 𝔛_{0} there is x^{∗} ∈ N such that ∥x∥ ≤ τx^{∗}(x). It is well known and easy to see that a sequence {𝔛_{j}}_{j⩾1} of subspaces of 𝔛 forms a finite dimensional decomposition with constant at most τ provided that for each n ∈ ℕ the space generated by {𝔛_{1}, …, 𝔛_{n}} can be normed by a set from S(𝔛_{n+1}^{⊥}) within τ_{n} > 1 where Πτ_{n} ≤ τ.

To help demystify Theorem 1, we examine more closely the operator T_{0} : L_{1} → ℓ_{∞} given above. This operator does more than just map the Rademacher functions {r_{n}} to the standard unit vectors {e_{n}} in ℓ_{∞} (which suffices to guarantee that it is not completely continuous). Let x_{n}^{∗} be the n^{th} unit vector of ℓ_{1}, viewed as an element in the dual of ℓ_{∞}. For the usual dyadic splitting of the unit interval, r_{n} is just the sum of the Haar functions in H_{n}, properly normalized. Thus 1 = ∥T_{0}r_{n}∥ = x_{n}^{∗}(T_{0}r_{n}) follows from the stronger condition that

x_{n}^{∗}(T_{0}h) = δ_{n,m} for each h ∈ H_{m} .

Note that T_{0}^{∗}x_{n}^{∗} is just r_{n}, which as a sequence in L_{1}^{∗} is weak*-null and equivalent to the unit vector basis of ℓ_{1}. Since T_{0} maps each element in H_{n} to e_{n}, the collection {sp T_{0}H_{n}} forms a finite dimensional decomposition. Theorem 1 states that each non-completely-continuous operator T on L_{1} behaves like the operator T_{0} in the sense that there is some dyadic splitting of some subset of [0, 1] so that the corresponding Haar system with T enjoys the above properties of the usual Haar system with T_{0}.

Theorem 1. Let Y be a subset of S(𝔛^{∗}) that norms 𝔛 within some fixed constant greater than one and let Y be a subspace of 𝔛^{∗} that contains Y. If the operator T : L_{1} → 𝔛 is not completely continuous and {τ_{n}}_{n≥0} is a sequence of numbers larger than 1, then there exist

(A) a dyadic splitting A = {A_{k}^{n}} (B) a sequence {x_{n}^{∗}}_{n≥0} in S(𝔛^{∗}) ∩ Y (C) a finite set {z_{n,i}^{∗}}_{i=1}^{p_{n}} in S(𝔛^{∗}) for each n ≥ 0

such that for the Haar system {h_{j}}_{j≥1} and the blocking {H_{n}}_{n≥0} corresponding to A, for some δ > 0, and each n, m ≥ 0,

(1) x_{n}^{∗}(T h) = δ · δ_{n,m} for each h ∈ H_{m}

(2) {T^{∗}x_{n}^{∗}} is weak*-null in L_{∞}

(3) {T^{∗}x_{n}^{∗}} is equivalent to the unit vector basis of ℓ_{1}

(4) {z_{n,i}^{∗}}_{i=1}^{p_{n}} norms sp(∪_{j=0}^{n}T H_{j}) within τ_{n}

(5) T H_{n+1} ⊂ ^{⊥}{z_{n,i}^{∗}}_{i=1}^{p_{n}} . Note that condition (3) implies that {x_{n}^{∗}} is also equivalent to the standard unit vector basis of ℓ_{1}. If Πτ_{n} is finite, then the last two conditions guarantee that {sp T H_{n}}_{n≥0} forms a finite dimensional decomposition with constant at most Πτ_{n}.

The proof uses the following two standard lemmas.

Lemma 2. Let E = sp {x_{i}}_{i=0}^{m} be a finite dimensional subspace of a Banach space 𝔛 and let Y be a total subspace of 𝔛^{∗}. For each ϵ > 0 there exists η > 0 such that if y^{∗} ∈ 𝔛^{∗} satisfies |y^{∗}(x_{i})| < η for each 1 ≤ i ≤ m, then there exists x^{∗} ∈ E^{⊥} of norm 0 or ∥y^{∗}∥ such that ∥x^{∗} − y^{∗}∥ < ϵ. Furthermore, if y^{∗} is in Y then x^{∗} can be taken to be in Y.

