The Kwapie´ n-Maurey theorems for type-2 spaces

Factorization Theory

7.4 The Kwapie´ n-Maurey theorems for type-2 spaces

This implies that n+l i=1

|x^∗(zi)|²= m j=1

|x^∗(xj)|², x^∗∈X^∗.

Then the vector-valued random variables n+l

i=1γ_iz_i andm

j=1γ_jx_j have the same distributions onX. As a consequence,

E n+l i=1

γ_iz_i²=E m j=1

γ_jx_j². (7.18) Now,

E n i=1

γizi²1/2

≤ 1 2

E n i=1

γ_iz_i+ n+l i=n+1

γ_iz_i²1/2

+1 2

E n i=1

γ_iz_i−

n+l

i=n+1

γ_iz_i²1/2

= E

n+l

i=1

γ_iz_i²1/2

= E

m j=1

γjxj²1/2

, which completes the proof.

Theorem 7.4.4.Let X andY be Banach spaces and E a closed subspace of X. Suppose T :E →Y is an operator. If X has type 2 and Y has cotype 2 then there is a Hilbert spaceH and operatorsS:X →H,R:H →Y so that RS ≤T₂(X)C₂(Y)T andRS|E=T.

Proof. We shall prove that for all sequences (z_i)ⁿ_i=1 in E and (x_j)^m_j=1 in X such that

n i=1

|x^∗(z_i)|²≤ m j=1

|x^∗(x_j)|², x^∗∈X^∗ (7.19) we have

ⁿ

i=1

T zi²1/2

≤T2(X)C2(Y)T^m

j=1

xj²1/2

,

and then we will appeal to the factorization criterion given by Theorem 7.3.4.

The key to the argument is to replace the Rademacher functions in the defi- nition of type and cotype by Gaussian random variables.

7.4 The Kwapie´n-Maurey theorems for type-2 spaces 189 On the one hand, for any (z_i)ⁿ_i=1 ⊂E, using the cotype-2 property of Y we have

n i=1

T z_i²≤C₂(Y)² E n i=1

ε_iT z_i².

Then, if for each N ∈ N we consider (ε_ki)_1≤i,k≤N, a sequence of N ×N Rademachers,

n i=1

T zi²≤ C2(Y)²

N E

N k=1

n i=1

εkiT zi²

=C₂(Y)² E n i=1

N k=1

ε_ki

√NT z_i².

Notice that for each 1 ≤ i ≤ n, the random variables ε_i1, ε_i2, . . . , ε_iN are independent and identically distributed, so by the Central Limit theorem, for each i the sequence ^PN_k=1ε_ik

√N

_∞

N=1 converges in distribution to a Gaussian, γ_i. Thus,

lim

N→∞E n

i=1

N k=1

εki

√NT z_i²=E n i=1

γ_iT z_i², and, therefore,

n i=1

T z_i²≤C₂(Y)² E n i=1

γ_iT z_i². (7.20) On the other hand, if we let (ε_i)^∞_i=1be a sequence of Rademachers indepen- dent of (γ_i)^∞_i=1, for any sequence (x_j)^m_j ⊂X, the symmetry of the Gaussians yields

E m j=1

γ_ix_j²=EEε m j=1

ε_jγ_jx_j²

≤T₂(X)²E m j=1

|γ_j|²x_j²

=T2(X)² m j=1

x_j²E|γ_j|²

=T2(X)² m j=1

xj². (7.21)

Suppose that the vectors (z_i)ⁿ_i=1 in E and (x_j)^m_j=1 in X satisfy equa- tion (7.19). Using Lemma 7.4.3 in combination with (7.18), (7.20), and (7.21) we obtain the inequality we need to apply Theorem 7.3.4:

n i=1

T z_i²≤C₂(Y)²E n i=1

γ_iz_i²

≤C2(Y)²T²E n i=1

γizi²

≤C₂(Y)²T²E m j=1

γ_jx_j²

≤C₂(Y)²T₂(X)²T² m j=1

x_j².

There is a quantitative estimate here that we would like to emphasize:

Definition 7.4.5.IfXandY are two isomorphic Banach spaces, theBanach- Mazur distancebetweenX andY, denotedd(X, Y), is defined by the formula

d(X, Y) = inf

TT⁻¹: T :X →Y is an isomorphism

.

The Banach-Mazur distance is not a distance in the real sense of the term since the triangle law does not hold, butdsatisfies a submultiplicative triangle inequality; that is,

d(X, Z)≤d(X, Y)d(Y, Z)

when X, Y, Z are all isomorphic. If X and Y are isometric then d(X, Y) = 1. The converse holds for finite-dimensional spaces but fails for infinite- dimensional spaces! (see the Problems).

In this language, Kwapie´n’s theorem (Theorem 7.4.1) really states:

Theorem 7.4.6.IfX is a Banach space of type 2 and cotype 2 then d(X, H)≤T₂(X)C₂(X)

for some Hilbert spaceH.

