The Rademacher Cotype of Operators from l_{∞}^{N}

S.J. Montgomery-Smith

Department of Mathematics, University of Missouri, Columbia, MO 65211.

M. Talagrand

Department of Mathematics, The Ohio State University, 231 W. 18th Avenue, Columbus, OH 43210. Equipe d’Analyse – Tour 46, Université Paris VI, 4 Place Jussieu, 75230 Paris Cedex 05.

ABSTRACT: We show that for any operator T : l_{∞}^{N} → Y, where Y is a Banach space, that its cotype 2 constant, K^{(2)}(T), is related to its (2, 1)-summing norm, π_{2,1}(T), by

K^{(2)}(T) ≤ c log log N π_{2,1}(T).

Thus, we can show that there is an operator T : C(K) → Y that has cotype 2, but is not 2-summing.

A.M.S. Classification: Primary 46B20, Secondary 60G99

Introduction

The notation we use in this paper is loosely based on that given in [L–T1], [L–T2] and

[P1].

We let ε_{1}, ε_{2}, … be independent Rademacher random variables, that is, Pr(ε_{s} = 1) =

Pr(ε_{s} = −1) = \frac{1}{2}. A linear operator T : X → Y is said to have (Rademacher) cotype p

(p ≥ 2) if there is a constant C < ∞ such that for all x_{1}, x_{2}, …, x_{S} in X we have

(∑_{s=1}^{S} ∥T (x_{s})∥^{p})^{\frac{1}{p}} ≤ C IE ∥∑_{s=1}^{S} ε_{s}x_{s}∥ .

The smallest value of C is called the (Rademacher) cotype p constant of T, and is denoted

by K^{(p)}(T). These definitions extend to spaces in the obvious way; a space X has cotype p

if its identity operator has cotype p.

THE RADEMACHER COTYPE OF OPERATORS FROM l_{∞}^{N}

We define the (p, q)-summing norm of a linear operator T : X → Y, denoted by π_{p,q}(T), to be the least number C such that for all x_{1}, x_{2}, …, x_{S} in X we have

(∑_{s=1}^{S} ∥T (x_{s})∥^{p})^{\frac{1}{p}} ≤ C sup \{(∑_{s=1}^{S} |⟨x^{∗}, x_{s}⟩|^{q})^{\frac{1}{q}} \} ,

where the supremum is taken over all x^{∗} in the unit ball of X^{∗}. We call a (p, p)-summing operator a p-summing operator, and write π_{p}(T) for π_{p,p}(T). We say that the operator is (p, q)-summing (p-summing) if π_{p,q}(T) < ∞ (respectively π_{p}(T) < ∞).

If 1 ≤ p < ∞, and 1 ≤ q ≤ ∞, then we let L_{p,q}(μ) denote the Lorentz space on the measure μ. We refer the reader to [H] or [L–T2] for details, but just note that the L_{p,1} norm may be calculated using

∥f∥_{p,1} = ∫_{0}^{∞} μ(|f| > t)^{\frac{1}{p}} dt = \frac{1}{p} ∫_{0}^{∞} s^{\frac{1}{p} −1}f^{∗}(s) ds,

where f^{∗} denotes the non-decreasing rearrangement of |f|.

The basic motivation behind this paper is in classifying operators from C(K) that factor through a Hilbert space, where C(K) denotes the continuous functions on the compact Hausdorff topological space, K. The first result in this direction is due to Grothendieck, which states that any bounded linear operator C(K) → L_{1} factors through Hilbert space. This was generalized by Maurey [Ma1], allowing L_{1} to be replaced by any space of cotype 2, to give the following result (see also [P1]).

Theorem 1. Let T : C(K) → Y be a linear operator, where Y is any Banach space. Then the following are equivalent:

i) T is 2-summing; ii) T factors through Hilbert space; iii) T factors through a space of cotype 2.

However, we are still left with the following question: if the operator T : C(K) → Y has cotype 2, does it follow that it factors through Hilbert space?