Proof of Lemma 2. Assume, without loss of generality, {x_{i}}_{i=0}^{m} is linearly independent. Consider the isomorphism l: E → ℓ_{1}^{m} that takes x_{i} to the i^{th} unit basis vector of ℓ_{1}^{m} and let P be a projection from 𝔛 onto E that is w(Y)-continuous, so that P^{∗}E^{∗} is a subspace of Y. Such a projection exists because Y is total. Then x̃^{∗} ≡ y^{∗} · (I_{𝔛} − P) is in E^{⊥} . It is easy to check that for η = \frac{ϵ}{3 ∥l∥ ∥P ∥}, if |y^{∗}(x_{i})| < η for each i, then ∥x̃^{∗} − y^{∗}∥ ≤ \frac{ϵ}{3}. If ∥x̃^{∗}∥ = 0, then let x^{∗} = x̃^{∗}. Otherwise, let x^{∗} = (∥y^{∗}∥ / ∥x̃^{∗}∥) x̃^{∗}. Then ∥x^{∗} − y^{∗}∥ ≤ 2∥x̃^{∗} − y^{∗}∥. Thus x^{∗} does what it should do.

Recall that the extreme points of B(L_{∞}) are just the ±1-valued measurable functions.

Lemma 3. If {f_{i}}_{i=0}^{n} is a finite subset of L_{1}, {α_{i}}_{i=0}^{n} are scalars, and

S = {g ∈ B(L_{∞}): ∫ f_{i}g dμ = α_{i} for each 0 ≤ i ≤ n} ,

then ext S = S ∩ ext B(L_{∞}), where ext denotes the extreme points of a set. Also, if S is non-empty then so is ext S.

Specifically, we use the following version of this extreme point argument lemma.

Lemma 3′. If F = {f_{1}, …, f_{n}} and there exists g in B(L_{∞}) ∩ F^{⊥} such that ∫ f_{0}g dμ ≥ α_{0} > 0, then there exists a ±1-valued function u in B(L_{∞}) ∩ F^{⊥} such that ∫ f_{0}u dμ = α_{0}.

Proof of Lemma 3. Consider, if there is one, a function g in S for which there exists a subset A of positive measure and ϵ > 0 such that −1 + ϵ < g1_{A} < 1 − ϵ. Since the set {f ∈ L_{∞}: | f | ≤ 1_{A}} ∩ {f_{i}}_{i=0}^{n} ^{⊥} is infinite dimensional, it contains a non-zero element h of norm less than ϵ. But then g ± h ∈ S and so g is not an extreme point of S. Thus ext S = S ∩ ext B(L_{∞}).

Since S convex and weak*-compact in L_{∞}, if S is non-empty then so is ext S. As for the last claim of the lemma, just note that if g ∈ B(L_{∞}) ∩ F^{⊥} satisfies

∫ f_{0}g dμ ≡ β ≥ α_{0} > 0, then \frac{α_{0}}{β} g is in the set S where α_{i} = 0 for i > 0. By the first part of the lemma, any extreme point u of S will do.

Although the proof of Theorem 1 is somewhat technical, the overall idea is simple. Since T is not completely continuous, we start by finding a weakly convergent sequence {g_{n}} in L_{1} and norm one functionals y_{n}^{∗} such that δ_{0} ≤ y_{n}^{∗} (T g_{n}). Each x_{n}^{∗} will be a small perturbation of some y_{j_{n}}^{∗}. Conditions (2) and (3) can be arranged by standard arguments.

