Classes of Banach spaces connected with the Lyapunov

convexity theorem

Dissertation

zur Erlangung der Doktorwürde des Fachbereichs Mathematik und Informatik

der Freien Universität Berlin

vorgelegt von O. I. Vladimirskaya

Berlin 1999

Contents

Preface 1

1 Vector Measures 3 1.1 Elementary properties . . . . . . . . . . . . . . . . . . . 3 1.2 Bartle-Dunford-Schwartz’ theorem . . . . . . . . . . . . . 6 1.3 Lyapunov’s convexity theorem . . . . . . . . . . . . . . . 9 1.4 l_{p}-valued measures . . . . . . . . . . . . . . . . . . . . . 13 1.5 The three-space problem . . . . . . . . . . . . . . . . . . 22

2 Lyapunov Trees 29 2.1 Lyapunov tree cotype . . . . . . . . . . . . . . . . . . . . 30 2.2 Lyapunov tree type . . . . . . . . . . . . . . . . . . . . . 37

3 Examples 41 3.1 Orlicz sequence spaces . . . . . . . . . . . . . . . . . . . 41 3.2 The A_{p}-property . . . . . . . . . . . . . . . . . . . . . . 50 3.3 Tsirelson-type spaces . . . . . . . . . . . . . . . . . . . . 57 3.4 Asymptotic l_{p} spaces . . . . . . . . . . . . . . . . . . . . 63 3.5 Tokarev’s space . . . . . . . . . . . . . . . . . . . . . . . 64

Bibliography 67

Résumé (Zusammenfassung) 71

Preface

Vector measures with values in finite-dimensional and infinite-dimen-sional spaces play an important role in various questions of analysis, operator theory, optimization theory, games and statistical decisions theory and represent an interesting object for study. Works of well known mathematicians as R. G. Bartle, N. Dunford, J. T. Schwartz, B. J. Pettis, I. Gelfand, A. Grothendieck and others were dedicated to vector measure theory. The present state of the theory is reflected in the monographs ”Vector measures” by J. Diestel and J. J. Uhl [7] and ”Vector measures and control systems” by I. Kluvanek and G. Knowles [20].

One of the classical results of vector measure theory is A. A. Lyapunov’s theorem ([25], [7, p. 264], [29, theorem 5.5]) which states that the range of every countably additive finite nonatomic vector measure valued in a finite dimensional space is a compact and convex set. This result, first obtained by A. A. Lyapunov in 1940 attracted the attention of many mathematicians. Different versions of this theorem appeared in the fifties and sixties and in 1966 J. Lindenstrauss [23] gave the shortest proof of the statement.

It is known that the Lyapunov theorem is false in the infinitedimensional case: for every infinite-dimensional Banach space there exists a nonatomic countably additive measure of bounded variation, with values in this space and defined on the measurable subsets of [0, 1], with nonconvex range ([7, p. 266], [16]). Nevertheless by additional restrictions on the measure and the space this theorem has some nontrivial generalizations that have been obtained by G. Knowles [21], D. Pecherskii [28], J. Elton [8], J. J. Uhl [7], V. M. Kadets [14], V. M. Kadets and M. M. Popov [16], V. M. Kadets and G. Schechtman [17].

However all known infinite-dimensional generalizations of the Lyapunov convexity theorem have a different nature.

The aim of this thesis is to develop infinite-dimensional generalizations of the Lyapunov theorem to clarify the situation.

The present work consists of three chapters. General information on vector measures, different approaches to generalizations of the Lyapunov convexity theorem and the three-space problem for the Lyapunov property are presented in Chapter 1. In Chapter 2 the notions of a Lyapunov tree and Lyapunov C-convexity are introduced. It is proved that Banach spaces having this property have the Lyapunov property. In Chapter 3 we find out which of the known Banach spaces have the Lyapunov property and which do not. Most of the constructed examples fit into the scheme: if a Banach space has no isomorphic copies of l_{2} then it has the Lyapunov property. However, we present an example of a Banach space that does not fit into this scheme.

The main results of the dissertation were published in the works [18], [35], [34]. The results included in the thesis were communicated in the Kharkov seminar on Banach space theory and on the following conferences: XXVIII scientific-technical conference of the Kharkov Academy of Municipal Economy [36] and VI international conference in honor of academician M. Kravchuk [37].

I would like to express my special gratitude to my scientific supervisor V. M. Kadets for his leadership and encouragement to conduct this research and D. Werner for useful consultations.

Chapter 1

Vector Measures

In this chapter we present a survey of known results about vector measures. For the convenience of the readers some of the results are given with proofs, but neither results nor proofs pretend to be ours. The only exception is the material of Section 1.5 that covers the results of our joint paper with V. M. Kadets [18].

1.1 Elementary properties

The results of this section may be found in [7].

Definition 1.1.1 A function μ from a field F of subsets of a set Ω to a Banach space X is called a finitely additive vector measure, or simply a vector measure, if whenever A_{1} and A_{2} are disjoint members of F then μ (A_{1} ∪ A_{2}) = μ (A_{1}) + μ (A_{2}).

If in addition μ ( ⋃_{n=1}^{∞} A_{n}) = ∑_{n=1}^{∞} μ (A_{n}) in the norm topology of X

for all sequences (A_{n}) of pairwise disjoint members of F such that ⋃_{n=1}^{∞} A_{n} ∈ F, then μ is termed a countably additive vector measure, or

simply μ is countably additive.

Example 1.1.2 Finitely additive and countably additive vector measures. Let Σ be a σ-field of subsets of a set Ω, λ be a nonnegative countably additive measure on Σ. Consider 1 ≤ p ≤ ∞. Define μ_{p} : Σ → L_{p} (Ω, Σ, λ) by the rule μ_{p} (A) = χ_{A} for each set A ∈ Σ (χ_{A} denotes the characteristic function of A). It is easy to see that μ_{p} is a vector measure, which is countably additive in the case 1 ≤ p < ∞ and fails to be countably additive in the case p = ∞.

Definition 1.1.3 A vector measure μ : F → X is said to be nonatomic if for every set A ∈ F with μ (A) ≠ 0 there exist A_{1},A_{2} ⊂ F \ Ø such that A_{1} ∪ A_{2} = A and μ (A_{i}) ≠ 0 (i = 1, 2).

Definition 1.1.4 Let μ : F → X be a vector measure. The variation of μ is the nonnegative function |μ| whose value on a set A ∈ F is given by

|μ| (A) = sup_{π} ∑_{B∈π} ∥μ (B)∥ ,

where the supremum is taken over all partitions π of A into a finite number of pairwise disjoint members of F. If |μ| (Ω) < ∞, then μ will be called a measure of bounded variation.

The semivariation of μ is the extended nonnegative function ∥μ∥ whose value on a set A ∈ F is given by

∥μ∥ (A) = sup {|x^{∗}μ| (A) : x^{∗} ∈ X^{∗}, ∥x^{∗}∥ ≤ 1} ,

where |x^{∗}μ| is the variation of the real-valued measure x^{∗}μ.

Example 1.1.5 A measure of bounded variation. Let μ_{1} be the measure discussed in Example 1.1.2. Since ∥μ_{1} (A)∥ ≤ λ (A), it is plain that |μ_{1}| (A) ≤ λ (A), so that μ_{1} is of bounded variation.

The next proposition presents two basic facts about the semivariation of a vector measure.

Proposition 1.1.6 Let μ : F → X be a vector measure. Then for A ∈ F one has

(a) ∥μ∥ (A) = sup_{{ε_{n}},π} ∑_{B_{n}∈π} ∥ε_{n}μ (B_{n})∥, where the supremum is taken

over all partitions π of A into a finite number of pairwise disjoint members of F and over all finite collections {ε_{n}} satisfying |ε_{n}| ≤ (b) sup{∥μ(B)∥: A ⊇ B ∈ F } ≤ ∥μ∥(A) ≤ 2 sup{∥μ(B)∥: A ⊇ B ∈

F }. Consequently a vector measure is of bounded semivariation on Ω if and only if its range is bounded in X.

It is easy to define the integral of a bounded measurable function with respect to a bounded vector measure. To this end, let F be a field of subsets of a set Ω and μ : F → X be a bounded vector measure. If f is a scalar-valued simple function, say f = ∑_{i=1}^{n} α_{i}χ_{A_{i}}, where α_{i}

are nonzero scalars and A_{1}, ..., A_{n} are pairwise disjoint members of F, define T_{μ} (f) = ∑_{i=1}^{n} α_{i}μ (A_{i}). It is easy to show that this formula defines

a linear operator T_{μ} from the space of simple functions of the above form into X. Moreover, if f is as above and β = sup {|f (ω)| : ω ∈ Ω}, then

∥T_{μ} (f)∥ = ∥∑_{i=1}^{n} α_{i}μ (A_{i})∥ = β ∥∑_{i=1}^{n} \frac{α_{i}}{β} μ (A_{i})∥ ≤ β ∥μ∥ (Ω) .

by Proposition 1.1.6 (a). Thus T_{μ} is a continuous linear operator, which acts on the space of simple functions on F, equipped with the supremum norm. Another look at the Proposition 1.1.6 (a) and the above calculations shows that

∥T_{μ}∥ = ∥μ∥ (Ω)  (1.1)

Then T_{μ} has a unique continuous linear extension, still denoted by T_{μ}, to B (μ), the space of all scalar-valued functions on Ω that are uniform limits of simple functions modeled on F. (Note that in the case F is a σ-field, B (μ) is precisely the familiar space of bounded F -measurable scalar-valued functions defined on Ω).

This discussion allows us to make

Definition 1.1.7 Let F be a field of subsets of a set Ω and μ : F → X be a bounded vector measure. For each f ∈ B (μ), the integral ∫ fdμ is defined by

∫ fdμ = T_{μ} (f) ,

where T_{μ} is as above.

It is easy to see that this integral is linear in f (and also in μ) and satisfies

∥∫ fdμ∥ ≤ ∥f∥_{∞} ∥μ∥ (Ω) .

Moreover, if x^{∗} ∈ X^{∗}, then x^{∗} ∫ fdμ = ∫ fdx^{∗}μ; indeed, for simple functions this equality is trivial and density of simple functions in B (μ) proves the identity for all f ∈ B (μ).

Definition 1.1.8 A family {μ_{τ} : F → X | τ ∈ T } of countably additive vector measures is said to be uniformly countably additive whenever for any sequence (A_{n}) of pairwise disjoint members of F, the series ∑_{n=1}^{∞} μ_{τ} (A_{n}) converges in norm uniformly in τ ∈ T.

There are many alternative and useful formulations of countable additivity. We present two of them.

Proposition 1.1.9 Any one of the following statements about a vector measure μ defined on a field F implies all the others.

(i) μ is countably additive.

(ii) The set {x^{∗}μ : x^{∗} ∈ X^{∗}, ∥x^{∗}∥ ≤ 1} is uniformly countably additive.

(iii) The set {|x^{∗}μ| : x^{∗} ∈ X^{∗}, ∥x^{∗}∥ ≤ 1} is uniformly countably additive.

1.2 Bartle-Dunford-Schwartz’ theorem

The results of this section may be found in [7].

Definition 1.2.1 Let F be a field of subsets of a set Ω, μ : F → X be a vector measure, and λ be a finite nonnegative real-valued measure on F. If lim_{λ(A)→0} μ (A) = 0, then μ is called λ-continuous and this is

denoted by μ ≪ λ. Sometimes we will say μ is absolutely continuous with respect to λ.

It should be noted that writing μ ≪ λ is not the same as saying μ vanishes on λ-null sets unless both λ and μ are countably additive and defined on a σ-field.

Theorem 1.2.2 (Pettis) Let Σ be a σ-field of subsets of a set Ω, μ : Σ → X be a countably additive vector measure, and λ be a finite nonnegative real-valued measure on Σ. Then μ is λ-continuous if and only if μ vanishes on sets of λ-measure zero.

Proof. To prove the sufficiency, suppose μ vanishes on sets of λmeasure zero, but \overline{lim}_{λ(A)→0} μ (A) > 0. Then there exist an ε > 0 and a

sequence (A_{n}) in Σ, such that

∥μ (A_{n})∥ ≥ ε and λ (A_{n}) ≤ 2^{−n}

for all n. For each n select an x_{n}^{∗} ∈ X^{∗} such that

∥x_{n}^{∗}∥ ≤ 1 and |x_{n}^{∗}μ (A_{n})| ≥ \frac{ε}{2}.

As μ is countably additive on Σ, then in accordance with Proposition 1.1.9 the set {|x^{∗}μ| : x^{∗} ∈ X^{∗}, ∥x^{∗}∥ ≤ 1} is uniformly countably addi-

tive. Now set B_{n} = ⋃_{i=n}^{∞} A_{i}. Evidently λ ( ⋂_{n=1}^{∞} B_{n}) = 0. Consequently μ

vanishes on every set D ∈ Σ that is contained in ⋂_{n=1}^{∞} B_{n} = B. It follows

that |x^{∗}μ| (B) = 0.

Let C_{1} = Ω \ B_{1} and C_{n+1} = B_{n} \ B_{n+1} for all n. Then (C_{n}) constitutes a sequence of pairwise disjoint members of Σ for which

B_{n−1} \ B = ⋃_{i=n}^{∞} C_{i}.

Also, since |x_{n}^{∗}μ| (B) = 0 for all n, one has

lim_{m} |x_{n}^{∗}μ| (B_{m}) = lim_{m} |x_{n}^{∗}μ| ( ⋃_{i=m}^{∞} C_{i}) = lim_{m} ∑_{i=m}^{∞} |x_{n}^{∗}μ| (C_{i}) = 0

uniformly in n, by the uniform countable additivity of the family {|x_{n}^{∗}μ|}. But now

|x_{n−1}^{∗}μ| (B_{n−1}) ≥ |x_{n−1}^{∗}μ| (A_{n−1}) ≥ |x_{n−1}^{∗}μ (A_{n−1})| ≥ \frac{ε}{2}.

This contradicts the last calculation and proves the sufficiency; the converse is transparent.

The following theorem is central to the theory of vector measures.

Theorem 1.2.3 (Bartle-Dunford-Schwartz) Let μ be a bounded countably additive vector measure defined on a σ-field Σ. Then there exists a nonnegative real-valued countably additive measure λ on Σ such that μ is absolutely continuous with respect to λ. If μ is nonatomic, then λ can be chosen nonatomic. (λ can be chosen such that 0 ≤ λ(A) ≤ ∥μ∥ (A) for all A ∈ Σ.)

Corollary 1.2.4 Let Σ be a σ-field, μ : Σ → X be a bounded countably additive vector measure, λ be a measure from Theorem 1.2.3. Then there exist an operator T : L_{∞} (λ) → X, which is continuous for the weak^{∗} topology on L_{∞} (λ) and weak topology on X such that T χ_{A} = μ (A) for any A ∈ Σ.

Proof. Define T : L_{∞} (λ) → X by T f = ∫_{Ω} fdμ. Then for each

x^{∗} ∈ X^{∗}, one has

x^{∗}T f = ∫ fdx^{∗}μ = ∫ f \frac{dx^{∗}μ}{dλ}dλ,

where dx^{∗}μ/dλ = g_{x^{∗}} ∈ L_{1} (λ) is the Radon-Nikodym derivative of x^{∗}μ with respect to λ. If (f_{α}) is a net in L_{∞} (λ) converging weak^{∗} to f_{0}, then for each x^{∗} ∈ X^{∗},

lim_{α} x^{∗}T f_{α} = lim_{α} ∫ f_{α}g_{x^{∗}}dλ = ∫ f_{0}g_{x^{∗}}dλ = x^{∗}T f_{0},

i.e., (T f_{α}) converges weakly to T f_{0}. Hence T is a weak^{∗} to weak continuous linear operator.

Corollary 1.2.5 (Bartle-Dunford-Schwartz) Let μ be a bounded countably additive vector measure on a σ-field Σ. Then μ has a relatively weakly compact range.

Proof. Let T be the operator from Corollary 1.2.4. It follows that T maps the weak^{∗} compact set {f ∈ L_{∞} (λ) : ∥f∥_{∞} ≤ 1} onto a weakly compact set R ⊆ X. But now

{μ (A) : A ∈ Σ} = {T (x_{A}) : A ∈ Σ} ⊆ {T (f) : ∥f∥_{∞} ≤ 1} ⊆ R,

and the proof is complete.

Corollary 1.2.6 Let μ and λ be as in Corollary 1.2.4. Then

\overline{co} (μ (Σ)) = \{∫_{Ω} fdμ : 0 ≤ f ≤ 1, f ∈ L_{∞} (λ)\} .

Proof. Let

U = {f ∈ L_{∞} (λ) : 0 ≤ f ≤ 1} and V = {∫_{Ω} fdμ : f ∈ U} .

Note that U is weak^{∗} compact and {1_{A} : A ∈ Σ} = ex U (where ex U is the set of the extreme points of U). By the Krein-Milman theorem U = \overline{co}^{w^{∗}} (ex U). Then a glance at Corollary 1.2.4 proves that V = \overline{co} (μ (Σ)).

The following theorem shows that the measure λ from the Bartle-Dunford-Schwartz theorem may be taken of the form |x^{∗}μ| for a certain x^{∗} ∈ X^{∗}.

Theorem 1.2.7 (Rybakov) Let μ : Σ → X be a countably additive vector measure. Then there is x^{∗} ∈ X^{∗} such that μ ≪ |x^{∗}μ|.

1.3 Lyapunov’s convexity theorem

The results of this section may be found in [7].

One of the most important results in the theory of vector measures is the Lyapunov convexity theorem which states that the range of a nonatomic vector measures with values in a finite dimensional space is compact and convex. As it was mentioned above, this theorem fails in every infinite dimensional Banach space.

Example 1.3.1 (Uhl) A nonatomic vector measure of bounded variation whose range is closed but nonconvex and noncompact.

Let Σ be the Borel sets in [0, 1] and λ be Lebesgue measure. Define μ : Σ → L_{1} (λ) by μ (A) = χ_{A}. If Π ⊂ Σ is a partition of [0, 1], it is evident that ∑_{A∈Π} ∥μ (A)∥_{1} = ∑_{A∈Π} λ (A) = 1. Since every L^{1}-convergent sequence contains an almost everywhere convergent subsequence, μ (Σ) is closed in L_{1} (λ). To see that μ (Σ) is not a convex set, note that \frac{1}{2}χ_{[0,1]} ∈ co μ (Σ) while if A ∈ Σ

∥μ (A) − \frac{1}{2}χ_{[0,1]}∥_{1} = [μ ([0, 1] \E) + μ (E)] /2 = \frac{1}{2}.

To see that μ (Σ) is not compact let A_{n} = {t ∈ [0, 1] : sin (2^{n}πt) > 0} for each positive integer n. A brief computation shows that

∥μ (A_{n}) − μ (A_{m})∥ = \frac{1}{2}

for m ≠ n. Hence μ (Σ) is not compact.

Example 1.3.1 suggests that nonatomicity may not be a very strong property of vector measures from the point of view of the Lyapunov theorem in the infinite dimensional context. Let us attempt to understand what nonatomicity means.

Let μ : Σ → R^{n} have the form

μ (A) = (λ_{1} (A) , …, λ_{n} (A)) , A ∈ Σ,

where each λ_{i} is a countably additive finite signed scalar measure on Σ. Set λ (A) = ∑_{k=1}^{n} |λ_{k}| (A). Then ∥μ∥ (A) → 0 if and only if λ (A) → 0.

Now if μ is nonatomic, then λ is nonatomic. Consequently, if A ∈ Σ and λ (A) > 0 the mapping on the infinite dimensional subspace {fχ_{A} : f ∈ L_{∞} (λ)} that takes f into ∫_{A} fdμ is never one-to-one. We will

see that in the infinite dimensional case, this latter condition is precisely what is needed to make the Lyapunov theorem work. Throughout, Σ is a σ-field of subsets of Ω and X is a Banach space.