One way one might tackle this problem is to consider the (2, 1)-summing norms of such operators. Jameson [J] showed that there is an operator T : l_{∞}^{N} → Y such that π_{2}(T) ≥ c^{−1}\sqrt{log N} π_{2,1}(T). Hence, if we can establish a strong relationship between the cotype 2 constants and the (2, 1)-summing norms of such operators, then we can answer the above question in the negative. To this end, we have the following — the main result of this paper.

MONTGOMERY-SMITH — TALAGRAND

Theorem 2. There is a constant c such that for any operator T : l_{∞}^{N} → Y, where Y is a Banach space, then the cotype 2 constant is bounded according to the relation:

K^{(2)}(T) ≤ c log log N π_{2,1}(T).

Corollary. There is an operator T : C(K) → Y, where Y is a Banach space, that has cotype 2, but does not factor through Hilbert space.

Finally, before embarking on the proof of this result, we point out that for p > 2, the above problems have been completely answered.

Theorem 3. Let T : C(K) → Y be a bounded linear operator, where Y is a Banach space. Then for all p > 2, the following are equivalent:

i) T is (p, 1)-summing; ii) T has Rademacher cotype p; iii) T factors through a space with Rademacher cotype p.

The implication (i) ⇔ (ii) is due to Maurey [Ma2]. The third equivalence follows from the fact that any (p, 1)-summing operator from C(K) factors through L_{p,1} (see [P2] or Theorem 5 below), and that L_{p,1} has Rademacher cotype p, (see [C]).

Theorem 4. If p > 2, then there is a bounded linear operator C(K) → L_{p} that is not p-summing.

We refer the reader to [K].

Proof of the Main Result To prove Theorem 2, we need the following two results. The first allows us to reduce questions about (p, 1)-summing operators from C(K) to the canonical embedding C(K) → L_{2,1}(K, μ) (μ a probability measure), and is due to Pisier (see [P2]).

Theorem 5. Let T : C(K) → Y be a (p, 1)-summing operator, where Y is a Banach space, and p ≥ 1. Then there is a Radon probability measure μ on K and a constant C ≤ p^{\frac{1}{p}} π_{p,1}(T) such that for all x ∈ C(K) we have ∥T x∥ ≤ C ∥x∥_{L_{p,1}(K,μ)}.

The second result is about Rademacher processes, and is due to the second named author (for the proof, see [Ld–T]). First we establish some more notation. If T is a bounded subset of IR^{S}, we write

r(T) = IE sup_{t∈T} |∑_{s=1}^{S} ε_{s}t(s)| .

If B is a subset of IR^{S}, we write N (T, B) for the minimal number of translates of B required to cover D. We write B_{1}^{S} for the unit ball of l_{1}^{S}, and B_{2}^{S} for the unit ball of l_{2}^{S}. From now on, we take all logarithms to base 2.

Theorem 6. There is a constant c_{1} such that if T is a bounded subset of IR^{S}, and ϵ > 0, then letting D = c_{1} r(T) B_{1}^{S} + ϵ B_{2}^{S}, we have

r(T) ≥ c_{1}^{−1}ϵ\sqrt{log N (T, D)}.

Now we will state the main result towards proving Theorem 2.

Proposition 7. There is a constant c_{2} such that if (Ω, F, μ) is a probability space with N atoms, and x_{1}, x_{2}, …, x_{S} ∈ L_{∞}(μ) are such that

IE ∥∑_{s=1}^{S} ε_{s}x_{s}∥_{∞} ≤ 1,

then

(∑_{s=1}^{S} ∥x_{s}∥_{L_{2,1}(μ)}^{2})^{\frac{1}{2}} ≤ c_{2} log log N.

Our first step in establishing this result is to restate Theorem 6 in a more suitable form.

Lemma 8. There is a constant c_{1} (the same one as in Theorem 6) such that the following holds. Suppose that (Ω, F, μ) is a measure space, with Ω finite, and x_{1}, x_{2}, …, x_{S} ∈ L_{∞}(μ) with

IE ∥∑_{s=1}^{S} ε_{s}x_{s}∥_{∞} ≤ 1.