Now the proof gets technical. We begin by finding a subset A_{1}^{0} where the L_{∞} function (T^{∗}y_{n}^{∗})g_{n}, which in the motivating example of T_{0} is the function r_{n}, is large in some sense. We then proceed by induction on the level n. Given a finite dyadic splitting up to n^{th}-level provides the subsets {H_{m}}_{m=0}^{n} of corresponding Haar functions. We need to split each A_{k}^{n} into 2 sets A_{2k−1}^{n+1} and A_{2k}^{n+1} (thus finding h_{2^{n}+k}) and find the desired functionals so that all works. It is easy to find the functionals to satisfy condition (4). In the search for x_{n+1}^{∗}, apply Lemma 2 to the set E given in (†) so that we need only to almost (within some η) satisfy (1-i′) for some y_{j}^{∗}; for then we can perturb y_{j}^{∗} to find x_{n+1}^{∗} that satisfies (1-i′) exactly. Next, for each A_{k}^{n}, apply Lemma 3′ with F as given in (‡) and f_{0} = T^{∗}y_{j}^{∗}1_{A_{k}^{n}} and g being a small perturbation of g_{j}1_{A_{k}^{n}}. All is set up so that such a perturbation exists for a j (dependent on n but independent of k) sufficiently large enough. Now Lemma 3′ gives that desired ±1-valued Haar-like function that yields the desired splitting of the (n+1)^{th}-level. The sets F_{k}^{n} are chosen exactly so that conditions (1-ii′), (1-iii′), and (5′) hold.

Proof of Theorem 1. Let T: L_{1} → 𝔛 be a norm one operator that is not completely continuous. Then there is a sequence {g_{n}} in L_{1} and a sequence {y_{n}^{∗}} in S(𝔛^{∗}) ∩ Y satisfying:

(a) ∥g_{n}∥_{L_{∞}} ≤ 1

(b) g_{n} is weakly null in L_{1}

(c) δ_{0} ≤ y_{n}^{∗} (T g_{n}) for some δ_{0} > 0 . Using (a), (b), and (c) along with Rosenthal’s ℓ_{1} theorem [cf. LT, Prop. 2.e.5], by passing to a further subsequence, we also have that

(d) {T^{∗}y_{n}^{∗}} is equivalent to the standard unit vectors basis of ℓ_{1}. Since B(L_{∞}) is weak* sequentially-compact in L_{∞}, by passing to a subsequence and considering differences we may assume that

(e) T^{∗}y_{n}^{∗} is weak*-null in L_{∞}, where (d) allows normalization of the new y_{n}^{∗}’s so as to keep them in S(𝔛^{∗}) and, used with care, (b) ensures that (c) still holds for some (new) positive δ_{0}. But {(T^{∗}y_{n}^{∗}) · g_{n}} is also in B(L_{∞}) and so, by passing to yet another subsequence, we have that

(f) {(T^{∗}y_{n}^{∗}) · g_{n}} → h weak* in L_{∞} for some h ∈ L_{∞}.

Since ∫ h dμ ≥ δ_{0}, the set A ≡ [h ≥ δ_{0}] has positive measure. We may assume, by replacing y_{n}^{∗} by −y_{n}^{∗} and g_{n} by −g_{n} when needed, that ∥T^{∗}y_{n}^{∗} |_{A} ∥_{L_{∞}} = ess sup T^{∗}y_{n}^{∗} |_{A}. So from (a) and (f) it follows that δ_{0} ⩽ lim inf ess sup T^{∗}y_{n}^{∗} |_{A} while from (e) it follows that lim sup μ[T^{∗}y_{n}^{∗} |_{A}⩾ δ_{0}−η] < μ(A) for each 0 < η < δ_{0}.

Thus, since the closure of the set

\{\frac{∫_{E} f dμ}{μ(E)} : E ⊂ A, E ∈ Σ^{+}\}

is the interval [ess inf f, ess sup f], there is a subset A_{1}^{0} of A with positive measure and j_{0} such that y_{j_{0}}^{∗}T (1_{A_{1}^{0}}) = δμ(A_{1}^{0}) for some positive δ less than δ_{0}, say δ ≡ δ_{0}−3ϵ. Put x_{0}^{∗} = y_{j_{0}}^{∗} and H_{0} ≡ {h_{1}} = {1_{A_{1}^{0}}/μ(A_{1}^{0})}.

We shall construct, by induction on the level n, a dyadic splitting of A_{1}^{0} along with the desired functionals. Towards this, take a decreasing sequence {ϵ_{n}}_{n≥0} of positive numbers such that ϵ_{0} < ϵ and ∑ ϵ_{n} < \frac{δ_{0}}{2K} where K is the basis constant of {T^{∗}y_{n}^{∗}}. The sequence {x_{n}^{∗}} will be chosen such that ∥x_{n}^{∗} − y_{j_{n}}^{∗}∥ ≤ ϵ_{n} for some increasing sequence {j_{n}}_{n} of integers, which will ensure conditions (2) and (3). Note that condition (1) is equivalent to the following 3 conditions holding

(1-i) x_{n}^{∗}(T h) = 0 for h ∈ H_{m} and 0 ≤ m < n (1-ii) x_{m}^{∗}(T h) = 0 for h ∈ H_{n} and 0 ≤ m < n (1-iii) x_{n}^{∗}(T h) = δ for h ∈ H_{n} for each n. Clearly these three conditions hold for n = 0. Fix n ≥ 0.