Theorem 1.3.2 (Knowles; Lyapunov convexity theorem in the weak topology) Let μ : Σ → X be a countably additive vector measure and λ be a finite nonnegative countably additive measure on Σ from Theorem 1.2.3. The following statements are equivalent.

(i) If A ∈ Σ and λ (A) > 0, then the operator f ↦ ∫_{A} fdμ on L_{∞} (λ) is

not one-to-one on the subspace of functions in L_{∞} (λ) vanishing off A.

(ii) For each A ∈ Σ, {μ (B ∩ A) : B ∈ Σ} is a weakly compact convex set in X.

(iii) If 0 ≠ f ∈ L_{∞} (λ), there exists a function g ∈ L_{∞} (λ) such that

∥fg∥_{∞} > 0 but ∫_{Ω} fgdμ = 0.

Proof. Note that (iii) implies (i). To show that (i) implies (iii) let f ∈ L_{∞} (λ), ∥f∥_{∞} > 0. Then there is an ε > 0 and A ∈ Σ such that |fχ_{A}| > ε and μ (A) > 0. According to (i), there is an h ∈ L_{∞} (λ) such that ∥hχ_{A}∥_{∞} > 0 and ∫_{A} hdμ = 0. Set g = h/f on A and g = 0 off A.

Then fg = h on A and so ∥fgχ_{A}∥_{∞} > 0. Also ∫_{Ω} fgdμ = ∫_{A} hdμ = 0.

Hence (iii) holds.

To check that (ii) implies (i), suppose (i) is false. Without loss of generality, we can assume that A = Ω, so that the operator f ↦ ∫_{A} fdμ (f ∈ L_{∞} (λ)) is one-to-one. It is not difficult to see that this

means that μ (Σ) = {∫_{Ω} χ_{A}dμ : A ∈ Σ} is a proper subset of the set

V = {∫_{Ω} fdμ : f ∈ L_{∞} (λ) , 0 ≤ f ≤ 1}. But by Corollary 1.2.6, V =

\overline{co} (μ (Σ)); thus μ (Σ) cannot be both closed and convex. Hence (ii) is false. To complete the proof, we shall verify that (iii) implies (ii) again. It is enough to show that μ (Σ) is convex and weakly compact. Let f ∈ L_{∞} (λ) be such that 0 ≤ f ≤ 1. Since by Corollary 1.2.4 the operator ∫_{Ω} (·) dμ is continuous for the weak^{∗} topology on L_{∞} (λ) and

the weak topology on X, the set

H = {g ∈ L_{∞} (λ) : 0 ≤ g ≤ 1, ∫_{Ω} fdμ = ∫_{Ω} gdμ}

is a weak^{∗} compact convex set in L_{∞} (λ) and therefore has extreme points. If we can show that the extreme points of H, denoted by ext (H), are all in {χ_{A} : A ∈ Σ} it will follow that there exists A ∈ Σ such that μ (A) = ∫_{Ω} fdμ. Then an appeal to Corollary 1.2.6 will yield the equalities

\overline{co} (μ (Σ)) = \{∫_{Ω} fdμ : f ∈ L_{∞} (λ) , 0 ≤ f ≤ 1\} = μ (Σ)

and prove that μ (Σ) is weakly compact and convex.

To this end, suppose f_{0} ∈ ext (H) but ∥f_{0} − χ_{A}∥ > 0 for each A ∈ Σ. A simple calculation shows that there exists f_{1} ∈ L_{∞} (λ) with ∥f_{1}∥_{∞} > 0 such that 0 ≤ f_{0} ± f_{1} ≤ 1. An appeal to (iii) gives us a g_{1} ∈ L_{∞} (λ), that may be selected with ∥g_{1}∥_{∞} ≤ 1 such that

∥f_{1}g_{1}∥_{∞} > 0 but ∫_{Ω} f_{1}g_{1}dμ = 0. Then f_{0} ± f_{1}g_{1} ∈ H; thus f_{0} is not an

extreme point of H. This completes the proof.

Note that in light of the remarks before Theorem 1.3.2 the classical Lyapunov theorem is contained in this theorem; in fact, the above proof of (iii) ⇒ (ii) in the case X = R^{n} is Lindenstrauss’s argument [23] to prove the Lyapunov theorem.

Theorem 1.3.3 (Lyapunov) Let Σ be a σ-field of subsets of Ω, X be a finite dimensional Banach space and μ : Σ → X be a countably additive vector measure. If μ is nonatomic, then the range of μ is a compact convex subset of X.

Corollary 1.3.4 Let Σ be the σ-field of Borel subsets of [0, 1]. If X is an infinite dimensional Banach space, then there is a countably additive vector measure of bounded variation μ : Σ → X and a set A ∈ Σ such that {μ (B ∩ A) : B ∈ Σ} is not a weakly compact convex set in X.

Proof. Let λ be Lebesgue measure on Σ. Select a sequence (f_{n}) in L_{1} (λ) such that ∥f_{n}∥_{1} = 1 and the only g ∈ L_{∞} (λ) with ∫_{[0,1]} f_{n}gdλ = 0

for all n is g = 0. Choose a sequence of pairs (x_{n}^{∗}, x_{n}) such that x_{n}^{∗} ∈ X^{∗}, x_{n} ∈ X, x_{m}^{∗} (x_{n}) = 0 if m ≠ n and x_{n}^{∗} (x_{n}) = 1 for all n. Define T : L_{∞} (λ) → X by

T (g) = ∑_{n=1}^{∞} x_{n} (2^{n} ∥x_{n}∥^{−1}) ∫_{[0,1]} f_{n}gdλ.

1.4. l_{p}-VALUED MEASURES 13

If T (g) = 0, then x_{n}^{∗}T (g) = (2^{n} ∥x_{n}∥^{−1}) ∫_{[0,1]} f_{n}gdλ = 0 for all n. Hence

T is one-to-one on L_{∞} (λ). To produce the advertised measure, define μ (A) = T (χ_{A}) for A ∈ Σ. It is not hard to see that μ is countably additive and T (.) = ∫_{[0,1]} (.) dμ. Hence μ violates (i) of Theorem 1.3.2.

By Theorem 1.3.2, μ is as advertised.

1.4 l_{p}-valued measures

The main results of this section are contained in [17].

Another way of generalizing the Lyapunov convexity theorem is considering the closure of the range of the vector measure in the norm topology.

Further by an X-valued vector measure we mean a countably additive vector measure μ defined on a σ-field Σ of subsets of a set Ω.

Definition 1.4.1 A nonatomic X-valued vector measure μ is said to be Lyapunov if the closure of its range is convex.

In 1969 Uhl [7] proved the following theorem.

Theorem 1.4.2 Let Σ be a σ-field of subsets of Ω and suppose X has the Radon-Nikodym property. If μ : Σ → X is a nonatomic and countably additive measure of bounded variation, then μ is a Lyapunov measure.

If we waive the requirement that the variation be bounded, then Uhl’s theorem no longer holds, in particular there exists a nonatomic measure with values in a Hilbert space such that the closure of its range is not convex. (It is enough to consider the measure μ_{2} from Example 1.1.2.)

In 1991 V. M. Kadets and M. M. Popov [16] obtained the following strengthening of this result.

Theorem 1.4.3 The following conditions are equivalent for the space X:

1. Each nonatomic X-valued measure of bounded variation is a Lyapunov measure;

2. There does not exist a sign-embedding of the space L_{1} [0, 1] in X.

Definition 1.4.4 A Banach space X is said to have the Lyapunov property if every nonatomic countably additive vector measure with values in X is a Lyapunov measure.

As already mentioned above, an infinite dimensional Hilbert space does not have the Lyapunov property. Consequently, all spaces containing isomorphic copies of the space l_{2} do not have the Lyapunov property; in particular, L_{p} [0, 1], C [0, 1], l_{∞} do not have the Lyapunov property. Nevertheless, spaces having the Lyapunov property exist. In 1992 V. M. Kadets and G. Schechtman proved that l_{p} (1 ≤ p < ∞, p ≠ 2) and c_{0} have the Lyapunov property [17]. Before proving this result we need some lemmas.

Lemma 1.4.5 If the measure μ is not a Lyapunov measure, then there exist a number ε > 0 and a set V ∈ Σ such that ∥μ (U) − \frac{1}{2}μ (V)∥ ≥ ε for any subset U ∈ Σ|_{V} (here and below, Σ|_{V} denotes the family of elements of the σ-algebra Σ that are subsets of the set V : the “restriction of Σ to V ”).

Proof. Assume the contrary, i.e., assume that for any ε > 0 and any V ∈ Σ there exists a U ⊂ V such that ∥μ (U) − \frac{1}{2}μ (V)∥ < ε. Let x and y be arbitrary elements of μ (Σ), x = μ (A), y = μ (B). We prove that (x + y) /2 ∈ \overline{μ (Σ)}. This means that μ is a Lyapunov measure, contrary to assumption. We choose U_{1}^{ε} ⊂ A \ B and U_{2}^{ε} ⊂ B \ A such that

∥\frac{1}{2}μ (A \ B) − μ (U_{1}^{ε})∥ < ε, ∥\frac{1}{2}μ (B \ A) − μ (U_{2}^{ε})∥ < ε.

Then ∥\frac{1}{2} (x + y) − μ (W^{ε})∥ ≤ 2ε for the set W^{ε} = U_{1}^{ε} ∪ U_{2}^{ε} ∪ (A ∩ B), i.e., (x + y) /2 can be approximated by values of the measure to any degree of accuracy. The lemma is proved.

Lemma 1.4.6 Suppose that λ is a nonatomic positive numerical measure on Σ, and μ : Σ → X is a vector valued measure that is absolutely continuous with respect to λ but is not a Lyapunov measure. Then there exist a number a > 0 and a subset Ω_{1} ∈ Σ with λ (Ω_{1}) > 0 having the following property : for any subsets A_{1}, A ∈ Σ|_{Ω_{1}}, A_{1} ⊂ A,

∥μ (A_{1}) − \frac{1}{2}μ (A)∥ ≥ aλ (A) .  (1.2)

Proof. Suppose that ε > 0 and V ∈ Σ are as in the statement of Lemma1.4.5. Take a = ε/λ (V) as the required number a. A set A ∈ Σ|_{V} is said to be a narrow if there exists a subset A_{1} ⊂ A such that the inequality converse to (1.2) holds. By construction, V is not a narrow set. It suffices for us to prove the existence of a set Ω_{1} ∈ Σ|_{V} with λ (Ω_{1}) > 0 that does not have narrow subsets. Assume the contrary, i.e., assume that the quantity

narr (A) = sup {λ (A_{1}) : A_{1} ∈ Σ|_{A}, A_{1} is narrow}

is positive for any A ∈ Σ|_{V} with λ (A) ≠ 0. We use induction to construct a chain V = V_{0} ⊃ V_{1} ⊃ V_{2} ⊃ ... of sets such that at each step the set W_{n} = V_{n} \ V_{n+1} is narrow and ”large”: λ (W_{n}) ≥ \frac{1}{2} narr (V_{n}). Then W = ⋃_{n=0}^{∞} V_{n} is a narrow set. Consequently, since V is not narrow, λ (Ω_{1}) ≠ 0 holds for the set Ω_{1} = V \ W = ⋂_{n=0}^{∞} V_{n}. At the same time, the sets W_{n} are disjoint and ∑_{n=1}^{∞} λ (W_{n}) = λ (W) < ∞; therefore, λ (W_{n}) → 0. We get that narr (Ω_{1}) ≤ narr (V_{n}) ≤ 2λ (W_{n}) → 0; i.e., narr (Ω_{1}) = 0, although λ (Ω_{1}) > 0. This contradiction proves the lemma.

Lemma 1.4.7 The following two conditions are equivalent for a Banach space X:

(i) X does not have the Lyapunov property.

(ii) There exist a triple (Ω, Σ, λ) with a nonatomic numerical measure λ on Σ and an operator T : L_{∞} (Ω, Σ, λ) → X such that

(a) T is continuous for the weak^{∗} topology on L_{∞} (Ω, Σ, λ) and the weak topology on X, and (b) there exists an ε > 0 such that ∥T f∥ ≥ ελ (supp f) for any “sign”

f ∈ L_{∞} (Ω, Σ, λ), i.e., any function taking only the values 0 and ±1.

Proof. We first show that (i) implies (ii). Let (Ω, Σ, λ) be a triple (Ω_{1}, Σ|_{Ω_{1}}, λ) as in Lemma 1.4.6, we get the existence of a measure μ : Σ → X and a number a > 0 such that the inequality (1.2) holds for any A_{1} ⊂ A ∈ Σ. According to Corollary 1.2.4, there exists a weak^{∗}- weak continuous operator T : L_{∞} (Ω, Σ, λ) → X connected with the measure μ by the relation

T (χ_{A}) = μ (A) ,  A ∈ Σ.  (1.3)

It remains to verify the condition (b). Suppose that f is a “sign”; then f = χ_{A_{1}} − χ_{A_{2}}, where A_{1}, A_{2} ∈ Σ and A_{1} ∩ A_{2} = Ø. Let A = A_{1} ∪ A_{2} and ε = 2a. We get

\begin{aligned}∥T f∥ & = ∥μ (A_{1}) − μ (A_{2})∥ = 2 ∥μ (A_{1}) − \frac{1}{2}μ (A)∥ \\ & ≥ 2aλ (A) = ε · λ (supp f) . \end{aligned}

(ii) implies (i). Suppose that the operator T satisfies the conditions (a) and (b). The measure μ is given by (1.3). It is easy to verify that μ is nonatomic and not a Lyapunov measure. The lemma is proved.

Remark 1.4.8 Since μ : Σ → X, μ (A) = T χ_{A} is absolutely continuous with respect to λ, the operator T will also have the following property:

(c) for every ε > 0 there is δ > 0 such that ∥T f∥ ≤ ε ∥f∥ for any f ∈ L_{∞} (Ω, Σ, λ) with λ (suppf) ≤ δ.

Let us introduce some additional symbols for the sequel. Let r_{i} = r_{i} (ω) be a sequence of independent random variables taking values +1 and −1 with probability \frac{1}{2}. Fix a number N ∈ N and denote the random variable

T (ω) = inf \{j : |∑_{i=1}^{j} r_{i} (ω)| ≥ \sqrt{N}\} .

Further define the stopped martingale ∑_{i=1}^{k} s_{i} by the rule

s_{i} (ω) = \{ \begin{matrix}r_{i} (ω) & if & i ≤ T (ω) \\ 0 & if & i > T (ω) \end{matrix}

The constructed martingale is evidently symmetric and its absolute value is bounded by \sqrt{N} + 1.

Lemma 1.4.9 Let k be a function of N. If N = o (k), then

lim_{N→∞} P (|∑_{i=1}^{k} s_{i}| < \sqrt{N}) = 0.

If k = o (N), then

lim_{N→∞} P (|∑_{i=1}^{k} s_{i}| < \sqrt{N}) = 1.

Proof. We have,

\begin{aligned}P (| ∑_{i=1}^{k} s_{i}| < \sqrt{N}) & = P ( max_{1≤j≤k} | ∑_{i=1}^{j} r_{i}| < \sqrt{N}) \\ & ≤ P (| ∑_{i=1}^{k} r_{i}| < \sqrt{N}) = P (|\frac{1}{\sqrt{k}} ∑_{i=1}^{k} r_{i}| < \sqrt{\frac{N}{k}} ) . \end{aligned}

By the central limit theorem, the last expression tends to zero if N = o (k). On the other hand, if k = o (N), then

P (|∑_{i=1}^{k} s_{i}| < \sqrt{N}) = 1 − P (|∑_{i=1}^{k} s_{i}| ≥ \sqrt{N}) ≥ 1 − e^{−N/2k} ⟶_{N→∞} 1.

The lemma is proved.

Lemma 1.4.10 Suppose that x, x_{n} ∈ l_{p}, 1 ≤ p < ∞ and x_{n} ⟶^{w} 0. Then for any ε > 0 there exists an index n ∈ N such that

∥x∥^{p} + ∥x_{n}∥^{p} − ε < ∥x + x_{n}∥^{p} < ∥x∥^{p} + ∥x_{n}∥^{p} + ε.

But if x, x_{n} ∈ c_{0}, then under the same assumptions it is possible to ensure that

max {∥x∥ , ∥x_{n}∥} − ε < ∥x + x_{n}∥ < max {∥x∥ , ∥x_{n}∥} + ε.

The method in the following theorem due to V. M. Kadets and G. Schechtman will often be used in the main results of the thesis.

Theorem 1.4.11 The spaces l_{p} with p ≠ 2, 1 ≤ p < ∞, and the space c_{0} have the Lyapunov property.

Proof. The case 1 ≤ p < 2. Fix N ∈ N for the present, and let k = [N/ ln N]. We assume that X = l_{p} fails the Lyapunov property, and we use Lemma 1.4.7. Let Ω, Σ, λ, ε, and T : L_{∞} → X be as in that lemma, with λ (Ω) = 1. We prove by induction on j that there exist functions {t_{j}}_{i=1}^{∞} ∈ L_{∞} (Ω, Σ, λ) such that for each j the functions {t_{i}}_{i=1}^{j} are jointly equidistributed with the {s_{i}}_{i=1}^{j} in Lemma 1.4.9 and

∥T (∑_{i=1}^{j} t_{i})∥^{p} > ∑_{i=1}^{j} ∥T (t_{i})∥^{p} − 1.  (1.4)

Indeed, everything is obvious for j = 1. Suppose that the assertion is already proved for j = m, and that {t_{i}}_{i=1}^{m} have been constructed that satisfy (1.4) with j = m. We consider the set A =

\{ω ∈ Ω : | ∑_{i=1}^{m} t_{i}| < \sqrt{N}\} and a sequence \{z_{n}\}_{n=1}^{∞} of functions indepen-

dent on A, equal to zero off A, and taking the values ±1 with probability λ (A) /2. For any n the system {{t_{i}}_{i=1}^{m} , z_{n}} is equidistributed with {s_{i}}_{i=1}^{m+1}. The sequence z_{n} tends to zero in the weak^{∗} topology; hence T z_{n} ⟶^{w} 0, and by Lemma 1.4.10 it is possible to choose n such that

∥T (∑_{i=1}^{m} t_{i}) + T (z_{n})∥^{p} > ∥∑_{i=1}^{m} T (t_{i})∥^{p} + ∥T (z_{n})∥^{p} − δ,

where δ is arbitrarily small. Choosing δ sufficiently small and letting t_{m+1} = z_{n}, we get the required inequality (1.4) with j = m + 1.

We now use the fact just proved. Suppose that {t_{i}}_{i=1}^{k} is equidistributed with {s_{i}}_{i=1}^{k} and subject to the condition (1.4) with j = k. Then

∥T ∥^{p} · ∥∑_{i=1}^{k} t_{i}∥^{p} ≥ ∥T (∑_{i=1}^{k} t_{i})∥^{p} ≥ ∑_{i=1}^{k} ∥T (t_{i})∥^{p} − 1.

In view of the condition (ii) of Lemma 1.4.7, which T satisfies by construction, we have ∥T (t_{i})∥ ≥ ε · λ (supp t_{i}) ≥ ε · λ (supp t_{k}) for i ≤ k.

Since {t_{i}}_{i=1}^{k} and {s_{i}}_{i=1}^{k} are equidistributed, this gives us by the choice of k that

\begin{matrix}(\sqrt{N} + 1)^{p} ∥T ∥^{p} + 1 ≥ ∑_{i=1}^{k} ∥T (t_{i})∥^{p} ≥ kε^{p} (P (s_{k} ≠ 0))^{p} ≥ \\ ≥ (N/ ln N − 1) · ε^{p} · (P (|∑_{i=1}^{k−1} s_{i}| < \sqrt{N}))^{p} . \end{matrix}

By Lemma 1.4.9, the last factor tends to 1; hence this inequality cannot hold for large N. This contradiction completes the analysis of the first case.