Then for all integers k, we may partition Ω into at most 2^{2^{k}} measurable sets, find y_{1}, y_{2}, …, y_{S}, z_{1}, z_{2}, …, z_{S} ∈ L_{∞}(μ), and find x̄_{1}, x̄_{2}, …, x̄_{S} ∈ L_{∞}(Ω, F′, μ) (where F′ denotes the algebra generated by the partition), such that x_{s} = x̄_{s} + y_{s} + z_{s},

IE ∥∑_{s=1}^{S} ε_{s}x̄_{s}∥_{∞} ≤ 1, ∥∑_{s=1}^{S} |y_{s}|∥_{∞} ≤ c_{1} and ∥(∑_{s=1}^{S} |z_{s}|^{2})^{\frac{1}{2}} ∥_{∞} ≤ c_{1} 2^{− \frac{k}{2}} .

Proof: Let T = \{ (x_{s}(ω))_{s=1}^{S} : ω ∈ Ω\}, and let ϵ = c_{1}^{−1}2^{− \frac{k}{2}} . If we apply Theorem 6, we see that there are 2^{2^{k}} translates, t_{l} + c_{1}(B_{1}^{S} + 2^{− \frac{k}{2}} B_{2}^{S}) (1 ≤ l ≤ 2^{2^{k}}), that cover T. We let the covering of Ω be the sets

A_{l} = \{ ω : (x_{s}(ω))_{s=1}^{S} ∈ t_{l} + c_{1}(B_{1}^{S} + 2^{− \frac{k}{2}} B_{2}^{S})\} ,

and if A_{l} is non-empty, we choose ω_{l} ∈ A_{l}. Define x̄_{s}(ω) = x_{s}(ω_{l}) if ω ∈ A_{l}. Now, if ω ∈ A_{l}, we know that (x_{s}(ω)−x̄_{s}(ω))_{s=1}^{S} ∈ c_{1}(B_{1}^{S} +2^{− \frac{k}{2}} B_{2}^{S}), that is, there are (y_{s}(ω))_{s=1}^{S} ∈ c_{1} B_{1}^{S} and (z_{s}(ω))_{s=1}^{S} ∈ c_{1} 2^{− \frac{k}{2}} B_{2}^{S}, with x_{s}(ω) = x̄_{s}(ω) + y_{s}(ω) + z_{s}(ω).

Lemma 9. There is a constant c_{3} such that if (Ω, F, μ) is a measure space with Ω finite, then the following hold.

i) If y ∈ L_{∞}(μ), then ∥y∥_{2,1} ≤ ∥y∥∞^{\frac{1}{2}} ∥y∥1^{\frac{1}{2}} .

ii) If the smallest atom is of size a, then for all z ∈ L_{∞}(μ) we have

∥z∥_{2,1} ≤ c_{3} (1 + \sqrt{log(μ(Ω)/a)}) ∥z∥_{2} .

iii) If there are N atoms, then for all z ∈ L_{∞}(μ) we have ∥z∥_{2,1} ≤ \sqrt{N} ∥z∥_{2}.

Proof: i) We have that

∥y∥_{2,1} = ∫_{0}^{∥y∥_{∞}} \sqrt{μ(|y| > t)} dt

≤ (∫_{0}^{∥y∥_{∞}} dt)^{\frac{1}{2}} (∫_{0}^{∥y∥_{∞}} μ(|y| > t) dt)^{\frac{1}{2}}

= ∥y∥∞^{\frac{1}{2}} ∥y∥1^{\frac{1}{2}} .