Suppose that we are given a finite dyadic splitting {A_{k}^{m}: m = 0, …, n and k = 1, …, 2^{m}} of A_{1}^{0} up to n^{th}-level, which gives the subsets {H_{m}}_{m=0}^{n} of corresponding Haar functions. Thus we can find a finite set {z_{n,i}^{∗}}_{i=1}^{p_{n}} in S(𝔛^{∗}) such that {z_{n,i}^{∗}}_{i=1}^{p_{n}} norms sp(∪_{j=0}^{n}T H_{j}) within τ_{n}. Suppose that we are also given {x_{m}^{∗}}_{m=0}^{n} in Y ∩ S(𝔛^{∗}) such that the three subconditions of (1) hold and if k = 1, 2, …, n then ∥x_{k}^{∗} − y_{j_{k}}^{∗}∥ ≤ ϵ_{k} for some j_{k}.

We shall find x_{n+1}^{∗} along with j_{n+1} > j_{n} such that ∥x_{n+1}^{∗} − y_{j_{n+1}}^{∗}∥ ≤ ϵ_{n+1} and we shall partition, for each 1 ≤ k ≤ 2^{n}, the set A_{k}^{n} into 2 sets A_{2k−1}^{n+1} and A_{2k}^{n+1} of equal measure (thus finding h_{2^{n}+k} and so finding the corresponding set {H_{n+1}}) such that (1-i′) x_{n+1}^{∗}(T h) = 0 for h ∈ H_{m} and 0 ≤ m < n + 1 (1-ii′) x_{m}^{∗}(T h) = 0 for h ∈ H_{n+1} and 0 ≤ m < n + 1 (1-iii′) x_{n+1}^{∗}(T h) = δ for h ∈ H_{n+1}

(5′) T H_{n+1} ⊂ ^{⊥}{z_{n,i}^{∗}}_{i=1}^{p_{n}} .

Towards this, apply Lemma 2 to

(†)  E ≡ {T h: h ∈ H_{m} , 0 ≤ m ≤ n}

and ϵ_{n+1} to find the corresponding η_{n+1}. Let

(‡)  F_{k}^{n} = {1_{A_{k}^{n}}} ∪ {T^{∗}x_{m}^{∗}1_{A_{k}^{n}}}_{m=0}^{n} ∪ {T^{∗}z_{n,i}^{∗}1_{A_{k}^{n}}}_{i=1}^{p_{n}} ⊂ L_{1}

and F_{n} = sp [ ∪_{k=1}^{2^{n}}F_{k}^{n} ].

Pick j ≡ j_{n+1} > j_{n} so large that for k = 1, …, 2^{n}

(g) |(T^{∗}y_{j}^{∗}) h| < η_{n+1} for all h ∈ ∪_{m=0}^{n}H_{m}

(h) |∫_{Ω} g_{j}f dμ| ≤ \frac{ϵ}{3}∥f∥ for all f in F_{n}

(i) ∫_{A_{k}^{n}} T^{∗}y_{j}^{∗} · g_{j} dμ ≥ (δ_{0} − ϵ) μ(A_{k}^{n}) . Condition (g) follows from (e), condition (h) follows from (b) and the fact that F_{n} is finite dimensional, condition (i) follows from (f) and the definition of A.

By Lemma 2 and (g), there is x_{n+1}^{∗} ∈ S(𝔛^{∗}) ∩ Y such that ∥x_{n+1}^{∗} − y_{j_{n+1}}^{∗}∥ is at most ϵ_{n+1} and x_{n+1}^{∗}T h = 0 for each h ∈ ∪_{m=0}^{n}H_{m}. Thus (1-i′) holds.