The case 2 < p < ∞. Assume that X = l_{p} fails the Lyapunov property. By analogy with the first case, we fix an N ∈ N, let k = [N ln N], and construct functions {t_{i}}_{i=1}^{k} equidistributed with {s_{i}}_{i=1}^{k} and satisfying the requirement

∥T (∑_{i=1}^{k} t_{i})∥^{p} < ∑_{i=1}^{k} ∥T (t_{i})∥^{p} + 1 ≤ ∥T ∥^{p} · k + 1.  (1.5)

We define the auxiliary function f_{N} by

f_{N} = \{ \begin{matrix}1 & if ∑_{i=1}^{k} t_{i} ≥ \sqrt{N} \\ −1 & if ∑_{i=1}^{k} t_{i} ≤ −\sqrt{N} \\ 0 & for the rest \end{matrix} .

and denote supp f_{N} by A. Since | ∑_{i=1}^{k} t_{i}| < \sqrt{N} + 1,

|f_{N} − \frac{1}{\sqrt{N}} ∑_{i=1}^{k} t_{i}| ≤ \frac{1}{\sqrt{N}} on A.

Consequently,

∥χ_{A} · (f_{N} − \frac{1}{\sqrt{N}} ∑_{i=1}^{k} t_{i})∥_{∞} ⟶_{N→∞} 0.  (1.6)

Further, according to Lemma 1.4.9, λ (Ω\A) → 0, and by Remark 1.4.8

∥T (χ_{Ω/A} · \frac{1}{\sqrt{N}} ∑_{i=1}^{k} t_{i})∥ ⟶_{N→∞} 0.  (1.7)

Comparing (1.6) and (1.7), we get that

∥T (\frac{1}{\sqrt{N}} ∑_{i=1}^{k} t_{i})∥ ≥ ∥T (f_{N})∥ + o (1) .  (1.8)

By (1.5) and the property (1.8) of T,

[ε · λ (A)]^{p} ≤ ∥T ∥^{p} · \frac{k}{(\sqrt{N})^{p}} + o (1) = o (1) .

This contradicts Lemma 1.4.9, which asserts that lim_{N→∞} λ (A) = 1.

The case X = c_{0} is analysed just like the preceding case. The theorem is proved.

There are simpler proofs for c_{0} and l_{1}, due to V. Kadets, which were never published before. We use the chance to present (with the kind permission of V. Kadets) these proofs too.

Proposition 1.4.12 c_{0} has the Lyapunov property.

Proof. Let μ : Σ → c_{0} be a nonatomic countably additive vector measure. According to Corollary 1.2.4 there is an operator T : L_{∞} (Ω, Σ, λ) → c_{0} such that μ (A) = T χ_{A} for all A ∈ Σ and which is continuous for the weak^{∗} topology on L_{∞} (Ω, Σ, λ) and the weak topology on c_{0}, where λ is a nonnegative nonatomic real-valued countably additive measure on Σ such that μ is absolutely continuous with respect to λ. By Lemma 1.4.5, it is sufficient to show that for every A and every ε > 0 there exists a set B ∈ Σ|_{A} such that

∥μ (B) − \frac{1}{2}μ (A)∥ ≤ ε.  (1.9)

Let {e_{i}}_{i=1}^{∞} be the unit vector basis of c_{0} and {f_{i}}_{i=1}^{∞} be the coordinate functionals on c_{0}. Then T x = ∑_{i=1}^{∞} f_{i} (T x) e_{i}. Denote T_{i}x = f_{i} (T x). Take an arbitrary A ∈ Σ and an ε > 0. Fix N ∈ N for the present. As μ is a nonatomic measure we can decompose A into the union of mutually disjoint sets A_{k} ∈ Σ|_{A} (k = 1, …, N) with μ (A_{k}) ≠ 0 for all k, λ (A_{k}) = λ (Ω) /N. Let r_{1}^{1} be a Rademacher function on A_{1}. Put

n_{1} = 1 and m_{0} = 0. Select m_{1} ∈ N such that ∥T r_{1}^{1} − ∑_{i=1}^{m_{1}} T_{i}r_{1}^{1}∥ ≤

\frac{1}{2^{N+1}} . Consider a sequence \{r_{n}^{2}\}_{n=1}^{∞} of Rademacher functions on A_{2}. By Corollary 1.2.5, T r_{n}^{2} ⟶_{n→∞} 0 in the weak topology. Hence there are a

number n_{2} such that ∥ ∑_{i=1}^{m_{1}} T_{i}r_{n_{2}}^{2}∥ ≤ \frac{1}{2^{N+3}} and a number m_{2} such that

∥ ∑_{i=m_{2}+1}^{∞} T_{i}r_{n_{2}}^{2}∥ ≤ \frac{1}{2^{N+3}} . Consequently

∥T r_{n_{2}}^{2} − ∑_{i=m_{1}+1}^{m_{2}} T_{i}r_{n_{2}}^{2}∥ ≤ \frac{1}{2^{N+2}} .

Continuing this process we obtain the sequences of numbers {n_{k}}_{k=1}^{N} and {m_{k}}_{k=0}^{N} and the set of functions {r_{n_{k}}^{k}}_{k=1}^{N} such that r_{n_{k}}^{k} is a Rademacher function on A_{k} and

∥T r_{n_{k}}^{k} − ∑_{i=m_{k−1}+1}^{m_{k}} T_{i}r_{n_{k}}^{k} ∥ ≤ \frac{1}{2^{N+k}}

for k = 1, …, N. Consequently

∥T (∑_{k=1}^{N} r_{n_{k}}^{k})∥ ≤ ∥∑_{k=1}^{N} ∑_{i=m_{k−1}+1}^{m_{k}} T_{i}r_{n_{k}}^{k} ∥+\frac{1}{2^{N}} = max_{1≤k≤N} ∥ ∑_{i=m_{k−1}+1}^{m_{k}} T_{i}r_{n_{k}}^{k} ∥+\frac{1}{2^{N}} .

If the maximum is reached at k_{0} then applying the equation (1.1) we have

∥T (∑_{k=1}^{N} r_{n_{k}}^{k})∥ ≤ ∥ ∑_{i=m_{k_{0}−1}+1}^{m_{k_{0}}} T_{i}r_{n_{k_{0}}}^{k_{0}} ∥+\frac{1}{2^{N}} ≤ ∥T |_{A_{k_{0}}} ∥+\frac{1}{2^{N}} ≤ ∥μ∥ (A_{k_{0}})+\frac{1}{2^{N}} .

As μ is absolutely continuous with respect to λ and λ (A_{k}) = λ (Ω) /N for k = 1, ..., N and by Proposition 1.1.6, the last sum tends to zero as N → ∞. Thus choosing N sufficiently large and taking B = ⋃_{k=1}^{N} {ω ∈ Ω : r_{n_{k}}^{k} = 1} we obtain the required inequality (1.9). The proposition is proved.

Before we prove that l_{1} has the Lyapunov property, let us recall that Schur has proved that in l_{1} the norm-convergence and the weakconvergence coincide. We will show that all Banach spaces, which have this property, have the Lyapunov property.

Definition 1.4.13 A Banach space X is said to have the Schur property if a sequence (x_{n}) of elements of X converges in the weak topology if and only if it converges in the norm topology.

Proposition 1.4.14 If a Banach space X has the Schur property, then it has the Lyapunov property.

Proof. Let μ be a nonatomic countably additive vector measure on a σ-field Σ and T : L_{∞} (λ) → X be the operator from Corollary 1.2.4. Take an arbitrary A ∈ Σ and consider on A the sequence of functions f_{n} = \frac{1+r_{n}}{2} , where (r_{n}) are Rademacher functions on A. As f_{n} ⟶_{n→∞} \frac{1}{2}χ_{A} in the weak^{∗} topology, then by Corollary 1.2.5 we have T f_{n} ⟶_{n→∞} \frac{1}{2}μ (A)

in the weak topology. It follows that ∥T f_{n} − \frac{1}{2}μ (A)∥ ⟶_{n→∞} 0. But f_{n} = χ_{A_{n}}, where A_{n} ∈ Σ. Thus we have

∥μ (A_{n}) − \frac{1}{2}μ (A)∥ ⟶_{n→∞} 0.

By Lemma 1.4.5, the proposition is proved.

1.5 The three-space problem

The results of this section have been published in [18], [5].

The “three-space problem” arises for every Banach space property. Namely, do a subspace Y and the quotient space X/Y have the property if X does and does X have the property if Y and X/Y do? Some of these problems can be solved easily in the case of the Lyapunov property. Indeed, if X has the Lyapunov property and Y ⊂ X , then Y has the Lyapunov property, but X/Y needs not have the Lyapunov property (for instance, X = l_{1} because the set of its quotient spaces contains all separable Banach spaces [24, p.108]). At the same time the answer to the last problem runs into difficulties. The purpose of this section is the positive solution of the above mentioned problem : let Y be a subspace of a Banach space X. If Y and X/Y have the Lyapunov property, then X has the Lyapunov property.

The following three lemmas are of technical nature.

Lemma 1.5.1 Let X have the Lyapunov property and μ : Σ → X be a nonatomic measure. Then there is a nonatomic nonnegative measure λ : Σ → R such that for every A ∈ Σ and n ∈ N there exists B_{n} ∈ Σ|_{A} satisfying the following inequalities:

∥μ (B_{n}) − \frac{1}{2}μ (A)∥ ≤ \frac{1}{2^{n}} and |λ (B_{n}) − \frac{1}{2}λ (A)| ≤ \frac{1}{2^{n}} .  (1.10)

Proof. By Theorem 1.2.7 there is a functional x^{∗} ∈ X^{∗} with ∥x^{∗}∥ = 1 for which μ ≪ |x^{∗}μ|. Put λ = |x^{∗}μ|. Let A ∈ Σ. In accordance with the Hahn decomposition theorem let us denote by Ω^{+}and Ω^{−}the positivity and negativity sets for x^{∗}μ respectively. Then λ (C) = x^{∗}μ (C ⋂ Ω^{+}) − x^{∗}μ (C ⋂ Ω^{−}) for any C ∈ Σ. Put A^{+} = A ⋂ Ω^{+}and A^{−} = A ⋂ Ω^{−}. Since X has the Lyapunov property, we can choose B_{n}^{+} ∈ Σ|_{A^{+}} and B_{n}^{−} ∈ Σ|_{A^{−}}such that

∥μ (B_{n}^{+}) − \frac{1}{2}μ (A^{+})∥ ≤ \frac{1}{2^{n+1}} and ∥μ (B_{n}^{−}) − \frac{1}{2}μ (A^{−})∥ ≤ \frac{1}{2^{n+1}} .

Then

|x^{∗}μ (B_{n}^{+}) − \frac{1}{2}x^{∗}μ (A^{+})| ≤ \frac{1}{2^{n+1}} and |x^{∗}μ (B_{n}^{−}) − \frac{1}{2}x^{∗}μ (A^{−})| ≤ \frac{1}{2^{n+1}} .

Define B_{n} = B_{n}^{+} ⋃ B_{n}^{−}. It is easy to check that for B_{n} the inequalities (1.10) are true.

Lemma 1.5.2 Let X have the Lyapunov property, μ : Σ → X be a nonatomic measure, λ be the measure from Lemma 1.5.1. Then for every A ∈ Σ with λ (A) ≠ 0 and ε > 0 there exist G′ ∈ Σ|_{A}, G′′ = A\G′such that

(i) λ (G′) = λ (G′′) = λ (A) /2,

(ii) ∥μ (G′) − \frac{1}{2}μ (A)∥ < ε.

Proof. Let us choose B_{n} as in Lemma 1.5.1. We can select C_{n} ∈ Σ|_{B_{n}} if λ (B_{n}) ≥ \frac{1}{2}λ (A) or C_{n} ∈ Σ|_{A\B_{n}}if λ (B_{n}) < \frac{1}{2}λ (A) for which λ (C_{n}) = |\frac{1}{2}λ (A) − λ (B_{n})| (because λ is a nonatomic real-valued measure). By (1.10), λ (C_{n}) →_{n→∞} 0 . Put Gn^{′} = B_{n} △C_{n}, Gn^{′′} = A\Gn^{′}. Then

we obtain λ (Gn^{′}) = λ (Gn^{′′}) = \frac{1}{2}λ (A) , and λ (Gn^{′}\B_{n}) →_{n→∞} 0, and

λ (B_{n}\Gn^{′}) →_{n→∞} 0. Since μ is absolutely continuous with respect to

λ the last condition implies that μ (B_{n}\Gn^{′}) →_{n→∞} 0 and μ (Gn^{′}\B_{n}) →_{n→∞} 0. Together with inequality (1.10) this gives us

∥μ (Gn^{′}) − \frac{1}{2}μ (A)∥ →_{n→∞} 0.

So for sufficiently large n the sets G′ = Gn^{′} and G′′ = Gn^{′′} will satisfy the conditions (i) and (ii).

Lemma 1.5.3 Under the conditions of Lemma 1.5.2 for every A ∈ Σ with λ (A) ≠ 0 and ε > 0 there exists a σ-algebra Σ′ ⊂ Σ|_{A} such that for every B ∈ Σ′ we have

∥μ (B) − λ (B) \frac{μ (A)}{λ (A)}∥ ≤ ελ (B)  (1.11)

and the measure λ is nonatomic on Σ′.

Proof. Take A ∈ Σ and ε > 0. Employing Lemma 1.5.2, we choose sets A_{1} ∈ Σ|_{A}, A_{2} = A\A_{1} with λ (A_{1}) = λ (A_{2}) = \frac{1}{2}λ (A) and ∥μ (A_{1}) − \frac{1}{2}μ (A)∥ ≤ \frac{1}{4}ε (note that

∥μ (A_{1}) − \frac{1}{2}μ (A)∥ = \frac{1}{2} ∥μ (A_{1}) − μ (A_{2})∥ = ∥μ (A_{2}) − \frac{1}{2}μ (A)∥ ≤ \frac{1}{4}ε).

Employing Lemma 1.5.2 twice (for A = A_{1} and A = A_{2}) we obtain A_{1,1} ∈ Σ|_{A_{1}}, A_{1,2} = A_{1}\A_{1,1}; A_{2,1} ∈ Σ|_{A_{2}}, A_{2,2} = A_{2}\A_{2,1} with λ (A_{1,1}) = λ (A_{1,2}) = λ (A_{2,1}) = λ (A_{2,2}) = \frac{1}{4}λ (A) and

∥μ (A_{1,1}) − \frac{1}{2}μ (A_{1})∥ ≤ \frac{ϵ}{16}, ∥μ (A_{2,1}) − \frac{1}{2}μ (A_{2})∥ ≤ \frac{ϵ}{16}.

When we continue this process, we obtain a tree of sets A_{i_{1},i_{2},...,i_{n}}, i_{k} ∈ {1, 2} , n ∈ N with

\begin{matrix}A_{i_{1},i_{2},...,i_{n+1}} ⊂ A_{i_{1},i_{2},...,i_{n}}, A_{i_{1},i_{2},...,i_{n},2} = A_{i_{1},i_{2},...,i_{n}}\A_{i_{1},i_{2},...,i_{n},1,} \\ ∥μ (A_{i_{1},i_{2},...,i_{n}}) − \frac{1}{2}μ (A_{i_{1},i_{2},...,i_{n−1}})∥ ≤ \frac{1}{4^{n}} ε, λ (A_{i_{1},i_{2},...,i_{n}}) = \frac{1}{2^{n}} λ (A) \end{matrix}

Let Σ′ be a σ-algebra generated by the sets A_{i_{1},i_{2},...,i_{n}}. We are going to show that the algebra Σ′ has the required property.

Let B = A_{i_{1},i_{2},...,i_{n}}. Then

∥μ (B) − \frac{λ(B)}{λ(A)}μ (A)∥ = ∥μ (A_{i_{1},i_{2},...,i_{n}}) − \frac{1}{2^{n}} μ (A)∥
≤ ∥μ (A_{i_{1},i_{2},...,i_{n}}) − \frac{1}{2}μ (A_{i_{1},i_{2},...,i_{n−1}})∥

+\frac{1}{2} ∥μ (A_{i_{1},i_{2},...,i_{n−1}}) − \frac{1}{2}μ (A_{i_{1},i_{2},...,i_{n−2}})∥ + ...

+\frac{1}{2^{n−1}} ∥μ (A_{i_{1}}) − \frac{1}{2}μ (A)∥
≤ \frac{1}{4^{n}} ε + \frac{1}{2} \frac{1}{4^{n−1}} ε + ... + \frac{1}{2^{n−1}} \frac{1}{4}ε
= ε (\frac{1}{2^{2n}} + \frac{1}{2^{2(n−1)}} + ... + \frac{1}{2^{n+1}} ) ≤ \frac{1}{2^{n}} ε
= ελ (A_{i_{1},i_{2},...,i_{n}}) .

Hence by the triangle inequality and the σ-additivity of μ and λ we get (1.11) for B = ⋃_{n=1}^{∞} B_{n}, where the B_{n} are disjoint sets of the form

A_{i_{1},i_{2},...,i_{k}}. Then we obtain (1.11) for every B ∈ Σ′ using approximation of λ (B) by bigger sets of the form B = ⋃_{n=1}^{∞} B_{n}.

Lemma 1.5.4 The statements of Lemmas 1.5.2 and 1.5.3 are valid for an arbitrary nonatomic measure λ.

Proof. Let μ : Σ → X, λ : Σ → R_{+} be nonatomic measures. Let ν be the measure which played the role of λ in Lemma 1.5.1. Consider two cases.

Case 1: λ ≪ ν.Take A ∈ Σ, n ∈ N. In view of Lemma 1.5.3, there is Σ′ ⊂ Σ|_{A} such that for any B ∈ Σ′

∥μ (B) − \frac{ν (B)}{ν (A)}μ (A)∥ ≤ \frac{1}{2^{n}} ν (B) ,  (1.12)

ν is a nonatomic measure with respect to Σ′. Applying the Lyapunov theorem to the measure σ : Σ′ → R^{2} : σ (A) = (ν (A) , λ (A)), we obtain sets G′, G′′ ∈ Σ′ such that λ (G′) = λ (G′′) = \frac{1}{2}λ (A) and

ν (G′) = ν (G′′) = \frac{1}{2}ν (A). Then (1.12) implies inequality (ii) which we need.

Case 2: λ ≪̸ ν. We decompose λ into the sum of absolutely continuous and strictly singular measures with respect to ν : λ = λ_{1} + λ_{2}. Then λ_{2} is concentrated on a ν-negligible set S. Now we consider A\S and S ⋂ A separately. By the case 1 chose G1^{′} ⊂ A\S so that

λ_{1} (G1^{′}) = \frac{1}{2}λ_{1} (A) , ∥μ (G1^{′}) − \frac{1}{2}μ (A)∥ ≤ \frac{1}{2^{n}} and G2^{′} ⊂ S ⋂ A with

λ_{2} (G2^{′}) = \frac{1}{2}λ (S ⋂ A). Because λ_{1} (G2^{′}) = 0, μ (G2^{′}) = 0 and

λ_{2} (G1^{′}) = 0 it is clear that G′ = G1^{′} ⋃ G2^{′} satisfies (ii). The proof of the Lemma 1.5.3 shows that if Lemma 1.5.2 is valid for an arbitrary λ then Lemma 1.5.3 is valid too for an arbitrary nonatomic measure λ.

The following statement is evident.

Lemma 1.5.5 If X is a Banach space, Y is a subspace of X, μ : Σ → X is a nonatomic measure, then μ̄ : Σ → X/Y (μ̄ (A)-the equivalence class of μ (A)) is a nonatomic measure too.

Theorem 1.5.6 Let Y be a subspace of a Banach space X. If Y and X/Y have the Lyapunov property, then X has the Lyapunov property too.