We have seen (Theorem 6.4.8) that ifp >2 every subspace of L_p which is isomorphic to a Hilbert space is necessarily complemented. Theorem 7.4.4 shows that this phenomenon is simply a consequence of the type-2 property:

Theorem 7.4.7 (Maurey). Let X be a Banach space of type 2. Let E be a closed subspace of X which is isomorphic to a Hilbert space. Then E is complemented inX.

Proof. Since E has cotype 2 the identity map on E can be extended to a projection ofX ontoE.

As we mentioned above, if we specialize the range space in Theorem 7.4.4 to be a Hilbert space then the assertion is a form of the Hahn-Banach theorem

7.4 The Kwapie´n-Maurey theorems for type-2 spaces 191 for Hilbert-space valued operators defined on a type-2 space. An interesting question is whether the extension property in Theorem 7.4.4 actually charac- terizes type-2 spaces:

Problem 7.4.8.SupposeX is a Banach space with the property that for every closed subspace E of X and every operatorT₀ :E →H (H a Hilbert space) there is a bounded extensionT :X→H. Must X be a space of type2?

For a partial positive solution of this problem we refer to [28].

Up to now the only spaces that we have considered in the context of type and cotype are theL_p-spaces (and their subspaces and quotients). It is worth pointing out that there are many other Banach spaces to which this theory can be applied. Perhaps the simplest examples are the “noncommutative”_p- spaces orSchatten ideals. These are ideals of operators on a separable Hilbert space which were originally introduced in 1946 by Schatten and studied in several papers by Schatten and von Neumann; an account is given in [202].

IfH is a separable (complex) Hilbert space we defineSp to be the set of compact operators A : H → H so that the positive operator (A^∗A)^p/2 has finite trace and we impose the norm

ASp= tr (A^∗A)^p/2.

It is not entirely obvious, but is true, that this is a norm and that the class of such operators forms a Banach space.

In many ways the structure ofSp resembles that ofp.Thus if 1≤p≤2, Sp has typepand cotype 2, while if 2≤p <∞, Sp has cotypepand type 2 (see [215], [65]). See [5] for the structure of subspaces of Sp.

Recently there has been considerable interest in noncommutative L_p- spaces but even to formulate the definition would take us too far afield.

Problems

7.1.For 1≤r, p <∞, prove that the space_r(_p) embeds inL_p if and only ifr=p.

7.2.Let p_n = 1 + _n¹. Consider the Banach space X =₂(²_p

n).Show that ²₁ does not embed isometrically intoX but thatd(X, X⊕2²₁) = 1.

7.3.Show that any reflexive quotient of aC(K) space has type two.

7.4. The weakL_p-spaces, L_p,∞.

Let (Ω, µ) be a probability measure space and 0 < p < ∞. A measurable functionf is said to belong toweakL_p, denotedL_p,∞,if

fp,∞= sup

t>0

tµ(|f|> t)^1/p<∞.

(a) Show thatL_p,∞is a linear space and that·p,∞is aquasi-normonL_p,∞, i.e., · p,∞satisfies the properties of a norm except the triangle law which is replaced by

f +gp,∞≤C(fp,∞+gp,∞), f, g∈L_p,∞, whereC≥1 is a constant independent off,g.

(b) Show that L_p,∞ is complete for this quasi-norm and hence becomes a quasi-Banach space.

(c) Show that ifp >1, · p,∞ is equivalent to the norm fp,∞,c= sup

t>0

sup

µ(A)=t

t^1/p−1

A

|f|dµ.

ThusLp,∞ can be regarded as a Banach space.

(d) Show thatL_p,∞⊂L_rwhenever 0< r < p.

7.5 (Nikishin [158]). (Continuation.) SupposeX is a Banach space of type pfor some 1≤p <2. Suppose 1≤r < pand T :X →L_r(µ) is a bounded linear operator.

(a) Show that for some suitable constantC we have the following estimate:

µ !^m

j=1

{|T x_j| ≥1}1/r

≤C ^m

j=1

x_j^p1/p

, x₁, . . . , x_m∈X.

(b) For any constant K > C consider a maximal family of disjoint sets of positive measure (E_i)_i∈I such that we can find x_i ∈ X with x_i ≤ 1 and

|T x_i| ≥K(µ(E_i)^−1/p) onE_i.Show that this collection is countable and that

i∈I

µ(Ei)≤C K

_p−r^rp .

(c) Show that given >0 there is a setEwithµ(E)>1−so that the map T_Ef =χ_ET f is a bounded operator fromX intoL_p,∞(µ).

This gives a “factorization” through weakL_p; it is possible to obtain a more elegant “change of density” formulation (see [186]). Note that ifX is an arbitrary Banach space andr <1 we get boundedness ofT_E into weakL₁. 7.6 (Jordan-von Neumann [96]). Show, without appealing to Kwapien’s theorem, that if a Banach space X has type 2 with T₂(X) = 1 then X is isometrically a Hilbert space. [Hint:For real scalars, define an inner product by (x, y) =¹₄(x+y²− x−y²).]

7.7.Let µ, ν be σ-finite measures. A linear operator T : Lp(µ) → Lr(ν), 0< r, p <∞, is said to be apositiveoperator iff ≥0 impliesT f ≥0.