Condition (h) gives that the L_{∞}-distance from g_{j} to

F_{n}^{⊥} ≡ {g ∈ L_{∞}: ∫_{Ω} fg dμ = 0 for each f ∈ F_{n}}

is at most \frac{ϵ}{3}. So there is g̃_{j} ∈ F_{n}^{⊥} ∩ B(L_{∞}) such that ∥g̃_{j} − g_{j}∥_{L_{∞}} is less than ϵ. Clearly g̃_{j}1_{A_{k}^{n}} ∈ F_{k}^{n} ^{⊥} ∩ B(L_{∞}) for each admissible k. By condition (i), for each admissible k,

∫_{Ω} (T^{∗}x_{n+1}^{∗}) · (g̃_{j}1_{A_{k}^{n}}) dμ ≥ δμ(A_{k}^{n})

and so, by Lemma 3, there exists a function u_{k}^{n} ∈ B(L_{∞}) ∩ F_{k}^{n} ^{⊥} such that

(∗)  ∫_{Ω} (T^{∗}x_{n+1}^{∗}) · (u_{k}^{n}) dμ = δμ(A_{k}^{n})

and u_{k}^{n} is of the form 1_{A_{2k−1}^{n+1}} − 1_{A_{2k}^{n+1}} for 2 disjoint sets A_{2k−1}^{n+1} and A_{2k}^{n+1} whose union is A_{k}^{n}. Furthermore, A_{2k−1}^{n+1} and A_{2k}^{n+1} are of equal measure since 1_{A_{k}^{n}} ∈ F_{k}^{n}. Since u_{k}^{n} ∈ F_{k}^{n} ^{⊥}, conditions (1-ii′) and (5′) hold. Condition (1-iii′) is just (∗).

Theorem 1 contains much information. For example, the next corollary crystallizes the role of the previously mentioned operator T_{0}.

Corollary 4. If the operator T : L_{1} → 𝔛 is not completely continuous, then there exist an isometry A and an operator B such that the following diagram commutes.

L_{1}  ⟶^{T}  𝔛
A ↑  ↓ B
L_{1}  ⟶^{T_{0}}  ℓ_{∞}

Furthermore, if 𝔛 is separable, then T_{0} and B may be viewed as operators into c_{0}.

Proof of Corollary 4. Let j_{1} be the natural injection of L_{1}(A) into L_{1}, let 𝔛_{0} be the norm closure of T (j_{1} L_{1} (A)), and let x̃_{n}^{∗} be the restriction of x_{n}^{∗} to 𝔛_{0}.

Since {T^{∗}x_{n}^{∗}} is weak*-null in L_{∞}, x̃_{n}^{∗} is weak*-null in 𝔛_{0}^{∗}. Thus the mapping U: ℓ_{1} → 𝔛_{0}^{∗} that take the n^{th} unit basis vector of ℓ_{1} to x̃_{n}^{∗} is weak* to weak* continuous and so U is the adjoint of the operator S: 𝔛_{0} → c_{0} where S(x) = (x̃_{n}^{∗} (x))_{n≥0}.

Consider the (commutative) diagram:

L_{1}  ⟶^{T}  𝔛
j_{1} ↑  ↑ j_{2}
L_{1}(A)  ⟶^{T_{A}}  𝔛_{0}
R ↑  ↓ S
L_{1}  ⟶  c_{0}  ⟶^{j_{3}}  ℓ_{∞}

where R: L_{1} → L_{1}(A) is the natural isometry that takes a usual Haar function h̃_{j} in L_{1} to the corresponding associated Haar function h_{j} in L_{1}(A), the maps j_{i} are the natural injections, and T_{A} is such that the upper square commutes.

For an arbitrary space 𝔛, since ℓ_{∞} is injective, the operator j_{3}S extends to an operator S̃: 𝔛 → ℓ_{∞}. For a separable space 𝔛, since c_{0} is separably injective, this extension S̃ may be view as taking values in c_{0}.

Let A = j_{1}R and B = \frac{1}{δ} S̃. Then BT A(h̃_{j}) = \frac{1}{δ} (x̃_{n}^{∗} (T h_{j}))_{n≥0}. Property 1 of Theorem 1 gives that BT A = T_{0}.

Corollary 4 says that, viewed as an operator into ℓ_{∞} (respectively, into c_{0}), T_{0} is universal for the class of non-completely-continuous operators from L_{1} into an arbitrary (respectively, separable) Banach space.