Proof. Let μ : Σ → X, λ : Σ → R_{+} be nonatomic measures, λ (Ω) = 1. Fix A ∈ Σ, λ (A) ≠ 0, and ε > 0. By Lemmas 1.5.4 and 1.5.5 there is a σ-algebra Σ′ ⊂ Σ|_{A} such that

∥μ (B) − \frac{λ (B)}{λ (A)}μ̄ (A)∥ < \frac{1}{2}ελ (B)  (1.13)

for all B ∈ Σ′ and λ is nonatomic on Σ′. Define σ : Σ′ → X by the rule σ (B) = μ (B) − \frac{λ(B)}{λ(A)}μ (A) . Now we show that there exist a σ-algebra Σ̃ ⊂ Σ′ and a nonatomic measure β : Σ̃ → Y such that

∥σ (B) − β (B)∥ < 2ε

for all B ∈ Σ̃. Indeed, by inequality (1.13) and nonatomicity of λ we can choose sets A_{1,}A_{2} ∈ Σ′ such that A = A_{1} ⋃ A_{2}, λ (A_{1}) = λ (A_{2}) = \frac{1}{2}λ (A) , and ∥σ (A_{1})∥ < ε\frac{1}{2}λ (A). Clearly, there is x ∈σ (A_{1}) such that ∥x∥ ≤ ε\frac{1}{2}λ (A) . Denote

α (A_{1}) = x, α (A_{2}) = −x,

Applying step by step Lemma 1.5.2 and Lemma 1.5.4, we get sets A_{i_{1},...,i_{k}} ∈ Σ′, k = 2, 3, ...; i_{1}, ..., i_{k} = 1, 2, such that A_{i_{1},...,i_{k−1}} = A_{i_{1},...,i_{k},1} ⋃ A_{i_{1},...,i_{k},2}, λ (A_{i_{1},...,i_{k}}) = \frac{1}{2}λ (A_{i_{1},...,i_{k−1}}) = \frac{1}{2^{k}} λ (A) , and

∥σ (A_{i_{1},...,i_{k−1},1}) − \frac{1}{2} σ (A_{i_{1},...,i_{k−1}})∥ < ε\frac{1}{2^{2k}} λ (A) . It is readily seen that

in every equivalence class σ (A_{i_{1},...,i_{k−1},1}) − \frac{1}{2} σ (A_{i_{1},...,i_{k−1}}) ⊂ X there

exists an element x_{i_{1},...,i_{k−1}} such that ∥x_{i_{1},...,i_{k−1}} ∥ ≤ ε\frac{1}{2^{2k}} λ (A). Put

\begin{matrix}α (A_{i_{1},...,i_{k−1},1}) = \frac{1}{2}α (A_{i_{1},...,i_{k−1}}) + x_{i_{1},...,i_{k−1}}, \\ α (A_{i_{1},...,i_{k−1},2}) = \frac{1}{2}α (A_{i_{1},...,i_{k−1}}) − x_{i_{1},...,i_{k−1}}. \end{matrix}

Evidently, α (A_{i_{1},...,i_{k−1},1}) ∈ σ̄ (A_{i_{1},...,i_{k−1},1}) and α (A_{i_{1},...,i_{k−1},2}) ∈

σ̄ (A_{i_{1},...,i_{k−1},2}) . By Σ̃ denote σ-algebra generated by the sets A_{i_{1},...,i_{k}}.

Iterating the inequality ∥α (A_{i_{1},...,i_{k}})∥ ≤ \frac{1}{2} ∥α (A_{i_{1},...,i_{k−1}})∥ + ε\frac{1}{2^{2k}} λ (A), we obtain ∥α (A_{i_{1},...,i_{k}})∥ ≤ ε\frac{1}{2^{k−1}} λ (A) = 2ελ (A_{i_{1},...,i_{k}}) . Let us extend α to Σ̃ and show that

∥α (B)∥ ≤ 2ελ (B)  (1.14)

for any B ∈ Σ̃. For this purpose we take B = ⋃_{I} A_{i_{1},...,i_{k}}, where I

is a finite set of indices and A_{i_{1},...,i_{k}} are mutually disjoint sets. Put

α (B) = ∑_{I} α (A_{i_{1},...,i_{k}}) . Clearly,

∥α (B)∥ ≤ ∑_{I} ∥α (A_{i_{1},...,i_{k}})∥ ≤ 2ε ∑_{I} λ (A_{i_{1},...,i_{k}}) = 2ελ (B) .

This proves that inequality (1.14) is valid for elements of the algebra S, generated by the sets A_{i_{1},...,i_{k}}. Now applying the Kluvanek-Uhl extension theorem [6, Chapter 6, Section 8, Addition 4] we obtain an extension of α to Σ̃. Thus, we have constructed a measure α : Σ̃ → X such that α (B) ∈ \overline{σ} (B) for any B ∈ Σ̃ and ∥α (B)∥ ≤ 2ελ (B) . α is a nonatomic measure because λ is nonatomic by construction. It is clear that β = α − σ has the required property. The statement is proved.

Let us complete the proof of the theorem. Since β : Σ̃ → Y is a nonatomic measure and Y has the Lyapunov property, we see that by Lemma 1.5.2 there is B ∈ Σ̃ such that λ (B) = \frac{1}{2}λ (A) and

∥β (B) − \frac{1}{2}β (A)∥ ≤ ε.

Thus we have

\begin{aligned}∥σ (B) − \frac{1}{2}σ (A)∥ & ≤ ∥σ (B) − β (B)∥ + ∥\frac{1}{2}σ (A) − \frac{1}{2}β (A)∥ \\ & + ∥β (B) − \frac{1}{2}β (A)∥ ≤ 3ε. \end{aligned}

Since σ (B) − \frac{1}{2}σ (A) = μ (B) − \frac{1}{2}μ (A) , the proof is completed. Let us stress an important particular case of this theorem.

Corollary 1.5.7 Let X = X_{1} ⊕ X_{2} where X_{1}, X_{2} have the Lyapunov property. Then X has the Lyapunov property too.

It is clear that by induction we can extend this property to any finite sum of Banach spaces.

Corollary 1.5.8 The spaces X_{p} from the work [19] (“twisted sums” of l_{p} spaces) have the Lyapunov property for p ≠ 2.

Lyapunov Trees

In this chapter we introduce a generalization of the well known notions of type and cotype, b- and C-convexity for a special type of trees. Applying the methods known from the theory of type and cotype (see [15]) we will establish the connection between generalizations of type and b-convexity, cotype and C-convexity. From these properties we will see that if in a Banach space there exists any norm restriction from above (below) on weakly null sequences then there exists a ”p-type estimation” from above (below), which is necessary for the proving of the Lyapunov property. The results obtained in this chapter will be used in Chapter 3.

First we introduce the notion of Lyapunov convergence. It will be convenient for proving a generalization of the Lyapunov theorem to some Banach spaces that will be considered in the next chapter.

Definition 2.0.1 Let X be a Banach space, T̃ ∈ L (X, c_{0}). By the c_{0}-T̃-weak topology on X we will mean the topology with the following determining system of neighborhoods of zero:

{x ∈ X : ∥T̃x∥_{∞} < ε, |f_{i} (x)| < ε, i = 1, ..., n} ,

where ε > 0, n ∈ N, and f_{i} ∈ X^{∗} (i = 1, ..., n).

It is not difficult to see that this topology determines the convergence: x_{n} converges to zero if and only if x_{n} ⟶_{n→∞}^{w} 0 and ∥T̃x_{n}∥_{∞} ⟶_{n→∞} 0.

Denote this convergence by c_{0}-T̃-weak convergence.

Definition 2.0.2 Let X be a Banach space. By a Lyapunov topology on X we will mean the weak topology or a c_{0}–T̃-weak topology on X.

In the sequel we fix once and for all a Lyapunov topology τ.

Definition 2.0.3 A Lyapunov tree X^{n} of length n in a Banach space X is a family of sequences {x_{m_{1},...,m_{i}}}_{i=1 m_{i}∈N}^{n} ⊂ X such that for given 0 ≤ i_{i} < i_{2} ≤ n for every fixed m_{1}, ..., m_{i_{1}}

x_{m_{1},...,m_{i_{1}},m_{i_{1}+1},...,m_{i_{2}}} ⟶_{m_{i_{1}+1},...,m_{i_{2}} →∞}^{τ} 0.

2.1 Lyapunov tree cotype

Definition 2.1.1 We will say that a Banach space X has Lyapunov tree cotype p with constant C if for every n ∈ N and for every Lyapunov tree X^{n} there exist numbers m_{1}^{0}, ..., m_{n}^{0} such that

∥∑_{i=1}^{n} x_{m_{1}^{0},...,m_{i}^{0}} ∥ ≥ C (∑_{i=1}^{n} inf_{m_{1},...,m_{i}} ∥x_{m_{1},...,m_{i}}∥^{p})^{1/p} .

Example 2.1.2 By Lemma 1.4.10 for every 1 ≤ p < ∞ the space l_{p} has Lyapunov tree cotype p with constant 1 for the weak topology.

Definition 2.1.3 A Banach space X is called Lyapunov tree C-convex if there exists n ∈ N such that

C^{τ} (n, X) = inf { sup_{m_{1},...,m_{n}} ∥∑_{i=1}^{n} x_{m_{1},...,m_{i}} ∥} > 1,

where the infimum is taken over all Lyapunov trees of length n with ∥x_{m_{1},...,m_{i}}∥ ≥ 1. (Recall that τ denotes a Lyapunov topology.)

Remark 2.1.4 If ∥x_{m_{1},...,m_{i}}∥ ≥ α for all elements of a Lyapunov tree X^{n}, then we have

sup_{m_{1},...,m_{n}} ∥∑_{i=1}^{n} x_{m_{1},...,m_{i}} ∥ ≥ α · C^{τ} (n, X) .

Lemma 2.1.5 The numbers C^{τ} (n, X) generate a semimultiplicative sequence, i.e.

C^{τ} (n · l, X) ≥ C^{τ} (n, X) · C^{τ} (l, X) .

Proof. Let us consider a Lyapunov tree {x_{m_{1},...,m_{i}}}_{i=1 m_{i}∈N}^{n·l} with ∥x_{m_{1},...,m_{i}}∥ ≥ 1 and take an arbitrary δ > 0. Introduce the Lyapunov tree of length n

{x_{m_{1},...,m_{i}}}_{i=1 m_{i}∈N}^{n} .

In accordance with the definition of C^{τ} (n, X) we can choose numbers

m_{1}^{1}, ..., m_{n}^{1} such that ∥ ∑_{i=1}^{n} x_{m_{1}^{1},...,m_{i}^{1}} ∥ ≥ C^{τ} (n, X) − δ. Denote y_{1} =

∑_{i=1}^{n} x_{m_{1}^{1},...,m_{i}^{1}}. Further, consider the Lyapunov tree of length n

{x_{m_{1},...,m_{i}}}_{i=1 m_{i}=m_{i}^{1}+1}^{n ∞}

and select m_{1}^{2}, ..., m_{n}^{2} such that ∥ ∑_{i=1}^{n} x_{m_{1}^{2},...,m_{i}^{2}} ∥ ≥ C^{τ} (n, X) − δ. Put

y_{2} = ∑_{i=1}^{n} x_{m_{1}^{2},...,m_{i}^{2}}. Continuing this process, we obtain the sequence

y_{s_{1}} = ∑_{i=1}^{n} x_{m_{1}^{s_{1}},...,m_{i}^{s_{1}}} such that

∥y_{s_{1}}∥ ≥ C^{τ} (n, X) − δ for all s_{1} and y_{s_{1}} ⟶_{s_{1}→∞}^{τ} 0.

For every s_{1} we consider the Lyapunov tree

{x_{m_{1}^{s_{1}},...,m_{n}^{s_{1}}, m_{n+1},...,m_{i}}}_{i=n+1 m_{i}∈N}^{2n} .

By analogy with above we construct a sequence

y_{s_{1},s_{2}} = ∑_{i=n+1}^{2n} x_{m_{1}^{s_{1}},...,m_{n}^{s_{1}}, m_{n+1}^{s_{2}},...,m_{i}^{s_{2}}} (s_{2} ∈ N)

such that

∥y_{s_{1},s_{2}}∥ ≥ C^{τ} (n, X) − δ for all s_{2} and y_{s_{1},s_{2}} ⟶_{s_{2}→∞}^{τ} 0.

It is not difficult to see that y_{s_{1},s_{2}} ⟶_{s_{1},s_{2}→∞}^{τ} 0. Continuing this process, we get the Lyapunov tree

{y_{s_{1},...,s_{i}}}_{i=1 s_{i}∈N}^{l}

satisfying the condition

∥y_{s_{1},...,s_{i}}∥ ≥ C^{τ} (n, X) − δ.

Then, in accordance with the Remark 2.1.4, we infer

sup_{s_{1},...,s_{l}} ∥∑_{i=1}^{l} y_{s_{1},...,s_{i}} ∥ ≥ (C^{τ} (n, X) − δ) · C^{τ} (l, X) .

As δ is arbitrary, the required inequality is proved.

Lemma 2.1.6 Let X^{n} be a Lyapunov tree of a Banach space X. Then for every k = 1, ..., n the following inequality holds

sup_{m_{1},...,m_{n}} ∥∑_{i=1}^{n} x_{m_{1},...,m_{i}} ∥ ≥ sup_{m_{1},...,m_{k}} ∥∑_{i=1}^{k} x_{m_{1},...,m_{i}} ∥ .

Proof. Take k ∈ {1, ..., n} and an arbitrary set of indices m_{1}^{0}, ..., m_{k}^{0}. Applying Mazur’s theorem (which states that if a_{i}, a ∈ X and a_{i} ⟶_{i→∞}^{τ} a then \underline{lim}∥a_{i}∥ ≥ ∥a∥) we obtain

sup_{m_{1},...,m_{n}} ∥∑_{i=1}^{n} x_{m_{1},...,m_{i}} ∥
≥ sup_{m_{k+1},...,m_{n}} ∥∑_{i=1}^{k} x_{m_{1}^{0},...,m_{i}^{0}} + ∑_{i=k+1}^{n} x_{m_{1}^{0},...,m_{k}^{0},m_{k+1},...,m_{i}} ∥
≥ \underline{lim}_{m_{k+1},...,m_{n}→∞} ∥∑_{i=1}^{k} x_{m_{1}^{0},...,m_{i}^{0}} + ∑_{i=k+1}^{n} x_{m_{1}^{0},...,m_{k}^{0},m_{k+1},...,m_{i}} ∥
≥ ∥∑_{i=1}^{k} x_{m_{1}^{0},...,m_{i}^{0}} ∥ .

This completes the proof.

Lemma 2.1.7 Let X be a Banach space, {x_{n}}_{n=1}^{∞}, x ∈ X, and x_{n} ⟶_{n→∞}^{τ} 0. Then for every ε > 0 there exists an n ∈ N such that

∥x + x_{n}∥ ≥ \frac{1}{2} ∥x_{n}∥ − ε.

Proof. Let ε > 0. Take a functional f ∈ X^{∗} such that ∥f∥ = 1 and f (x) = ∥x∥. Choose n ∈ N such that f (x_{n}) ≤ 2ε, then

∥x + x_{n}∥ ≥ |f (x + x_{n})| ≥ ∥x∥ − 2ε.

It follows that

∥x_{n}∥ ≤ ∥x + x_{n}∥ + ∥x∥ ≤ 2 ∥x + x_{n}∥ + 2ε.

The lemma is proved.

Theorem 2.1.8 Let X be a Banach space. If there exists some n_{1} such that C^{τ} (n_{1}, X) > n_{1}^{1/p}, where 1 ≤ p < ∞, then X has Lyapunov tree cotype p.

Proof. Let us choose ε > 0 such that C^{τ} (n_{1}, X) ≥ n_{1}^{1/(p−ε)}. Put n_{k} = n_{1}^{k} for k = 0, 1, 2.... By Lemma 2.1.5, C^{τ} (n_{k}, X) ≥ n_{1}^{k/(p−ε)}. Denote D = ∑_{k=0}^{∞} n_{1}^{−εk}. We will show that X has Lyapunov tree cotype

p with the constant \frac{1}{2}D^{−1/(p−ε).} Take an arbitrary Lyapunov tree X^{n}. Define

α_{i} = inf_{m_{1},...,m_{i}} ∥x_{m_{1},...,m_{i}}∥ (i=1, ..., n) .

Decompose the set of indices {1, ..., n} into the union of mutually disjoint sets

A_{k} = \{j : \frac{( ∑_{i=1}^{n} α_{i}^{p})^{1/p}}{n_{k}} ≥ α_{j} > \frac{( ∑_{i=1}^{n} α_{i}^{p})^{1/p}}{n_{k+1}} \} ,

where k = 0, 1, 2, .... Define a_{k} as the number of elements of A_{k}. Then

∑_{j=1}^{n} α_{j}^{p} = ∑_{k=0}^{∞} ∑_{j∈A_{k}} α_{j}^{p} ≤ ∑_{k=0}^{∞} a_{k} · \frac{∑_{i=1}^{n} α_{i}^{p}}{n_{k}^{p}} .

34 CHAPTER 2. LYAPUNOV TREES Consequently

∑_{k=0}^{∞} \frac{a_{k}}{n_{k}^{p}} ≥ 1.

It follows that there exists a number k = k_{0} such that a_{k_{0}} ≥ [^{\frac{n_{k_{0}}^{p−ε}}{D}} ]. Indeed, if a_{k} < [\frac{n_{k}^{p−ε}}{D} ] for all k, then

\frac{a_{k}}{n_{k}^{p}} < \frac{n_{k}^{p−ε}}{D · n_{k}^{p}} = \frac{1}{D · n_{1}^{εk}} and ∑_{k=0}^{∞} \frac{a_{k}}{n_{k}^{p}} < 1.

Fix θ > 0 and construct a tree of length a_{k_{0}} as follows. Reindex the elements of the set A_{k_{0}} = {j_{1} < ... < j_{a_{k_{0}}} }. Fix arbitrary j_{1} −1 indices: m_{1}^{1}, ..., m_{j_{1}−1}^{1}. By Lemma 2.1.7 we can choose a number m_{j_{1}}^{1} such that

∥∑_{i=1}^{j_{1}} x_{m_{1}^{1},...,m_{i}^{1}} ∥ ≥ \frac{1}{2} ∥x_{m_{1}^{1},...,m_{j_{1}}^{1}} ∥ − θ.

Put y_{1} = ∑_{i=1}^{j_{1}} x_{m_{1}^{1},...,m_{i}^{1}}. Take m_{1}^{2}, ..., m_{j_{1}−1}^{2} satisfying the condition m_{1}^{2} > m_{1}^{2}. Using Lemma 2.1.7, we find a number m_{j_{1}}^{2} such that

∥∑_{i=1}^{j_{1}} x_{m_{1}^{2},...,m_{i}^{2}} ∥ ≥ \frac{1}{2} ∥x_{m_{1}^{2},...,m_{j_{1}}^{2}} ∥ − θ.

Denote y_{2} = ∑_{i=1}^{j_{1}} x_{m_{1}^{2},...,m_{i}^{2}}. Continuing this process we obtain the sequence {y_{s_{1}}}_{s_{1}=1}^{∞} such that y_{s_{1}} ⟶_{s_{1}→∞}^{τ} 0. For each s_{1} we fix arbitrary m_{j_{1}+1}^{1}, ..., m_{j_{2}−1}^{1} and select m_{j_{2}}^{1} so that

∥ ∑_{i=j_{1}+1}^{j_{2}} x_{m_{1}^{s_{1}},...,m_{j_{1}}^{s_{1}},m_{j_{1}+1}^{1},...,m_{i}^{1}} ∥ ≥ \frac{1}{2} ∥x_{m_{1}^{s_{1}},...,m_{j_{1}}^{s_{1}},m_{j_{1}+1}^{1},...,m_{j_{2}}^{1}} ∥ − θ.