ii) We have

∥z∥_{2,1} = \frac{1}{2} ∫_{0}^{∞} \frac{z^{∗}(s)}{\sqrt{s}} ds

≤ \sqrt{a} ∥z∥_{∞} + \frac{1}{2} ∫_{a}^{μ(Ω)} \frac{z^{∗}(s)}{\sqrt{s}} ds

≤ ∥z∥_{2} + \frac{1}{2} (∫_{a}^{μ(Ω)} \frac{ds}{s})^{\frac{1}{2}} (∫_{a}^{μ(Ω)}(z^{∗}(s))^{2} ds)^{\frac{1}{2}}

≤ c_{3} (1 + \sqrt{log(μ(Ω)/a)}) ∥z∥_{2} .

iii) Let B_{1}, B_{2}, …, B_{N} be the atoms of Ω arranged so that z^{∗}(n), the value of |z| on B_{n}, is in non-increasing order. Also, let z^{∗}(N + 1) = 0. Then

∥z∥_{2,1} = ∑_{n=1}^{N} ( ∑_{m=1}^{n} μ(B_{m}))^{\frac{1}{2}} (z^{∗}(n) − z^{∗}(n + 1))

≤ \sqrt{N} (∑_{n=1}^{N} ∑_{m=1}^{n} μ(B_{m})(z^{∗}(n) − z^{∗}(n + 1))^{2})^{\frac{1}{2}}

≤ \sqrt{N} (∑_{n=1}^{N} μ(B_{n})(z^{∗}(n))^{2})^{\frac{1}{2}}

= \sqrt{N} ∥z∥_{2} .

as desired.

We remark that Lemma 9(i) is a well known interpolation result, and is true for all measure spaces.

Lemma 10. If (Ω, F, μ) is a probability space with Ω finite, then

(∑_{s=1}^{S} ∥y_{s}∥_{2,1}^{2})^{\frac{1}{2}} ≤ ∥∑_{s=1}^{S} |y_{s}|∥_{∞} .

Proof: This follows straight away from Lemma 9(i).

Lemma 10 is also well known (and true for all probability spaces). In fact it is a reformulation of the statement that the canonical embedding C(Ω) → L_{2,1}(μ) has (2, 1)- summing norm equal to 1.

Lemma 11. There is a constant c_{4} such that, if (Ω, F, μ) is a probability space with at most N atoms, then

(∑_{s=1}^{S} ∥z_{s}∥_{2,1}^{2})^{\frac{1}{2}} ≤ c_{4}\sqrt{log N} ∥(∑_{s=1}^{S} |z_{s}|^{2})^{\frac{1}{2}} ∥_{∞} .

Proof: Let A ⊂ Ω be the union of those atoms of measure less than \frac{1}{N^{2}} , so that μ(A) ≤ \frac{1}{N}. By Lemma 9(ii), we have that ∥z_{s}χ_{Ω\A}∥_{2,1} ≤ c_{3}\sqrt{log N} ∥z_{s}∥_{2}, and by Lemma 9(iii), we have that ∥z_{s}χ_{A}∥_{2,1} ≤ \sqrt{N} ∥z_{s}χ_{A}∥_{2}. Thus, we have that

(∑_{s=1}^{S} ∥z_{s}∥_{2,1}^{2})^{\frac{1}{2}} ≤ (∑_{s=1}^{S} ∥z_{s}χ_{Ω\A}∥_{2,1}^{2})^{\frac{1}{2}} + (∑_{s=1}^{S} ∥z_{s}χ_{A}∥_{2,1}^{2})^{\frac{1}{2}}

≤ c_{3}\sqrt{log N} (∑_{s=1}^{S} ∥z_{s}∥_{2}^{2})^{\frac{1}{2}} + \sqrt{N μ(A)} ∥(∑_{s=1}^{S} |z_{s}|^{2})^{\frac{1}{2}} ∥_{∞}

≤ c_{4}\sqrt{log N} ∥(∑_{s=1}^{S} |z_{s}|^{2})^{\frac{1}{2}} ∥_{∞} ,

as desired.