Theorem 5. There does not exist a universal operator for the class of noncompletely-continuous operator.

The proof of the nonexistence of such an operator uses the existence of a factorization through a reflexive space for a weakly compact operator.

Proof. Suppose that there did exist a universal non-completely-continuous operator, say T_{1} : 𝔛 → Z where 𝔛 and Z are Banach spaces. Then there is a sequence {x_{n}} in 𝔛 of norm one elements that converge weakly to zero but whose images {T_{1}x_{n}} are uniformly bounded away from zero. Furthermore, by passing to a subsequence, we also have that {T_{1}x_{n}} is a basic sequence in Z.

The first step of the proof uses T_{1} to construct a “nice” universal non-completely continuous operator. By Corollary 7 in [DFJP], there exists a reflexive space Y with a normalized unconditional basis {y_{n}} such that the map S: Y → 𝔛 that sends y_{n} to x_{n} is continuous. Consider the map U: Z → ℓ_{∞} that sends z to (z_{n}^{∗}(z)) where {z_{n}^{∗}} is a bounded sequence in Z^{∗} such that {T_{1}x_{n}, z_{n}^{∗}} is a biorthogonal system. The map I_{Y} ≡ UT_{1}S sends y_{n} to the n^{th} unit vector of ℓ_{∞}. The reflexivity of Y guarantees that I_{Y} is not completely continuous. Since I_{Y} factors through the universal operator T_{1}, the operator I_{Y} must also be universal. We now work with this “nice” operator I_{Y}.

For any linearly independent finite set {x_{k}}_{k=1}^{n}, let D{x_{k}}_{k=1}^{n} be the norm of the operator from the span of {x_{k}}_{k=1}^{n} to ℓ_{1}^{n} that sends x_{k} to the k^{th} unit vector of ℓ_{1}^{n}. Set d_{n} = D{y_{k}}_{k=1}^{n}. Reflexivity of Y gives that d_{n} tends to infinity. Let T be a (reflexive) Tsirelson-like space with normalized unconditional basis {t_{n}} such that for all finite subsets F of natural numbers,

D\{t_{n}\}_{n∈F} ≤ max \{2, \sqrt{d_{|F |}} \} ,

where | F | is the cardinality of F. For example, {t_{n}} can just be an appropriately chosen subsequence of the usual basis of the usual Tsirelson space [cf. CS, Chapter I]. Consider the non-completely-continuous map I_{T} : T → ℓ_{∞} that sends t_{n} to the n^{th} unit vector of ℓ_{∞}. By the universality of I_{Y}, there exists maps A and B such that the following diagram commutes.

T  ⟶^{I_{T}}  ℓ_{∞}
A ↑  ↓ B
Y  ⟶^{I_{Y}}  ℓ_{∞}

Since each I_{Y}(y_{n}) is of norm one, there exists δ > 0 such that δ < ∥I_{T} Ay_{n}∥ for each n. Each Ay_{n} is of the form

Ay_{n} = ∑_{m=1}^{∞} α_{n,m} t_{m}

and so there is a sequence {m(n)}_{n} of natural numbers such that δ <| α_{n,m(n)} |. Since {y_{n}} tends weakly to zero, for each m the set of all n for which m(n) = m is finite. Thus by replacing Y with the closed span of a suitable subsequence of {y_{n}}, we may assume that the m(n)’s are distinct.

Let T_{∗} be the subspace of T spanned by {t_{m(n)}}_{n}. Since {y_{n}} and {t_{m(n)}} are both unconditional bases, by the diagonalization principle [cf. LT, Prop. 1.c.8], the correspondence y_{n} ↦ α_{n,m(n)}t_{m(n)} extends to an operator D: Y → T_{∗}. Since {t_{m(n)}} is an unconditional basis and δ <| α_{n,m(n)} |, the correspondence α_{n,m(n)} t_{m(n)} ↦ t_{m(n)} extends to an operator M: T_{∗} → T_{∗}.

By the definition of d_{n}, there exists a sequence {β_{i}^{n}}_{i=1}^{n} such that ∑_{i=1}^{n} | β_{i}^{n} |= 1 and

∥ ∑_{i=1}^{n} β_{i}^{n}y_{i}∥_{Y} = \frac{1}{d_{n}} .