Define y_{s_{1},1} = ∑_{i=j_{1}+1}^{j_{2}} x_{m_{1}^{s_{1}},...,m_{j_{1}}^{s_{1}},m_{j_{1}+1}^{1},...,m_{i}^{1}}. By the same method as above we get the sequence y_{s_{1},s_{2}} ⟶_{s_{2}→∞}^{τ} 0. It is clear that y_{s_{1},s_{2}} ⟶_{s_{1},s_{2}→∞}^{τ} 0. Continuing this process we construct the Lyapunov

tree {y_{s_{1},...,s_{i}}}_{i=1 s_{i}∈N}^{a_{k_{0}}}. Applying Lemma 2.1.6, we have

sup_{m_{1},...,m_{n}} ∥ ∑_{i=1}^{n} x_{m_{1},...,m_{i}} ∥ ≥ sup_{m_{1},...,m_{ia_{k_{0}}}} ∥ ∑_{i=1}^{i_{a_{k_{0}}}} x_{m_{1},...,m_{i}} ∥

≥ sup_{s_{1},...,s_{a_{k_{0}}}} ∥ ∑_{i=1}^{a_{k_{0}}} y_{s_{1},...,s_{i}} ∥

≥ inf_{s_{1},...,s_{a_{k_{0}}}} ∥y_{s_{1},...,s_{i}}∥ · C^{τ} (a_{k_{0}}, X)

≥ (\frac{1}{2} min_{j∈A_{k_{0}}} α_{j} − θ) · C^{τ} ([^{\frac{n_{k_{0}}^{p−ε}}{D}} ] , X) .

Note that D = n_{1}^{ln D/ ln n_{1}} and put γ = ln D/ ln n_{1}, then

\begin{aligned}C^{τ} ([^{\frac{(n_{k_{0}})^{p−ε}}{D}} ] , X) & = C^{τ} ([n_{1}^{k_{0}(p−ε)−γ}] , X) ≥ C^{τ} (n_{1}^{[k_{0}(p−ε)−γ]}, X) \\ & ≥ n1^{\frac{[k_{0}(p−ε)−γ]}{p−ε}} ≥ n1^{\frac{[−γ]}{p−ε}} ≥ D^{−1/(p−ε)}. \end{aligned}

As θ is arbitrary and by the definition of A_{k}, we obtain the required inequality

sup_{m_{1},...,m_{n}} ∥∑_{i=1}^{n} x_{m_{1},...,m_{i}} ∥ ≥ \frac{1}{2}D^{−1/(p−ε)} (∑_{i=1}^{n} α_{i}^{p})^{1/p} .

The theorem is proved.

Theorem 2.1.9 If a Banach space X has Lyapunov tree cotype p ∈ [1, 2), then X has the Lyapunov property.

Proof. Fix N ∈ N for the present, and let k = [N/ ln N]. We assume that X fails the Lyapunov property and use Lemma 1.4.7. Let Ω, τ, λ, ε, and T : L_{∞} → X be as in that lemma, with λ (Ω) = 1 and ∥T ∥ ≤ 1. Note that we can select a σ-algebra Σ (see the proof of Lemma 1.5.3) such that L_{1} (Ω, Σ, λ) is separable. We shall prove by induction that there exist sequences {r_{n_{1},...,n_{i}}}_{n_{i}∈N} ⊂ L_{∞} (Ω, Σ, λ) (i = 1, 2, ...) such that for every j {T r_{n_{1},...,n_{i}}}_{i=1 n_{i}∈N}^{j} generates a Lyapunov tree and for any {n_{i}, ..., n_{j}} the functions {r_{n_{1},...,n_{i}}}_{i=1}^{j} are jointly equidistributed with the {s_{i}}_{i=1}^{j} in Lemma 1.4.9.

Indeed, everything is obvious for j = 1. Suppose that the assertion is already proved for j = m, and that {r_{n_{1},...,n_{i}}}_{i=1 n_{i}∈N}^{j} have been constructed that satisfy the required condition with j = m. For every {n_{1}, ..., n_{m}} we consider the set A_{n_{1},...,n_{m}} =

\{ω ∈ Ω : | ∑_{i=1}^{m} r_{n_{1},...,n_{i}} (ω)| < \sqrt{N}\} and a sequence \{z_{n_{1},...,n_{m},n_{m+1}}\}_{n_{m+1}=1}^{∞}

of functions independent on A_{n_{1},...,n_{m}}, equal to zero off A_{n_{1},...,n_{m}}, and taking the values ±1 with probability λ (A_{n_{1},...,n_{m}}) /2 such

that T z_{n_{1},...,n_{m},n_{m+1}} ⟶_{n_{m+1}→∞}^{τ} 0. For any {n_{1}, ..., n_{m+1}} the system

{{r_{n_{1},...,n_{i}}}_{i=1}^{m} , z_{n_{1},...,n_{m},n_{m+1}}} is equidistributed with {s_{i}}_{i=1}^{m+1}. The sequence z_{n_{1},...,n_{m},n_{m+1}} tends to zero in the weak^{∗} topology as n_{m+1} → ∞;

hence T z_{n_{1},...,n_{m},n_{m+1}} ⟶_{n_{m+1}→∞}^{w} 0. As L_{1} (Ω, Σ, λ) is separable, the

weak^{∗} topology on L_{∞} (Ω, Σ, λ) is metrizable on bounded sets. Let ρ be a metric defining the weak^{∗} convergence on the unit ball of L_{∞} (Ω, Σ, λ). Then for every set {n_{1}, ..., n_{m}} we can choose a subsequence {r_{n_{1},...,n_{m},n_{m+1}}}_{n_{m+1}=1}^{∞} of {z_{n_{1},...,n_{m},n_{m+1}}}_{n_{m+1}=1}^{∞} such that

sup_{n_{m+1}} ρ (r_{n_{1},...,n_{m},n_{m+1}}, 0) ≤ \frac{1}{∑_{i=1}^{m} n_{i}}

(and ∥T̃T r_{n_{1},...,n_{m},n_{m+1}} ∥_{∞} ≤ 1/∑_{i=1}^{m} n_{i} in the case τ is a c_{0}-T̃-weak topology). Hence, for given 0 ≤ i_{1} < i_{2} ≤ m + 1 and for every fixed n_{1}, ..., n_{i_{1}}, we obtain

ρ (r_{n_{1},...,n_{i_{1}},n_{i_{1}+1},...,n_{i_{2}}}, 0) ⟶_{n_{i_{1}+1},...,n_{i_{2}}→∞} 0

(and ∥T̃T r_{n_{1},...,n_{m},n_{m+1}} ∥_{∞} ⟶_{n_{i_{1}+1},...,n_{i_{2}}→∞} 0 in the case τ is a c_{0}-T̃-weak

topology). It follows that {T r_{n_{1},...,n_{i}}}_{i=1 n_{i}∈N}^{m+1} is a Lyapunov tree.

We now use the fact just proved. Suppose that j = k. Since the Banach space X has Lyapunov tree cotype p with constant C, there exist numbers n_{1}^{0}, ..., n_{k}^{0} such that

∥∑_{i=1}^{k} T r_{n_{1}^{0},...,n_{i}^{0}} ∥^{p} ≥ C^{p} (∑_{i=1}^{k} inf_{n_{1},...,n_{i}} ∥T r_{n_{1},...,n_{i}}∥^{p}) .

In view of the condition (ii) from the Lemma1.4.7, which T satisfies by construction, it follows that ∥T r_{n_{1},...,n_{i}}∥ ≥ ε · λ (supp r_{n_{1},...,n_{i}}) ≥ ε·λ (supp r_{n_{1},...,n_{k}}) for i ≤ k. Since {r_{n_{1},...,n_{i}}}_{i=1}^{k} and {s_{i}}_{i=1}^{k} are equidistributed, this gives us by the choice of k that

\begin{matrix}(\sqrt{N} + 1)^{p} ∥T ∥^{p} + 1 ≥ ∥∑_{i=1}^{k} T r_{n_{1}^{0},...,n_{i}^{0}} ∥^{p} ≥ kε^{p}C^{p} (P (s_{k} ≠ 0))^{p} ≥ \\ ≥ (N/ ln N − 1) ε^{p}C^{p} (P (|∑_{i=1}^{k−1} s_{i}| < \sqrt{N}))^{p} . \end{matrix}

By Lemma 1.4.9, the last factor tends to 1; hence this inequality cannot hold for large N. This contradiction completes the proof.

2.2 Lyapunov tree type

Definition 2.2.1 We will say that a Banach space X has Lyapunov tree type p > 1 with constant C if for every n ∈ N and for every Lyapunov tree X^{n} there exist numbers m_{1}^{0}, ..., m_{n}^{0} such that

∥∑_{i=1}^{n} x_{m_{1}^{0},...,m_{i}^{0}} ∥ ≤ C (∑_{i=1}^{n} sup_{m_{1},...,m_{i}} ∥x_{m_{1},...,m_{i}}∥^{p})^{1/p} .

Definition 2.2.2 A Banach space X is called Lyapunov tree b-convex if there exists n ∈ N such that

b^{τ} (n, X) = sup { inf_{m_{1},...,m_{n}} ∥∑_{i=1}^{n} x_{m_{1},...,m_{i}} ∥} < n,

where the supremum is taken over all Lyapunov trees X^{n} of length n with ∥x_{m_{1},...,m_{i}}∥ ≤ 1.

Remark 2.2.3 If ∥x_{m_{1},...,m_{i}}∥ ≤ α for all elements of a Lyapunov tree X^{n}, then we have

inf_{m_{1},...,m_{n}} ∥∑_{i=1}^{n} x_{m_{1},...,m_{i}} ∥ ≤ α · b^{τ} (n, X) .

Lemma 2.2.4 The numbers b^{τ} (n, X) generate a semimultiplicative sequence, i.e.

b^{τ} (n · l, X) ≤ b^{τ} (n, X) · b^{τ} (l, X) .

Proof. The lemma is proved by analogy with Lemma 2.1.5.

Theorem 2.2.5 Let X be a Lyapunov tree b-convex Banach space. If there exists n_{1} such that b^{τ} (n_{1}, X) < n_{1}^{1/p}, where 1 < p < ∞, then X has the Lyapunov tree type p.

Proof. Let us choose n_{1} > 1 and ε > 0 such that b^{τ} (n_{1}, X) ≤ n_{1}^{1/(p+ε)}. Put n_{k} = (n_{1})^{k} (k = 0, 1, 2, ...). By Lemma 2.2.4, b^{τ} (n_{k}, X) ≤ n_{1}^{k/(p+ε)}.

Take an arbitrary Lyapunov tree X^{n}. Define

α_{i} = sup_{m_{1},...,m_{i}} ∥x_{m_{1},...,m_{i}}∥ (i = 1, ..., n) .

Decompose the set of indices {1, ..., n} into the union of mutually disjoint sets

A_{k} = \{j : \frac{( ∑_{i=1}^{n} α_{i}^{p})^{1/p}}{n_{k}} > α_{j} ≥ \frac{( ∑_{i=1}^{n} α_{i}^{p})^{1/p}}{n_{k+1}} \} ,

where k = 0, 1, 2, .... Define a_{k} as the number of elements of A_{k}. Then

∑_{j=1}^{n} α_{j}^{p} = ∑_{k=0}^{∞} ∑_{j∈A_{k}} α_{j}^{p} ≥ ∑_{k=0}^{∞} a_{k} · \frac{∑_{i=1}^{n} α_{i}^{p}}{n_{k+1}^{p}} .

Consequently,

∑_{k=0}^{∞} \frac{a_{k}}{n_{k+1}^{p}} ≤ 1.

It follows that for every k the inequality a_{k} ≤ n_{1}^{[p(k+1)]+1} holds. Indeed, if there exists a number k_{0} such that a_{k_{0}} > n_{k_{0}}^{[p(k_{0}+1)]+1}, then

\frac{a_{k_{0}}}{n_{k_{0}+1}^{p}} > 1 and ∑_{k=0}^{∞} \frac{a_{k}}{n_{k+1}^{p}} > 1.

It is not difficult to see that for every k and for every fixed numbers {m_{i}}_{i∉A_{k}} the sequences {x_{m_{1},...,m_{i}}}_{i∈A_{k} m_{i}∈N} generate a Lyapunov tree of length a_{k}. It follows that for every fixed set {m_{i}}_{i∉A_{k}} and for every δ > 0 we can select {m_{i}}_{i∈A_{k}} such that

∥ ∑_{i∈A_{k}} x_{m_{1},...,m_{i}} ∥ ≤ sup_{m_{i}:i∈A_{k}} ∥x_{m_{1},...,m_{i}}∥ · (b^{τ} (a_{k}, X) + δ)  (2.1)

Consequently we can find numbers m_{1}^{0}, ..., m_{n}^{0} such that condition (2.1) holds for all k. Then we have

\begin{aligned}inf_{m_{1},...,m_{n}} ∥ ∑_{i=1}^{n} x_{m_{1},...,m_{i}} ∥ & ≤ ∥ ∑_{i=1}^{n} x_{m_{1}^{0},...,m_{n}^{0}} ∥ \\ & ≤ ∑_{k=0}^{∞} ^{\frac{(∑_{i=1}^{n} α_{i}^{p})^{1/p}}{n_{k}}} · (b^{τ} (a_{k}, X) + δ) . \end{aligned}

Estimate

\begin{aligned}∑_{k=0}^{∞} \frac{b^{τ} (a_{k},X)}{n_{k}} & ≤ ∑_{k=0}^{∞} ^{\frac{b^{τ}(n_{1}^{[p(k+1)]+1},X)}{n_{k}}} ≤ ∑_{k=0}^{∞} n_{1}^{\frac{[p(k+1)]+1}{p+ε} −k} \\ & ≤ n_{1}^{1+1/(p+ε)} ∑_{k=0}^{∞} n_{1}^{−ε/(p+ε)(k+1)}. \end{aligned}

It is clear that the series converges. Denote its sum by C. As δ is arbitrary, we obtain the required inequality

inf_{m_{1},...,m_{n}} ∥∑_{i=1}^{n} x_{m_{1},...,m_{i}} ∥ ≤ C (∑_{i=1}^{n} α_{i}^{p})^{1/p} .

The theorem is proved.

Theorem 2.2.6 If a Banach space X has Lyapunov tree type p > 2 with constant C, then X has the Lyapunov property.

Proof. Fix N ∈ N for the present, and let k = [N ln N]. We assume that X has not the Lyapunov property and use Lemma 1.4.7. Let Ω, τ, λ, ε, and T : L_{∞} → X be as in that lemma, with λ (Ω) = 1 and ∥T ∥ ≤ 1. By analogy with the proof of Theorem 2.1.9 we construct a Lyapunov tree of length k and choose n_{1}^{0}, ..., n_{k}^{0} such that

∥∑_{i=1}^{k} T r_{n_{1}^{0},...,n_{i}^{0}} ∥^{p} ≤ C^{p} (∑_{i=1}^{k} inf_{n_{1},...,n_{i}} ∥T r_{n_{1},...,n_{i}}∥^{p}) ≤ C^{p} ∥T ∥^{p} · k.

The rest of the proof is similar to the proof of Theorem 1.4.11.

Examples

3.1 Orlicz sequence spaces

As it has been shown in Section 1.4 the spaces l_{p} (1 ≤ p < ∞, p ≠ 2) and c_{0} have the Lyapunov property. The next class of Banach spaces for which it is natural to consider this property is the class of Orlicz sequence spaces. The introduction of Orlicz functions has been inspired by the obvious role played by the functions t^{p} in the definition of the space l_{p}. It is quite natural to try to replace t^{p} by a more general function M and then to consider the set of scalars {a_{n}}_{n=1}^{∞} for which the series ∑_{n=1}^{∞} M (|a_{n}|) converges. W. Orlicz [27] has checked the restrictions which have to be imposed on the function M in order to make this set of sequences into a suitable Banach space. His study led to the following definition.

Definition 3.1.1 An Orlicz function M is a continuous, nondecreasing, and convex function defined for t ≥ 0, with M (0) = 0.

We shall consider vector - valued Orlicz sequence spaces. With any Orlicz function M and sequence {X_{n}} of Banach spaces we associate the space (X_{1} ⊕ X_{2} ⊕ ...)_{M} of all sequences x = (x_{1}, x_{2}, ...) , where x_{i} ∈

X_{i} (i = 1, 2, ...) , such that ∑_{n=1}^{∞} M (\frac{∥x_{n}∥}{ρ}) < ∞ for some ρ > 0. The

space (X_{1} ⊕ X_{2} ⊕ ...)_{M} equipped with the norm

∥x∥ = inf \{ρ > 0 : ∑_{n=1}^{∞} M (\frac{∥x_{n}∥}{ρ}) ≤ 1\}

is a Banach space. If X_{n} = R (n = 1, 2, ...) we get the Orlicz sequence space l_{M}.

We will show that all Orlicz sequence spaces containing no isomorphic copies of l_{2} have the Lyapunov property. For this aim we recall some definitions and statements (that can be found in [24]) concerning them.

Proposition 3.1.2 ([24], p. 139) Let M_{1} and M_{2} be Orlicz functions. Then the following assertions are equivalent:

(i) l_{M_{1}} = l_{M_{2},} i.e. both spaces consist of the same sequences and the identity mapping is an isomorphism between l_{M_{1}}and l_{M_{2}}.

(ii) M_{1} and M_{2} are equivalent at zero, i.e. there exist constants k > 0, K > 0, and t_{0} > 0 such that for all 0 < t < t_{0}, we have K^{−1}M_{2}(k^{−1}t) ≤ M_{1} (t) ≤ KM_{2} (kt) .

Anyway, without loss of generality we may assume that M (1) = 1. We shall be interested in Orlicz spaces not containing isomorphic

copies of l_{2} space because l_{2} is not a Lyapunov space.

Definition 3.1.3 An Orlicz function M is said to satisfy the Δ_{2}- condition at zero if M is nondegenerate and sup_{0<t≤1} \frac{M(2t)}{M(t)} < ∞.

Proposition 3.1.4 ([24], p. 138) For an Orlicz function M the following conditions are equivalent:

(i) M satisfies the Δ_{2}-condition at 0;

(ii) l_{M} contains no subspace isomorphic to l_{∞}.

In the sequel we suppose M to satisfy the Δ_{2}-condition.

Remark 3.1.5 If M is an Orlicz function, x ∈ (X_{1} ⊕ X_{2} ⊕ ...)_{M} , and ε > 0, then evidently by the definition of the norm ∥x∥ > ε if and only

if ∑_{m=1}^{∞} M (\frac{∥x_{m}∥}{ε}) > 1.

Proposition 3.1.6 ([24], p. 143) The space l_{p} or c_{0} if p = ∞ is isomorphic to a subspace of an Orlicz sequence space l_{M} if and only if p ∈ [α_{M}, β_{M}] , where

α_{M} = sup \{q : sup_{0<λ, t≤1} \frac{M(λt)}{M(λ)t^{q}} < ∞\} ,
β_{M} = inf \{p : inf_{0<λ, t≤1} \frac{M(λt)}{M(λ)t^{p}} > 0\}

are the Boyd indices.

A detailed exposition of the basic properties of Orlicz sequence spaces is given in [24]. Now we prove some lemmas. The following lemma is a strengthening of Lemma 1.5.2.

Lemma 3.1.7 Under the conditions of Lemma 1.5.2 for every A ∈ Σ, λ (A) ≠ 0 there exist Gn^{′} ∈ Σ|_{A}, Gn^{′′} = A\Gn^{′} (n = 1, 2, ...) such that

(i) λ (Gn^{′}) = λ (Gn^{′′}) = \frac{1}{2}λ (A) ;

(ii) r_{n}=χ_{G′_{n}}-χ_{G′′_{n}} are independent random variables on the measure space (A, Σ|_{A}, \frac{1}{λ(A)}λ (·));

(iii) ∥μ (Gn^{′}) − \frac{1}{2}μ (A)∥ ≤ \frac{1}{2^{n}}.