Proof of Proposition 7: Without loss of generality, we may suppose that N = 2^{2^{k}}. We prove the result by induction over k. Suppose that Ω has 2^{2^{k+1}} atoms. Apply Lemma 8 to cover Ω by 2^{2^{k}} subsets, and to give x̄_{1}, x̄_{2}, …, x̄_{S}, y_{1}, y_{2}, …, y_{S}, z_{1}, z_{2}, …, z_{S} as described in the lemma. Then, by the triangle inequality

(∑_{s=1}^{S} ∥x_{s}∥_{2,1}^{2})^{\frac{1}{2}} ≤ (∑_{s=1}^{S} \left∥{ \atop x̄_{s}}\right∥_{2,1}^{2})^{\frac{1}{2}} + (∑_{s=1}^{S} ∥y_{s}∥_{2,1}^{2})^{\frac{1}{2}} + (∑_{s=1}^{S} ∥z_{s}∥_{2,1}^{2})^{\frac{1}{2}} .

By the induction hypothesis,

(∑_{s=1}^{S} \left∥{ \atop x̄_{s}}\right∥_{2,1}^{2})^{\frac{1}{2}} ≤ c_{2}k.

By Lemmas 10 and 11 we have that

(∑_{s=1}^{S} ∥y_{s}∥_{2,1}^{2})^{\frac{1}{2}} ≤ c_{1}

and

(∑_{s=1}^{S} ∥z_{s}∥_{2,1}^{2})^{\frac{1}{2}} ≤ c_{1}c_{4} 2^{− \frac{k}{2}} (1 + \sqrt{log 2^{2^{k}}}) ≤ 2c_{1}c_{4}.

Hence

(∑_{s=1}^{S} ∥x_{s}∥_{2,1}^{2})^{\frac{1}{2}} ≤ c_{2}(k + 1),

as required, taking c_{2} = 1 + 2c_{1}c_{4}.

To prove the main result is now easy.

Proof of Theorem 2: By Theorem 5, it is sufficient to show that for any probability measure μ on {1, 2, …, N}, the cotype 2 constant of the canonical embedding l_{∞}^{N} → L_{2,1}(μ) is bounded by some universal constant times log log N. But this is precisely what Proposition 7 says.

Final Remarks There is a similar result for Gaussian cotype (see [Mo2]).

Theorem 12. There is a constant c such that, for any operator T : l_{∞}^{N} → Y, where Y is a Banach space, the Gaussian cotype 2 constant, β^{(2)}(T), is bounded according to the relation:

β^{(2)}(T) ≤ c\sqrt{log log N} π_{2,1}(T).

This result is the best possible, as is implicitly shown in [T].

Theorem 13. There is a constant c such that for any integer N, there is an operator T : l_{∞}^{N} → Y, where Y is a Banach space, such that

β^{(2)}(T) ≥ c^{−1}\sqrt{log log N} π_{2,1}(T).

Since the Rademacher cotype 2 constant is greater than a constant times the Gaussian cotype 2 constant, we have the following corollary.

Corollary. There is a constant c such that for any integer N, there is an operator T : l_{∞}^{N} → Y, where Y is a Banach space, such that

K^{(2)}(T) ≥ c^{−1}\sqrt{log log N} π_{2,1}(T).

We also have the following, the result originally stated in [T].

Corollary. There is an operator T : C(K) → Y, where Y is a Banach space, that is (2, 1)-summing, but does not have Rademacher cotype 2.

If we write R_{N} for the supremum of K^{(2)}(T)/π_{2,1}(T) over all T : l_{∞}^{N} → Y, then we have shown that c^{−1}\sqrt{log log N} ≤ R_{N} ≤ c log log N. Clearly, we are left with the following problem.

Open Question. What is the asymptotic behavior of R_{N}?

Acknowledgements

The main result of this paper originally appears in the Ph.D. thesis of the first named author [Mo1], and he would like to express his thanks to his advisor, D.J.H. Garling, and to the Science and Engineering Council who financed his studies. He would also like to express his gratitude to G.J.O. Jameson who first suggested the problem to him, and gave him much encouragement.

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