By the choice of T, for large n,

\frac{1}{\sqrt{d_{n}}} ≤ ∥ ∑_{i=1}^{n} β_{i}^{n} t_{m(i)}∥_{T_{∗}} .

Since MD: Y → T_{∗} maps y_{n} to t_{m(n)},

∥ ∑_{i=1}^{n} β_{i}^{n}t_{m(i)}∥_{T_{∗}} ≤ ∥MD∥ ∥ ∑_{i=1}^{n} β_{i}^{n}y_{i}∥_{Y} .

This gives that

\frac{1}{\sqrt{d_{n}}} ≤ \frac{∥MD∥}{d_{n}} ,

which cannot be since d_{n} tends to infinity.

The first two paragraphs of the proof of Theorem 5 yield part (a) of the next proposition. Part (b) follows from similar considerations and the Gurarii-James theorem [Ja, Thm. 2].

Proposition 6.

(a) Let 𝔖 be the collection of all formal identity operators into ℓ_{∞} from reflexive sequence spaces for which the unit vectors form a normalized unconditional basis. Then 𝔖 is universal for the class of all non-completely-continuous operators.

(b) The collection {I : ℓ_{p} → ℓ_{∞}; 1 < p < ∞} of formal identity operators is universal for the class of all non-completely-continuous operators whose domain is superreflexive.

Recall that a Banach space 𝔛 has the Radon-Nikodým Property (RNP) [respectively, is strongly regular, has the Complete Continuity Property (CCP)] if each bounded linear operator from L_{1} into 𝔛 is representable [respectively, strongly regular, completely continuous]. The books [DU], [GGMS], and [T] contain splendid surveys of these properties. Here we only recall that a representable operator is strongly regular and a strongly regular operator is completely continuous. The first paragraph of the proof of Theorem 1 uses elementary methods to construct, from an operator T: L_{1} → 𝔛 that is not completely continuous, a copy of ℓ_{1} in the closed span of a norming set of 𝔛. On a much deeper level, the following fact is well-known.

Fact. The following are equivalent.

(1) ℓ_{1} embeds into 𝔛.

(2) L_{1} embeds into 𝔛^{∗}.

(3) 𝔛^{∗} fails the CCP.

(4) 𝔛^{∗} is not strongly regular.

The well-known equivalence of (1) and (2) was shown by Pełczyński [P, for separable 𝔛] and Hagler [H, for non-separable 𝔛]. The other downward implications follow from the definitions. Bourgain [B] used a non-strongly-regular operator into a dual space to construct a copy of ℓ_{1} in the pre-dual. Here the authors wish to formalize the following essentially known fact which, to the best of our knowledge, has not appeared in print as such.

Fact. The following are equivalent.

(1) 𝔛 has trivial type.

(2) 𝔛 fails super CCP.

(3) 𝔛 is not super strongly regular.

Proof. To see that (1) implies (2), recall that 𝔛 has trivial type if and only if ℓ_{1} is finitely representable in 𝔛 and that L_{1} is finitely representable in ℓ_{1}. Thus, if 𝔛 has trivial type, then L_{1} is finitely representable in 𝔛 and so 𝔛 cannot have the super CCP. Property (3) formally follows from (2). Towards seeing that (3) implies (1), consider a space 𝔛 that is not strongly regular. From the above fact it follows that ℓ_{1} embeds into 𝔛^{∗}. Thus 𝔛^{∗} has trivial type, which implies the same for 𝔛.

References

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their applications, Studia Math. 29 (1968), 275–326. [LT]. J. Lindenstrauss and L.Tzafriri, Classical Banach Spaces I, Springer-Verlag, New

York-Berlin, 1977. [P]. A. Pełczyński, On Banach spaces containing L_{1}(μ), Studia Math. 30 (1968), 231–246. [RS]. Lawrence H. Riddle and Elias Saab, On functions that are universally Pettis inte-

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307, Amer. Math. Soc., Providence, R.I., 1984.

Maria Girardi University of South Carolina, Department of Mathematics, Columbia, SC 29208

E-mail address: girardi@math.sc.edu

W. B. Johnson Department of Mathematics, Texas A&M University, College Station, TX 77843

E-mail address: johnson@math.tamu.edu