Proof. We shall construct the sets Gn^{′}, Gn^{′′} (n = 1, 2, ...) by induction on n. Let n = 1. By Lemma 1.5.2 there is a G1^{′} ∈ Σ|_{A} such that

∥μ (G1^{′}) − \frac{1}{2}μ (A)∥ ≤ \frac{1}{2} and λ (G1^{′}) = \frac{1}{2}λ (A) .  (3.1)

Let n = k + 1. Suppose Gj^{′}, Gj^{′′} (j = 1, ..., k), satisfying the required conditions, have been constructed. By independence of {r_{j}}_{j=1}^{k} there

are mutually disjoint sets {D_{i}}_{i=1}^{2^{k}} ⊂ Σ|_{A} such that

\begin{aligned}r_{1} & = & ∑_{i=1}^{2^{k−1}} χ_{D_{i}} − ∑_{i=2^{k−1}+1}^{2^{k}} χ_{Di}; \\ r_{2} & = & ∑_{i=1}^{2^{k−2}} χ_{D_{i}} − ∑_{i=2^{k−2}+1}^{2^{k−1}} χ_{D_{i}} + ∑_{i=2^{k−1}+1}^{3·2^{k−2}} χ_{D_{i}} − ∑_{i=3·2^{k−2}+1}^{2^{k}} χ_{D_{i}}; \\ ... & ... & ............................................................................. \\ r_{k} & = & ∑_{i=1}^{2^{k}} (−1)^{i+1} χ_{D_{i}}. \end{aligned}

By Lemma 1.5.2, there exists Di^{′} ∈ Σ|_{D_{i}} such that

∥μ (Di^{′}) − \frac{1}{2}μ (D_{i})∥ ≤ \frac{1}{4^{k+1}} and λ (Di^{′}) = \frac{1}{2}λ (D_{i}) (k = 1, ..., 2^{k}) .

Put Gk+1^{′} = ⋃_{i=1}^{2^{k}} Di^{′}. It is easy to verify that Gk+1^{′}, Gk+1^{′′} satisfy conditions

(i)-(iii).

Lemma 3.1.8 Let X be a Banach space, r > 0, F : X → R be a convex function, which is bounded in the ball B (0, 2r) = {x ∈ X : ∥x∥ ≤ 2r}. Then F is uniformly continuous in the ball B (0, r) of the space X.

Proof. Let |F (x)| ≤ C for any x ∈ X with ∥x∥ ≤ 2r. Take arbitrary a, b ∈ B (0, r) with F (b) ≥ F (a) . Consider the straight line A = {θa + (1 − θ) b : θ ∈ R} passing through a and b. Choose e ∈ A, ∥e∥ ≤ 2r, so that ∥a − e∥ ≥ r and b is between a and e. Since F is convex, we have

\frac{F (b) − F (a)}{∥b − a∥} ≤ \frac{F (e) − F (a)}{∥e − a∥} ≤ \frac{2C}{r}

and F is uniformly continuous.

Remark 3.1.9 In the sequel we shall use Lemma 3.1.8 for the function

F (x) = ∑_{n=1}^{∞} M (∥x_{n}∥), where x = (x_{1}, x_{2}, ...) ∈ (X_{1} ⊕ X_{2} ⊕ ...)_{M} , which

is bounded in the ball B (0, 2) by the Δ_{2}-condition and the following lemma.

Lemma 3.1.10 Let M be an Orlicz function satisfying the Δ_{2}- condition, 0 < δ ≤ 1, x = (a_{1}, a_{2}, ...) ∈ (X_{1} ⊕ X_{2} ⊕ ...)_{M} . Then we have the following properties:

(i) if ∥x∥ ≤ δ, then ∑_{m=1}^{∞} M (∥a_{m}∥) ≤ δ,

(ii) if ∑_{m=1}^{∞} M (∥a_{m}∥) ≤ δ, then ∥x∥ ≤ 2δ^{\frac{1}{log_{2} C}}, where C is the constant

from the Δ_{2}−condition.

Proof. (i) Using the definition of an Orlicz function we get

∑_{m=1}^{∞} M (∥a_{m}∥) = ∑_{m=1}^{∞} M (δ \frac{1}{δ} ∥a_{m}∥) ≤ δ ∑_{m=1}^{∞} M (\frac{∥a_{m}∥}{δ}) ≤ δ.

(ii) By the Δ_{2}-condition, it follows that there is a constant C > 0 such that M (2t) ≤ CM (t) for any t ≤ 1. Define α = ∥x∥_{M} . Suppose

\frac{1}{2^{n}} ≤ α ≤ \frac{1}{2^{n−1}} . Let us remark that 1 = ∑_{m=1}^{∞} M (\frac{1}{α} ∥a_{m}∥), M is a

nondecreasing function, and M (1) = 1. Hence,

1 = ∑_{m=1}^{∞} M (\frac{∥a_{m}∥}{α}) ≤ C^{n} ∑_{m=1}^{∞} M (∥a_{m}∥) ≤ C^{n}δ.

Therefore 1 ≤ C^{n}δ, i.e. n ≥ \frac{log_{2} \frac{1}{δ}}{log_{2} C}. So we have obtained α ≤ 2δ^{\frac{1}{log_{2} C}}.

Theorem 3.1.11 Let M be an Orlicz function satisfying the Δ_{2}- condition, 2 ∉ [α_{M}, β_{M}] , {X_{n}}_{n=1}^{∞} be a sequence of Banach spaces such that X_{n} has the Lyapunov property for all n. Then (X_{1} ⊕ X_{2} ⊕ ...)_{M} has the Lyapunov property.

Proof. Let us consider two cases.

Case 1: β_{M} < 2. First we fix N ∈ N and k = [\frac{N}{ln N} ]. We shall prove ad absurdum. Assume that (X_{1} ⊕ X_{2} ⊕ ...)_{M} fails the Lyapunov property. By Lemma 1.4.7 there are (Ω, Σ, λ), ε > 0 and T : L_{∞} → X satisfying the conditions (a) and (b) of the condition (ii) of this lemma. Without loss of generality we may assume that ∥T ∥ ≤ \frac{1}{2} and ε ≤ 1. Denote T x = (T_{1}x, T_{2}x, ...). Let us show by induction on j that there exist functions {t_{i}}_{i=1}^{∞} ∈ L_{∞} such that for every j the functions {t_{i}}_{i=1}^{j} are jointly equidistributed with {s_{i}}_{i=1}^{j} from Lemma 1.4.9 and

\begin{matrix}∑_{m=1}^{∞} M (∥T_{m} ( ∑_{i=1}^{j} t_{i})∥) > ∑_{i=1}^{j} ∑_{m=1}^{∞} M (∥T_{m}t_{i}∥) − 1, \\ ∥T_{m} ( ∑_{i=1}^{j} t_{i})∥ < 1 for any m = 1, 2, ... . \end{matrix}  (3.2)

For j = 1 there is nothing to prove. Suppose that for j = n {t_{i}}_{i=1}^{n} satisfying (3.2) have been constructed. Now consider A =

\{ω ∈ Ω : | ∑_{i=1}^{n} t_{i}| < \sqrt{N}\} and put a_{m} = T_{m} ( ∑_{i=1}^{n} t_{i}) (m = 1, 2, ...) . Take

an arbitrary δ > 0. In accordance with Lemmas 3.1.8, 3.1.10, and Remark 3.1.9 there is θ ∈ (0, 1) such that

| ∑_{m=1}^{∞} M (∥x_{m}∥) − ∑_{m=1}^{∞} M (∥y_{m}∥)| ≤ δ  (3.3)

if ∥x∥ ≤ 1, ∥y∥ ≤ 1 and ∥x − y∥ ≤ θ. Combining the Δ_{2}−condition and Lemma 3.1.10, we obtain

∑_{m=N_{1}+1}^{∞} M (∥a_{m}∥) ≤ δ, ∥a_{m}∥ ≤ δ (m = N_{1} + 1, ...) ,  (3.4)
∥(0, ..., 0, a_{N_{1}+1}, a_{N_{1}+2}, ...)∥_{M} ≤ θ,  (3.5)

for some N_{1} ∈ N. By Theorem 1.5.6, (X_{1} ⊕ X_{2} ⊕ ... ⊕ X_{N_{1}})_{M} has the Lyapunov property. It is clear that μ (B) = (T_{1}χ_{B}, ..., T_{N_{1}}χ_{B}) is a nonatomic measure. Hence, by Lemma 3.1.7 there exists a se-

quence Gn^{′} ∈ Σ|_{A}, Gn^{′′} = A\Gn^{′}, λ (Gn^{′}) = λ (Gn^{′′}) = \frac{1}{2}λ (A) such

that the functions z_{n} = χ_{G′} − χ_{G′′} are independent on A and ∥(T_{1}z_{n}, ..., T_{N_{1}}z_{n})∥_{M} →_{n→∞} 0. By continuity of M there is a θ_{1} > 0 such that if |t| , |u| ≤ 1 and |t − u| ≤ θ_{1} then |M (t) − M (u)| ≤ \frac{δ}{N_{1}} . Select n_{0} ∈ N so that

∥(T_{1}z_{n_{0}}, ..., T_{N_{1}}z_{n_{0}})∥_{M} ≤ δ,  (3.6)
∥T_{i}z_{n_{0}}∥ ≤ θ_{1}, i = 1, 2, ..., N_{1}.  (3.7)

Combining (3.3)-(3.7) and Lemma 3.1.8, and Remark 3.1.9, we infer

| ∑_{m=1}^{∞} M (∥a_{m} + T_{m}z_{n_{0}}∥) − ∑_{m=1}^{∞} M (∥a_{m}∥) − ∑_{m=1}^{∞} M (∥T_{m}z_{n_{0}}∥)|

≤ | ∑_{m=1}^{N_{1}} [M (∥a_{m} + T_{m}z_{n_{0}}∥) − M (∥a_{m}∥)]|

+ ∑_{m=1}^{N_{1}} M (∥T_{m}z_{n_{0}}∥) + ∑_{m=N_{1}+1}^{∞} M (∥a_{m}∥)

+ | ∑_{m=N_{1}+1}^{∞} [M (∥a_{m} + T_{m}z_{n_{0}}∥) − M (∥T_{m}z_{n_{0}}∥)]| ≤ 4δ.

It is evident that

∥a_{m} + T_{m}z_{n_{0}}∥ ≤ [ \begin{matrix}∥a_{m}∥ + θ_{1} & if & m = 1, ..., N_{1} \\ δ + \frac{1}{2} & if & m = N_{1} + 1, ... \end{matrix}

Putting t_{n+1} = z_{n_{0}} and fitting δ sufficiently small we arrive at (3.2) for j = n + 1.

Notice that the condition β_{M} < 2 means the existence of 1 < p < 2 and C > 0 such that

Cx^{p}M (t) ≤ M (tx)  (3.8)

for any t, x ∈ (0, 1] .

Now we employ the proved fact and this inequality. Let {t_{i}}_{i=1}^{k} be jointly equidistributed with {s_{i}}_{i=1}^{k} and meet requirement (3.2). Introduce the sets of indices

J_{1}^{i} = {m ∈ N : ∥T_{m}t_{i}∥ ≤ ελ (supp t_{k})} ,
J_{2}^{i} = {m ∈ N : ∥T_{m}t_{i}∥ > ελ (supp t_{k})} ,

where i = 1, ..., k. Then employing (3.2), we get

\begin{aligned}∑_{m=1}^{∞} M (∥T_{m} ∑_{i=1}^{k} t_{i}∥) & ≥ ∑_{i=1}^{k} ∑_{m=1}^{∞} M (∥T_{m}t_{i}∥) − 1 \\ & = ∑_{i=1}^{k} ( ∑_{m∈J_{1}^{i}}^{∞} M (∥T_{m}t_{i}∥) + ∑_{m∈J_{2}^{i}}^{∞} M (∥T_{m}t_{i}∥)) − 1. \end{aligned}

If J_{2}^{i} = Ø, then by (3.8) we obtain

∑_{m=1}^{∞} M (∥T_{m}t_{i}∥) ≥ ∑_{m=1}^{∞} Cε^{p}λ (supp t_{k})^{p} M (\frac{∥T_{m}t_{i}∥}{ελ(supp t_{k})})
≥ Cε^{p}λ (supp t_{k})^{p} .

In the last inequality we used Remark 3.1.5 and property (b) of the map T. If J_{2}^{i} ≠ Ø, then

∑_{m=1}^{∞} M (∥T_{m}t_{i}∥) ≥ ∑_{m∈J_{2}^{i}}^{∞} M (∥T_{m}t_{i}∥)
≥ ελ (supp t_{k}) ≥ Cε^{p}λ (supp t_{k})^{p} .

Thus, we get

∑_{m=1}^{∞} M (∥T_{m} (∑_{i=1}^{k} t_{i})∥) ≥ C [\frac{N}{ln N} ] ε^{p} (λ (|∑_{i=1}^{k−1} s_{i}| < \sqrt{N}))^{p} − 1.

(3.9)

Now we consider two cases. Let 1 < ∥T ( ∑_{i=1}^{k} t_{i})∥ . Applying the ∥_{.}∥_{M} -

definition, (3.8), and (3.9), we get

1 = ∑_{m=1}^{∞} M (∥T_{m} ( ∑_{i=1}^{k} t_{i})∥ · _{\frac{1}{∥T(∑_{i=1}^{k} t_{i})∥}}) ≥

≥ C_{\frac{1}{∥T(∑_{i=1}^{k} t_{i})∥^{p}}} ( ∑_{m=1}^{∞} M (∥T_{m} ( ∑_{i=1}^{k} t_{i})∥)) ≥

≥ C_{\frac{1}{∥T(∑_{i=1}^{k} t_{i})∥^{p}}} (Cε^{p} [\frac{N}{ln N} ] (λ (| ∑_{i=1}^{k−1} s_{i}| < \sqrt{N}))^{p} − 1) .

Whence,

\begin{aligned}∥T ∥^{p} (\sqrt{N} + 1)^{p} + C & ≥ ∥T ( ∑_{i=1}^{k} t_{i})∥^{p} + C \\ & ≥ (\frac{N}{ln N} − 1) C^{2}ε^{p}λ (| ∑_{i=1}^{k−1} s_{i}| < \sqrt{N})^{p} \end{aligned}

for all N ∈ N. By Lemma 1.4.9, the multiplier λ (| ∑_{i=1}^{k−1} s_{i}| < \sqrt{N}) →_{N→∞}

1. It follows that this inequality is not true for large N. Let

∥T ( ∑_{i=1}^{k} t_{i})∥ ≤ 1. Then ∑_{m=1}^{∞} M (∥T_{m} ( ∑_{i=1}^{k} t_{i})∥) ≤ 1. Employing (3.9),

we arrive at a contradiction in the same way. Thus case 1 is finished.

Case 2: α_{M} > 2. Equivalently, there exist p > 2 and C > 0 such that

M (tx) ≤ Cx^{p}M (t)  (3.10)

for any 0 < t, x ≤ 1.

By analogy with case 1 we fix N ∈ N and k = [N ln N], choose the mapping T, construct functions {t_{i}}_{i=1}^{k} jointly equidistributed with {s_{i}}_{i=1}^{k} such that

∑_{m=1}^{∞} M (∥T_{m} (∑_{i=1}^{k} t_{i})∥) ≤ ∑_{i=1}^{k} ∑_{m=1}^{∞} M (∥T_{m}t_{i}∥) + 1 ≤ k + 1. (3.11)

Let f_{N} be as in the proof of Theorem 1.4.11:

f_{N} = \{ \begin{matrix}1 & if ∑_{i=1}^{k} t_{i} ≥ \sqrt{N} \\ −1 & if ∑_{i=1}^{k} t_{i} ≤ −\sqrt{N} \\ 0 & for the rest \end{matrix} .

Let us stress that

∥T (∑_{i=1}^{k} t_{i})∥ ≤ ∥T ∥ (\sqrt{N} + 1) ≤ \frac{\sqrt{N} + 1}{2}.  (3.12)

Put α = ∥T (\frac{1}{\sqrt{N}} ∑_{i=1}^{k} t_{i})∥ , α_{m} = ∥T_{m} (\frac{1}{\sqrt{N}} ∑_{i=1}^{k} t_{i})∥. Then α < 1 and

α_{m} < 1 for large N. If we combine (1.8) ,(3.12) and property (b) of the operator T, we get α > ελ (supp f_{N}) ≥ \frac{ε}{2} for large N. By the Δ_{2}−condition there is a constant C > 0 such that M (\frac{1}{α}α_{m}) ≤ C_{1}M (α_{m}). Consequently, by (3.10) and (3.11)

1 = ∑_{m=1}^{∞} M (\frac{α_{m}}{α} ) ≤ C_{1} ∑_{m=1}^{∞} M (α_{m}) = C_{1} ∑_{m=1}^{∞} M (∥T_{m} ( ∑_{i=1}^{k} t_{i}) \frac{1}{\sqrt{N}} ∥)
≤ C_{1} ∑_{m=1}^{∞} M (∥T_{m} ( ∑_{i=1}^{k} t_{i})∥) C (\frac{1}{\sqrt{N}} )^{p}
≤ C_{1}C (\frac{1}{\sqrt{N}} )^{p} (k + 1) = C_{1}C (\frac{1}{\sqrt{N}} )^{p} ([N ln N] + 1)

for sufficiently large N ∈ N. This contradiction concludes the proof.

Remark 3.1.12 Let {X_{n}} be a sequence of Banach spaces and {M_{n}}_{n=1}^{∞} be Orlicz functions satisfying the uniform Δ_{2}-condition (i.e., there exists a constant C such that sup_{0<t≤1} \frac{M_{n}(2t)}{M_{n}(t)} < C for

all n ∈ N). Then the previous theorem is valid for the modular space (∑ X_{n})_{M_{n}} of all sequences x = (x_{1}, x_{2}, ...) , where x_{n} ∈ X_{n}

(n = 1, 2, ...) , such that ∑_{n=1}^{∞} M_{n} (\frac{∥x_{n}∥}{ρ}) < ∞ for some ρ > 0 and

∥x∥ = inf \{ρ > 0 : ∑_{n=1}^{∞} M_{n} (\frac{∥x_{n}∥}{ρ}) ≤ 1\}.

3.2 The A_{p}-property

In this section we shall show that Lorentz sequence spaces, Schreier’s space, and Baernstein’s spaces not containing isomorphic copies of l_{2} have the Lyapunov property.

First let us summarize some material concerning these spaces.

The Lorentz sequence spaces were introduced in connection with some problems of analysis and interpolation theory. Let 1 ≤ p < ∞. For any a = (a_{1}, a_{2}, ...) ∈ c_{0}\l_{1}, a_{1} ≥ a_{2} ≥ ... ≥ 0, let d(a, p) = {x =

(x_{1}, x_{2}, ...) ∈ c_{0} :sup_{σ∈π} ∞ ; ∑_{n=1} |x_{σ(n)}|^{p} a_{n} < ∞}, where π is the set of all

permutations of the natural numbers N. Then d(a, p) with the norm

∥x∥ = (sup_{σ∈π} ∞ ; ∑_{n=1} |x_{σ(n)}|^{p} a_{n})^{\frac{1}{p}} for x ∈ d(a, p) is a Banach space and the

sequence of unit vectors {e_{n}}_{n=1}^{∞} is a symmetric basis of d(a, p). For the basic properties of the Lorentz spaces d(a, p) we refer to ([9], [10]). In particular, it is known that every infinite-dimensional subspace of d(a, p) has a complemented subspace isomorphic to l_{p} [1].

In 1930, J. Schreier [30] introduced the notion of “admissibility” while producing a counterexample to a question of S. Banach and S. Saks. These two had just shown [3] that in the spaces L_{p} [0, 1] (where p > 1) each weakly convergent sequence contains a subsequence whose arithmetic means converges in norm. They went on to ask whether such a thing held in the space of continuous functions C [0, 1] . J. Schreier showed that this was not the case. A slight variation in his original concept produces the space which is called Schreier’s space in [4].

Definition 3.2.1 A finite subset E = {n_{1} < n_{2} < ... < n_{k}} of N is said to be admissible if k ≤ n_{1}. We denote by L the class of all admissible subsets of N.

Let c_{00} denote the vector space of all real sequences which are eventually 0. Schreier’s space S is the ∥.∥_{S}-completion of c_{00}, where ∥x∥_{S} =sup_{E∈L} ∑_{k∈E} |x (k)|, where x (k) is a k-th coordinate of x, and x ∈ c_{00}.

It is known [4] that this space is not reflexive with a 1-unconditional basis.

3.2. THE A_{p}-PROPERTY 51

In 1972, A. Baernstein [2] produced a reflexive variant of S which contains a weakly null sequence such that no subsequence is strongly Cesaro convergent.

If E and F are finite non-void subsets of N, we write ”E < F ” for max E < min F. For x ∈ c_{00} we write Ex to indicate the vector defined by:

(Ex) (k) = \{ \begin{matrix}x (k) , & if k ∈ E \\ 0, & otherwise. \end{matrix}

Fix 1 < p < ∞. For x ∈ c_{00}, we define

∥x∥_{B_{p}} = sup \{( n ; ∑_{k=1} ∥E_{k}x∥_{l_{1}}^{p})^{\frac{1}{p}} : E_{k} ∈ L, E_{1} < E_{2} < ... < E_{n}, n ∈ N\} .

Baernstein’s space B_{p} is ∥.∥_{B_{p}}-completion of c_{00}. It is known that every infinite-dimensional subspace of B_{p} has a complemented subspace isomorphic to l_{p}. See [4] for details.

The idea of the proofs here is very close to Theorem 1.4.11. But the new trick which we employ, namely considering an auxiliary c_{0}-valued operator, allows us to generalize the Lyapunov theorem for more spaces.

Definition 3.2.2 A Banach space X with a basis is said to have the A_{p}-property (X ∈ A_{p}) if for some basis {e_{n}}_{n=1}^{∞} in X there exists a map T̃ ∈ L(X, c_{0}) such that for every x = ∑_{i=1}^{N} x (i) e_{i} ∈ X, with x (i) ≠ 0, (1 ≤ i ≤ N), and every ε > 0 there exists δ > 0 such that for any y = ∑_{i=N+1}^{M} y (i) e_{i} ∈ X with ∥T̃y∥_{∞} ≤ δ

|∥x + y∥^{p} − ∥x∥^{p} − ∥y∥^{p}| < ε

if 1 ≤ p < ∞, or

|∥x + y∥ − max {∥x∥ , ∥y∥}| < ε

if p = ∞.

Lemma 3.2.3 If X ∈ A_{p} and T̃ is from Definition 3.2.2 then for every x ∈ X, x ≠ 0, for every sequence x_{n} ⟶^{w}_{n→∞} 0 in X with ∥T̃x_{n}∥_{∞} ⟶_{n→∞} 0, and for every ε > 0 there exists n such that

|∥x + x_{n}∥^{p} − ∥x∥^{p} − ∥x_{n}∥^{p}| < ε

52 CHAPTER 3. EXAMPLES

if 1 ≤ p < ∞, or

|∥x + x_{n}∥ − max {∥x∥ , ∥x_{n}∥}| < ε

if p = ∞.

Proof. Let {e_{n}}_{n=1}^{∞} be a basis in X and let T̃ ∈ L(X, c_{0}), for which the conditions of Definition 3.2.2 are satisfied. Fix x = ∞ ; ∑_{i=1} x (i) e_{i} ∈ X,

x ≠ 0, x_{n} = ∞ ; ∑_{i=1} x_{n} (i) e_{i} ∈ X, (n = 1, 2, ...), x_{n} →_{n→∞}^{w} 0, ∥T̃x_{n}∥_{∞} →_{n→∞} 0,

ε > 0. Put C =sup_{n} ∥x_{n}∥. Choose θ ∈ (0, 1) such that for any a, b ∈ [0, ∥x∥ + 1 + C] with |a − b| ≤ θ the inequality |a^{p} − b^{p}| < \frac{ε}{4} holds.

Select N ∈ N such that ∥ ∞ ; ∑_{i=N+1} x (i) e_{i}∥ ≤ \frac{θ}{4}. Denote x^{ε} = N ; ∑_{i=1} x^{ε} (i) e_{i},

where

x^{ε} (i) = \{ \begin{matrix}x (i) & if x (i) ≠ 0 \\ \frac{θ}{4N} & if x (i) = 0 \end{matrix} .

It is obvious that

|∥x∥^{p} − ∥x^{ε}∥^{p}| < \frac{ε}{4}.  (3.13)

Find δ > 0 for x^{ε} and \frac{ε}{4} according to Definition 3.2.2. There is n ∈ N such that ∥T̃x_{n}∥_{∞} ≤ δ and ∥ N ; ∑_{i=1} x_{n} (i) e_{i}∥ ≤ \frac{θ}{4}. Select M ∈ N such that

∥ ∞ ; ∑_{i=M+1} x_{n} (i) e_{i}∥ ≤ \frac{θ}{4}. Put x_{n}^{ε} = M ; ∑_{i=N+1} x_{n} (i) e_{i}. It is easy to see that

∥x_{n} − x_{n}^{ε}∥ ≤ \frac{θ}{2} and ∥T̃x_{n}^{ε}∥_{∞} ≤ δ and consequently

|∥x_{n}∥^{p} − ∥x_{n}^{ε}∥^{p}| < \frac{ε}{4},  (3.14)
|∥x^{ε} + x_{n}^{ε}∥^{p} − ∥x^{ε}∥^{p} − ∥x_{n}^{ε}∥^{p}| < \frac{ε}{4}.  (3.15)

Note that |∥x + x_{n}∥ − ∥x^{ε} + x_{n}^{ε}∥| ≤ θ. Therefore

|∥x + x_{n}∥^{p} − ∥x^{ε} + x_{n}^{ε}∥^{p}| ≤ \frac{ε}{4}.  (3.16)

Combining (3.13)-(3.16), we get

|∥x + x_{n}∥^{p} − ∥x∥^{p} − ∥x_{n}∥^{p}| ≤ |∥x^{ε} + x_{n}^{ε}∥^{p} − ∥x^{ε}∥^{p} − ∥x_{n}^{ε}∥^{p}|

+ |∥x + x_{n}∥^{p} − ∥x^{ε} + x_{n}^{ε}∥^{p}| + |∥x∥^{p} − ∥x^{ε}∥^{p}|

+ |∥x_{n}∥^{p} − ∥x_{n}^{ε}∥^{p}| < ε.

The case p = ∞ can be shown in the same way.

Theorem 3.2.4 If X ∈ A_{p} (p ≠ 2), then there exist a Lyapunov topology τ on X, n′ ∈ N, and if 1 ≤ p < 2 there is p′ ∈ [p, 2) such that

C^{τ} (n′, X) > (n′)^{1/p′},

if 2 < p < ∞ there is p′ ∈ (2, p] such that

b^{τ} (n′, X) < (n′)^{1/p′}.

Proof. Case 1 ≤ p < 2. Let τ be the T̃−c_{0} weak topology on X, where T̃ is from Definition 3.2.2, and {f_{i}}_{i=1}^{∞} be the coordinate functionals. Take an arbitrary n ∈ N, a Lyapunov tree X^{n} with ∥x_{m_{1},...,m_{i}}∥ > 1. We will show that there exist {m_{i}^{0}}_{i=1}^{n} such that

∥∑_{i=1}^{n} x_{m_{1}^{0},...,m_{i}^{0}} ∥^{p} ≥ ∑_{i=1}^{n} ∥x_{m_{1}^{0},...,m_{i}^{0}} ∥^{p} − 1  (3.17)

Let m_{1}^{0} = m_{1}^{1}. Consider the sequence {x_{m_{1}^{0},m_{2}}}_{m_{2}=1}^{∞}. By the definition

of a Lyapunov tree x_{m_{1}^{0},m_{2}} ⟶^{w}_{m_{2}→∞} 0 and ∥T̃x_{m_{1}^{0},m_{2}} ∥_{∞} ⟶_{m_{2}→∞} 0. Then in

accordance with Lemma 3.2.3 we can choose m_{2}^{0} such that

∥∑_{i=1}^{2} x_{m_{1}^{0},m_{i}^{0}} ∥^{p} > ∑_{i=1}^{2} ∥x_{m_{1}^{0},m_{i}^{0}} ∥^{p} − \frac{1}{2}.

Now consider the sequence {x_{m_{1}^{0},m_{2}^{0},m_{3}}}_{m_{3}=1}^{∞}. Analogously choose m_{3}^{0} such that

∥∑_{i=1}^{3} x_{m_{1}^{0},m_{2}^{0},m_{i}^{0}} ∥^{p} > ∑_{i=1}^{2} ∥x_{m_{1}^{0},m_{i}^{0}} ∥^{p} + ∥x_{m_{3}^{0}} ∥^{p} − \frac{1}{4}.

It follows that

∥∑_{i=1}^{3} x_{m_{1}^{0},m_{2}^{0},m_{i}^{0}} ∥^{p} > ∑_{i=1}^{3} ∥x_{m_{1}^{0},m_{2}^{0},m_{i}^{0}} ∥^{p} − (\frac{1}{2} + \frac{1}{4}) .

Continuing this process we obtain {m_{i}^{0}}_{i=1}^{n} satisfying (3.17). As ∥x_{m_{1},...,m_{i}}∥ > 1 (i = 1, ..., n) we get

∥∑_{i=1}^{n} x_{m_{1}^{0},...,m_{i}^{0}} ∥^{p} ≥ n − 1,

and consequently

∥∑_{i=1}^{n} x_{m_{1}^{0},...,m_{i}^{0}} ∥^{p} ≥ (n − 1)^{1/p} .

It follows that C^{τ} (n, X) ≥ (n − 1)^{1/p} for all n. It is not difficult to see that there exist p′ ∈ [p, 2) and n′ such that (n′ − 1)^{1/p} > (n′)^{1/p′} and

consequently C^{τ} (n′, X) > n^{′ \frac{1}{p′}}.

Case 2 < p < ∞ is proved by analogy with the previous one.

Corollary 3.2.5 If X ∈ A_{p} (p ≠ 2), then X has the Lyapunov property.

Proof. The proof follows immediately from the last theorem and Theorems 2.1.8, 2.1.9, 2.2.5, and 2.2.6.

In Lemmas 3.2.6, 3.2.7, 3.2.8 below the natural coordinate embedding of the corresponding sequence space into c_{0} plays the role of T̃ from Definition 3.2.2.

Lemma 3.2.6 d (a, p) ∈ A_{p}.

Proof. Note that if x = (x_{1}, x_{2}, ...) ∈ d (a, p), ∥x∥ = ( ∑_{n=1}^{∞} x̂_{n}^{p}a_{n})^{\frac{1}{p}},

where (x̂_{1}, x̂_{2}, ...) is an enumeration of {|x_{n}|}_{n=1}^{∞} such that x̂_{1} ≥ x̂_{2} ≥ ....

Fix x = N ; ∑_{i=1} x_{i}e_{i} ∈ d (a, p), where x_{i} ≠ 0 (i = 1, 2, ..., N), ε ∈ (0, 1).

Without loss of generality we may assume that x_{1} ≥ x_{2} ≥ ... > 0. Since

a = (a_{1}, a_{2}, ...) ∈ c_{0}, there exists k > N such that k+N ; ∑_{i=k} a_{i} < \frac{ε}{2}. Take

δ = min \{\frac{ε}{2k}, min_{1≤i≤N} x_{i}\}. Let y = M ; ∑_{i=N+1} y_{i}e_{i} ∈ d (a, p) with ∥y∥_{∞} ≤ δ.

We may assume that y_{N+1} ≥ y_{N+2} ≥ ... ≥ 0. Further we estimate the number |∥x + y∥^{p} − ∥x∥^{p} − ∥y∥^{p}| :

\begin{aligned}|∥x + y∥^{p} − ∥x∥^{p} − ∥y∥^{p}| & = M ; ∑_{i=N+1} y_{i}^{p} (a_{i−N} − a_{i}) \\ & = N+k ; ∑_{i=N+1} y_{i}^{p} (a_{i−N} − a_{i}) + M ; ∑_{i=N+k+1} y_{i}^{p} (a_{i−N} − a_{i}) . \end{aligned}

Since 1 ≥ a_{1} ≥ a_{2} ≥ ... ≥ 0, we obtain

N+k ; ∑_{i=N+1} y_{i}^{p} (a_{i−N} − a_{i}) ≤ N+k ; ∑_{i=N+1} y_{i}^{p} ≤ \frac{ε}{2}.

An application of ∥y∥_{∞} ≤ 1 yields

\begin{aligned}M ; ∑_{i=N+k+1} y_{i}^{p} (a_{i−N} − a_{i}) & ≤ M ; ∑_{i=N+k+1} (a_{i−N} − a_{i}) \\ & = N+k ; ∑_{i=k+1} a_{i}− M ; ∑_{i=M−N+1} a_{i} < \frac{ε}{2}. \end{aligned}

So, |∥x + y∥^{p} − ∥x∥^{p} − ∥y∥^{p}| < ε.

Lemma 3.2.7 S ∈ A_{∞}.

Proof. Let x = N ; ∑_{i=1} x_{i}e_{i} ∈ S, where x_{i} ≠ 0 (i = 1, 2, ..., N), ε > 0.

Denote δ = \frac{1}{N} min_{1≤i≤N} |x_{i}|. Choose y = M ; ∑_{i=N+1} y_{i}e_{i} ∈ S with ∥y∥_{∞} ≤ δ.

It is evident that ∥x∥ , ∥y∥ ≤ ∥x + y∥, consequently max {∥x∥ , ∥y∥} ≤ ∥x + y∥.

We shall prove that ∥x + y∥ ≤ max {∥x∥ , ∥y∥}. Let k ≤ N, E = {n_{1} < ... < n_{k}} be an admissible set, i.e. k ≤ n_{1}. If n_{k} ≤ N, then

∑_{i∈E} |x_{i}| + ∑_{i∈E} |y_{i}| =∑_{i∈E} |x_{i}| ≤ ∥x∥ .

56 CHAPTER 3. EXAMPLES If n_{k} > N, then

∑_{i∈E} |x_{i}| + ∑_{i∈E} |y_{i}| ≤∑_{i∈E} |x_{i}| + min_{1≤i≤N} |x_{i}| ≤ ∑_{i∈E′} |x_{i}| ≤ ∥x∥ ,

where E′ = \{k_{0}\} ∪ \{i ∈ E : i ≤ N\}, k_{0} ∈ \overline{k, N}\E. Let k > N, then

∑_{i∈E} |x_{i}| + ∑_{i∈E} |y_{i}| =∑_{i∈E} |y_{i}| ≤ ∥y∥ .

Thus ∥x + y∥ ≤ max {∥x∥ , ∥y∥} . Lemma 3.2.8 B_{p} ∈ A_{p}. Proof. Let x = N ; ∑_{i=1} x_{i}e_{i} ∈ B_{p}, x ≠ 0, ε > 0. Select θ ∈ (0, 1) such that for any a, b ∈ [0, ∥x∥_{l_{1}} + 1] with |a − b| ≤ θ the inequality |a^{p} − b^{p}| < ε holds. Take δ = \frac{θ}{N}. Choose y = M ; ∑_{i=N+1} y_{i}e_{i} ∈ S with ∥y∥_{∞} ≤ δ. Evidently ∥x∥^{p} + ∥y∥^{p} ≤ ∥x + y∥^{p}. We shall prove that ∥x + y∥^{p} ≤ ∥x∥^{p} + ∥y∥^{p} + ε. Let E_{1} < E_{2} < ... < E_{n,} E_{i} ∈ L, and

∥x + y∥^{p} = n ; ∑_{i=1} ∥E_{i} (x + y)∥_{l_{1}}^{p} .

We introduce the sets of indices I_{1} = \{i ∈ \overline{1, n} : E_{i} ∩ \overline{N + 1, M} = Ø\} ,

I_{2} = \{i ∈ \overline{1, n} : E_{i} ∩ \overline{1, N} = Ø\} ,
i_{0} = \overline{1, n}\ (I_{1} ∪ I_{2}) .

Then

∥x + y∥^{p} =∑_{i∈I_{1}} ∥E_{i}x∥_{l_{1}}^{p} + ∑_{i∈I_{2}} ∥E_{i}y∥_{l_{1}}^{p} + ∥E_{i_{0}} (x + y)∥_{l_{1}}^{p} .

Let us estimate the last item

∥E_{i_{0}} (x + y)∥_{l_{1}}^{p} = (∥(E_{i_{0}} ∩ \overline{1, N}) x∥_{l_{1}} + ∥(E_{i_{0}} ∩ \overline{N + 1, M}) y∥_{l_{1}})^{p}
< ∥E_{i_{0}}x∥_{l_{1}}^{p} + ε.

This implies that

∥x + y∥^{p} = ∑_{i∈I_{1}∪i_{0}} ∥E_{i}x∥_{l_{1}}^{p} + ∑_{i∈I_{2}} ∥E_{i}y∥_{l_{1}}^{p} + ε ≤ ∥x∥^{p} + ∥y∥^{p} + ε.

The lemma is proved.

Corollary 3.2.9 The Lorentz sequence spaces d (a, p) (p ≠ 2), the Baernstein spaces B_{p} (p ≠ 2) and the Schreier space S have the Lyapunov property.

3.3 Tsirelson-type spaces

The results of this section are contained in [34].

One of the historical concerns of the structure theory of Banach spaces has been whether there are any ”fundamental” spaces which embed isomorphically in every infinite dimensional Banach space. The spaces c_{0} or l_{p} (1 ≤ p < ∞) were hoped to have this property, because all classical Banach spaces do indeed contain a copy of c_{0} or l_{p} (1 ≤ p < ∞). Also Orlicz spaces have this property despite the fact that the definition of an Orlicz space is not a priori connected to any l_{p} or c_{0}. This hope was destroyed by B. S. Tsirelson’s construction of a reflexive Banach space with a monotone unconditional Schauder basis and no embedded copies of c_{0} or any l_{p} [33]. T. Figiel and W. B. Johnson continued the research and obtained a reflexive Banach space with a symmetric basis that has the same property. The following construction of Tsirelson’s space is due to Figiel and Johnson [4].

We inductively define a sequence of norms {∥·∥_{N}}_{N=1}^{∞} as follows: for x ∈ c_{00}, let

\{ \begin{matrix}∥x∥_{0} = ∥x∥_{∞} , \\ ∥x∥_{N+1} = ∥x∥_{N} ∨ \frac{1}{2} max \{∑_{j=1}^{n} ∥E_{j}x∥_{N} : n ∈ N, n ≤ E_{1} < ... < E_{n}\} \end{matrix}

for N ≥ 0. It is easily seen that the ∥·∥_{N} are norms on c_{00}, that they increase with N, and that

∥x∥_{N} ≤ ∥x∥_{l_{1}}

for all x ∈ c_{00} and for all N. Thus, for each x ∈ c_{00}, lim_{N} ∥x∥_{N} exists and is majorized by ∥x∥_{l_{1}}. We denote lim_{N} ∥x∥_{N} by ∥x∥ and easily confirm that it norms c_{00}. Tsirelson’s space T is the ∥·∥-completion of c_{00}.

Note that from the definition of the norm it follows that

∥x∥ = max \{∥x∥_{∞} , \frac{1}{2} sup \{∑_{j=1}^{n} ∥E_{j}x∥ : n ∈ N, n ≤ E_{1} < ... < E_{n}\}\}

(3.18)

for each x ∈ T.

This construction has been developed further and some Banach spaces of Tsirelson-type were obtained in order to solve some longstanding problems of Banach space theory. Below we present the results of Schlumprecht, Maurey, and Gowers.

Definition 3.3.1 A space (Y, ∥·∥) is said to be λ − distortable if there exists an equivalent norm |∥·∥| such that for every infinite dimensional subspace Z ⊂ Y the quantity

sup {|∥y∥| / |∥z∥| : ∥y∥ = ∥z∥ = 1}

is at least λ.

In 1991 T. Schlumprecht [30] constructed an example of a space that is λ-distortable for every λ. Before giving his definition let us fix some notation.

T. Schlumprecht defines a class of functions f : [1, ∞) → [1, ∞), which we call F, as follows. The function f is a member of F if it satisfies the following five conditions:

(i) f (1) = 1 and f (x) < x for every x > 1.

(ii) f is strictly increasing and tends to infinity.

(iii) lim_{x→∞} x^{−q}f (x) = 0 for every q > 0.

(iv) The function x/f (x) is concave and not decreasing.

(v) f (xy) ≤ f (x) f (y)for every x, y ≥ 1. It is easily verified that f (x) = log_{2} (x + 1) satisfies these conditions, as does the function \sqrt{log_{2} (x + 1)}.

The support of a vector x = ∑_{n=1}^{∞} a_{n}e_{n} ∈ c_{00} is the set of n ∈ N for

which a_{n} ≠ 0. An interval of integers is a subset of N of the form {a, a + 1, ..., b} for some a, b ∈ N. We shall also define the range of a vector, written by the ran (x), to be a smallest interval containing its support. We shall write x < y to mean ran (x) < ran (y). If x_{1} < ... < x_{n}, we shall say that x_{1}, ..., x_{n} are successive.

Now let f be the function x ↦ log_{2} (x + 1) as above. Define the sequence of norms {∥·∥_{N}}_{N=0}^{∞} on c_{00} as follows. Fix x ∈ c_{00} and put

\{ \begin{matrix}∥x∥_{0} = ∥x∥_{∞} , \\ ∥x∥_{N+1} = ∥x∥_{N} ∨ sup \{f (n)^{−1} ∑_{j=1}^{n} ∥E_{j} (x)∥_{N} : n ≥ 2, E_{1} < ... < E_{n}\} . \end{matrix}

Schlumprecht’s space S is the completion of the space c_{00} equipped with the norm ∥x∥ = lim_{N} ∥x∥_{N} for every x ∈ c_{00}. It is not difficult to show that the constructed norm satisfies the following condition

∥x∥ = max {∥x∥_{∞} , sup f (n)^{−1} ∑_{j=1}^{n} ∥E_{j} (x)∥ : n ≥ 2, E_{1} < ... < E_{n}} .

(3.19)

In 1993 W. T. Gowers and B. Maurey [12] constructed a Banach space that does not contain any unconditional basic sequence. The definition of the space resembles that of Schlumprecht’s space. First we shall need a certain amount of preliminary notation.

Let Q be the set of scalar sequences with finite support, rational coordinates, and maximum at most one in modulus. Let J ⊂ N be an infinite set such that, if m < n and m, n ∈ J, then log log log n ≥ 4m^{2}. Let us write J in increasing order as {j_{1}, j_{2}, ...}. We shall also assume that f (j_{1}) > 256. (Recall that f (x)=log_{2} (x + 1).) Now let K ⊂ J be the set {j_{1}, j_{3}, ...}, and let L ⊂ N be the set of integers j_{2}, j_{4}, ....

Let σ be an injection from the collection of finite sequences of successive elements of Q to L such that, if z_{1}, ..., z_{s} is such a sequence,

S = σ (z_{1}, ..., z_{s}), and z = ∑_{i=1}^{s} z_{i}, then (1/20) f (S^{1/40})^{1/2} ≥ |supp (z)|.

We shall use the injection σ to define special functionals in an arbitrary normed space of the form (c_{00}, ∥·∥).

If (c_{00}, ∥·∥) is a normed space on the finitely supported sequences and m ∈ N, let A_{m}^{∗} (X) be the set of functionals of the form

f (m)^{−1} ∑_{i=1}^{m} f_{i} such that f_{1} < ... < f_{m} and ∥f_{i}∥ ≤ 1 for each i. If

k ∈ N, let Γ_{k}^{X} be the set of sequences g_{1}, ..., g_{k} such that g_{i} ∈ Q for each i, g_{1} ∈ A_{j_{2k}}^{∗} (X) and g_{i+1} ∈ A_{σ(g_{1},...,g_{i})}^{∗} (X) for each 1 ≤ i ≤ k − 1. We call these special sequences. Let B_{k}^{∗} (X) be the set of functionals of

the form f (k)^{−1/2} ∑_{j=1}^{k} g_{j} such that (g_{1}, ..., g_{k}) ∈ Γ_{k}^{X}. These are specials

functionals.

The definition of the norm is as the limit of sequences of norms. Define X_{0} = (c_{00}, ∥·∥_{0}) by ∥x∥_{0} = ∥x∥_{∞}, and for N ≥ 0 let

∥x∥_{X_{N+1}} = sup {f (n)^{−1} ∑_{i=1}^{n} ∥E_{i}x∥_{X_{N}} : 2 ≤ n ∈ N, E_{1} < ... < E_{n}}
∨ sup {|g (Ex)| : k ∈ K, g ∈ B_{k}^{∗} (X_{N}) , E ⊂ N}

Note that this is an increasing sequence of norms, because the sets B_{k}^{∗} (X_{N}) increase as N increases (and more generally, if ∥x∥_{Y} ≤ ∥x∥_{Z} for every x ∈ c_{00}, then B_{k}^{∗} (Y) ⊂ B_{k}^{∗} (Z)). They are also all bounded above by the l_{1}-norm. Define ∥x∥ = lim_{N→∞} ∥x∥_{X_{N}}. The ∥·∥-completion of the c_{00} is called Gowers-Maurey’s space GM.

The constructed norm satisfies the following condition

∥x∥ = ∥x∥_{∞} ∨ sup {f (n)^{−1} ∑_{i=1}^{n} ∥E_{i}x∥ : 2 ≤ n ∈ N, E_{1} < ... < E_{n}}
∨ sup {|g (Ex)| : k ∈ K, g ∈ B_{k}^{∗} (X) , E ⊂ N} .

(3.20)

W. T. Gowers modified the construction of GM to give an example of an infinite-dimensional Banach space that does not contain c_{0}, l_{1} or an infinite-dimensional reflexive Banach space [11].

Let Q be as above. Let J ⊂ N be a set such that if m < n and m, n ⊂ J, then log log log log log n ≥ 1000m. Let us also suppose that f (j) > 10^{103} for every j ∈ J, where f (x) = \sqrt{log_{2} (x + 1)}. Let σ be an injection from the set of finite sequences of successive elements of Q to J.

Let X = (c_{00}, ∥·∥) be a normed space such that the standard basis is bimonotone. For every m ∈ N define A_{m}^{∗} (X) to be the set of functionals of the form f (m)^{−1} ∑_{i=1}^{m} x_{i}^{∗}, where x_{1}^{∗}, ..., x_{m}^{∗} are successive members of

c_{00} and ∥x_{i}^{∗}∥ ≤ 1 for each i. A special sequence of functionals on X is defined to be a sequence of the form z_{1}^{∗}, ..., z_{M}^{∗}, where M ∈ N, the z_{i}^{∗} are successive, z_{i}^{∗} ∈ A_{m}^{∗} ∩ Q for some m ∈ J and for 2 ≤ i ≤ M, we have z_{i}^{∗} ∈ A_{σ(z_{1}^{∗},...,z_{i−1}^{∗})}^{∗} ∩ Q. A special functional on X is defined to

be a functional of the form E ( ∑_{i=1}^{M} z_{i}^{∗}) such that z_{1}^{∗}, ..., z_{M}^{∗} is a special

sequence. To any special sequence we associate a sequence of integers n_{1}, ..., n_{M} ∈ J such that z_{1}^{∗} ∈ A_{n_{1}}^{∗} and n_{i} = σ (z_{1}^{∗}, ..., z_{i−1}^{∗}) for 2 ≤ i ≤ M. The first number n_{1} is not necessary uniquely determined, but

n_{2}, ..., n_{M} certainly are. Given a special functional z^{∗} = E ( ∑_{i=1}^{M} z_{i}^{∗}),

we say that Z ⊂ J is associated to z^{∗} if we can pick such a sequence n_{1}, ..., n_{M} associated to the sequence z_{1}^{∗}, ..., z_{M}^{∗} and Z consists of those n_{i} for which E ∩ ran (z_{i}^{∗}) ≠ Ø. A collection of special functionals w_{1}^{∗}, ..., w_{N}^{∗} is called disjoint if we can choose for them disjoint associated sets Z_{1}, ..., Z_{N}.

We are now ready to define our norm. We shall define it as a limit of a sequence of norms on c_{00}. First let X_{0} be defined by ∥x∥_{0} = ∥x∥_{∞}. For N ≥ 0, define X_{N} by

\begin{aligned}∥x∥_{X_{N+1}} = & sup \{f (n)^{−1} ∑_{j=1}^{n} ∥E_{j}x∥_{X_{N}} : n ≥ 2, E_{1} < ... < E_{n}\} \\ & ∨ sup (∑_{j=1}^{M} |x_{j}^{∗} (x)|^{2})^{1/2} , \end{aligned}

where the second supremum ranges over all sequences x_{1}^{∗}, ..., x_{M}^{∗} of disjoint special functionals on X_{N}.

It is easy to check that every x ∈ G satisfies the equation

\begin{aligned}∥x∥ = & ∥x∥_{∞} ∨ sup \{f (n)^{−1} ∑_{j=1}^{n} ∥E_{j}x∥ : n ≥ 2, E_{1} < ... < E_{n}\} \\ & ∨ sup (∑_{j=1}^{M} |x_{j}^{∗} (x)|^{2})^{1/2} . \end{aligned}

(3.21)

We will show that the above mentioned spaces have the Lyapunov property. To this end we will give some preliminary information.

Definition 3.3.2 If X is a ∥·∥-completion of c_{00} such that (e_{i}) is a normalized monotone basis of X, f ∈ F or f = const, and every x ∈ X satisfies the inequality

∥x∥ ≥ sup {f (n)^{−1} ∑_{j=1}^{n} ∥E_{j}x∥ : n ∈ N, n ≤ E_{1} < ... < E_{n}}

then we shall say that X satisfies an admissible lower f-estimate.

Note that by (3.18)-(3.21) Tsirelson’s space has an admissible lower \frac{1}{2}-estimate, Schlumprecht’s and Gowers-Maurey’s spaces have a log_{2} (x + 1)-estimate, Gowers’s space has a \sqrt{log_{2} (x + 1)} -estimate.

Theorem 3.3.3 If X is a Banach space that satisfies an admissible lower f-estimate then there exist a Lyapunov topology τ on X and n such that C^{τ} (n, X) > n^{1/p} for some p ∈ [1, 2).

Proof. Let τ be the weak topology on X. Define the coordinate functionals by f_{j}. Take an arbitrary n ∈ N and consider a Lyapunov tree X^{n} with ∥x_{m_{1},...,m_{i}}∥ ≥ 1 and take an arbitrary ε > 0. Then by the definition of a Lyapunov tree we can find numbers m_{1}^{0}, ..., m_{n}^{0} such that

∥x_{m_{1}^{0},...,m_{k}^{0}} − ∑_{i=p_{k}+1}^{p_{k+1}} f_{i} (x_{m_{1}^{0},...,m_{k}^{0}}) e_{i}∥ < \frac{ε}{n},

where n − 1 ≤ p_{1} < p_{2} < ... < p_{n+1}, for k = 1, ..., n. Then

∥∑_{k=1}^{n} x_{m_{1}^{0},...,m_{k}^{0}} ∥ > ∥∑_{k=1}^{n} ∑_{i=p_{k}+1}^{p_{k+1}} f_{i} (x_{m_{1}^{0},...,m_{k}^{0}}) e_{i}∥ − ε

≥ f (n)^{−1} ∑_{k=1}^{n} ∥∑_{i=p_{k}+1}^{p_{k+1}} f_{i} (x_{m_{1}^{0},...,m_{k}^{0}}) e_{i}∥ − ε

≥ f (n)^{−1} ∑_{k=1}^{n} (∥x_{m_{1}^{0},...,m_{k}^{0}} ∥ − \frac{ε}{n}) − ε

≥ f (n)^{−1} (n − ε) − ε.

As ε is arbitrary, we get

sup_{m_{1},...,m_{i}} ∥∑_{k=1}^{n} x_{m_{1},...,m_{k}} ∥ > n · f (n)^{−1} .

For f = const everything is obvious. If f ∈ F, then from the condition (iii), it follows that there is n such that

C^{τ} (n, X) > n^{1/p},

where p < 2. This completes the proof.

From this theorem and Theorems 2.1.9 and 2.1.8 we have the main result of this section.

Corollary 3.3.4 Tsirelson’s space, Schlumprecht’s space, Gowers-Maurey’s space, Gowers’s space have the Lyapunov property.

3.4. ASYMPTOTIC l_{p} SPACES 63

3.4 Asymptotic l_{p} spaces

In 1993 Vitali D. Milman and Nicole Tomczak-Jaegermann introduced the class of asymptotic l_{p}-spaces and showed that every Banach space with bounded distortions contains a subspace from this class [26]. We will prove that the asymptotic l_{p} spaces (where 1 ≤ p < ∞, p ≠ 2) have the Lyapunov property.

Definition 3.4.1 A Banach space X with a normalized basis {x_{i}} is said to be an asymptotic l_{p} space, for some 1 ≤ p < ∞ (resp., an asymptotic c_{0} space) if there exists a constant C such that for any n ∈ N there is N ∈ N such that normalized successive blocks N < z_{1} < z_{2} < ... < z_{n} of {x_{i}} are C-equivalent to the unit vector basis in l_{p}^{n} (resp., in c_{0}).

Let us mention that we will say that two basis sequences {x_{i}} and {e_{i}} are C-equivalent, for some constant C, if for any finite sequence of scalars {a_{i}} we have

C^{−1} ∥∑_{i} a_{i}x_{i}∥ ≤ ∥∑_{i} a_{i}e_{i}∥ ≤ C ∥∑_{i} a_{i}x_{i}∥ .

Theorem 3.4.2 Let X be an asymptotic l_{p} space. Then if 1 ≤ p < 2, then for every p < p′ < 2 there exists n ∈ N such that

C^{w} (n, X) > n^{\frac{1}{p′}};

and if 2 < p < ∞, then for every 2 < p′ < p there exists n ∈ N such that

b^{w} (n, X) < n^{\frac{1}{p′}} .

Proof. Case 1: 1 ≤ p < 2. Let w be the weak topology of X, {x_{i}} be a normalized basis of X, {e_{i}} be the unit vector basis in l_{p}. Take p′ ∈ (p, 2) and fix for the present n ∈ N and a Lyapunov tree Y^{n} with ∥y_{m_{1},...,m_{i}}∥ ≥ 1 and an arbitrary ε > 0. For n choose N = N (n) ∈ N from Definition 3.4.1. By the definition of a Lyapunov tree we can find numbers m_{1}^{0}, ..., m_{n}^{0} and blocks N < z_{1} < z_{2} < ... < z_{n} such that {x_{i}}

∥y_{m_{1}^{0},...,m_{k}^{0}} − z_{k}∥ < \frac{ε}{n}

for k = 1, ..., n. Then we have

∥∑_{k=1}^{n} y_{m_{1}^{0},...,m_{k}^{0}} ∥ > ∥∑_{k=1}^{n} z_{k}∥ − ε ≥ C^{−1} ∥∑_{k=1}^{n} e_{k}∥ − ε = C^{−1}n^{\frac{1}{p}} − ε.

As ε is arbitrary, we obtain

C^{w} (n, X) ≥ C^{−1}n^{\frac{1}{p}}

and, consequently,

C^{−1}n^{\frac{1}{p}} > n^{\frac{1}{p′}}

for large enough n.

The case 2 < p < ∞ is proved in analogy with the previous one.

Corollary 3.4.3 If 1 ≤ p < ∞, p ≠ 2 and X is an asymptotic l_{p} space, then X has the Lyapunov property.

Proof. The proof follows immediately from Theorems 2.1.8, 2.1.9, 2.2.5, 2.2.6.

3.5 Tokarev’s space

We have seen that most of the constructed examples obey the scheme: if a Banach space in some class contains no isomorphic copies of l_{2} then it has the Lyapunov property. However, in this section we present an example of a Banach space that does not contain isomorphic copies of l_{2} and does not have the Lyapunov property.

In 1984 by E. V. Tokarev obtained an example of a symmetric function space that contains no isomorphic copies of l_{p} (1 ≤ p < ∞) or c_{0} [32]. We will recall this construction.

Definition 3.5.1 A symmetric function space E is a Banach space of measurable functions such that:

1. if x ∈ E, |y| ≤ |x|, then y ∈ E and ∥y∥_{E} ≤ ∥x∥_{E},

2. if the functions |x| and |y| are equimeasurable, then ∥x∥_{E} = ∥y∥_{E}.

Let us consider the Marcinkiewicz space M_{p} defined by the function ψ_{p} (t) = t (− ln t)^{1/p}, i.e., the space of functions x (t) which are summable on [0, 1] with finite norm

∥x∥_{M_{p}} = sup_{0<h≤1} A_{h} (x) ; A_{h} (x) = \frac{1}{ψ_{p} (h)} ∫_{0}^{h} x^{∗} (t) dt,

where x^{∗} is the nondecreasing rearrangement of x and such that lim_{h→0} A_{h} (x) = 0 for all x ∈ M_{p}.

Further we will suppose that p > 2. Denote (− ln t)^{−1/p} by φ_{p} (t) and note that ∥χ_{A}∥_{M_{p}} = φ_{p} (mes A) .

Choose a sequence of natural numbers n_{k}; n_{1} < n_{2} < ... so that for every k the lacunarity inequality n_{k}^{−1} ∑_{i=1}^{k−1} n_{i} + n_{k} ∑_{1=k+1}^{∞} n_{i}^{−1} < 2^{−k−1} holds and consider the space Y_{k} = n_{k}M_{p} + n_{k}^{−1}L_{∞} that consists of all summable functions x such that

∥x∥_{k} = inf {n_{k} ∥f∥_{M_{p}} + n_{k}^{−1} ∥g∥_{L_{∞}} : f + g = x} < ∞.

It is known (see [13] and [22]) that ∥χ_{A}∥_{k} = min {n_{k}φ_{p} (mes A) ; n_{k}^{−1}} .

Denote by E_{W} the set of all summable functions f with finite norm |||f||| = ∥∑ ∥f∥_{k} e_{k}∥_{W}, where W is an arbitrary reflexive Banach space with an unconditional basis (e_{k}) not containing isomorphic copies of l_{p} (1 < p < ∞), for example Tsirelson’s space.

For us it is important that E_{W} contains no isomorphic copies of l_{2}. From the proof of the result of Tokarev it follows that if we take W = l_{p} (p ≠ 2), then E_{W} contains no isomorphic copies of l_{2}.

Theorem 3.5.2 E_{W} does not have the Lyapunov property.

Proof. Consider the measure μ : Σ → E_{W} such that μ (A) = χ_{A} and check that it is countably additive. Indeed, let A_{1} ⊃ A_{2} ⊃ ... and mes (A_{n}) ⟶_{n→∞} 0. Take an arbitrary ε > 0. The series ∑_{k=1}^{∞} \frac{1}{n_{k}} converges, so we can find k_{0} ∈ N such that ∑_{k=k_{0}+1}^{∞} \frac{1}{n_{k}} ≤ \frac{ε}{2}. Select N_{0} ∈ N so that ∑_{k=1}^{k_{0}} n_{k}φ_{p} (mes (A_{n})) ≤ \frac{ε}{2} for all N ≥ N_{0}. Then it is plain that

|||χ_{A_{N}} ||| ≤ ε

for all N ≥ N_{0}.

Now we will show that the closure of μ (Σ) is not convex. Actually, it is clear that μ ([0, 1]) ∈ \overline{μ (Σ)}. On the other hand, \frac{1}{2}μ ([0, 1]) = \frac{1}{2}χ_{[0,1]} ∉ \overline{μ (Σ)}, as μ (Σ) involves only functions with values ±1. This completes the proof.

Theorem 3.5.3 There exists a Banach space X such that l_{2} ⊄ X and X fails the Lyapunov property.

Proof. By Tokarev’s results and Theorem 3.5.2, E_{W} is such a Banach space.

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Résumé (Zusammenfassung)

Der Konvexitätssatz von Lyapunov besagt, daß der Wertebereich eines in einen endlichdimensionalen Raum wirkenden nicht-atomaren Maßes eine konvexe und kompakte Menge ist. Das Theorem gilt nur für endlichdimensionale Räume. In dieser Arbeit werden unendlichdimensionale Versionen dieses Satzes untersucht.

Sei Σ eine σ-Algebra von Teilmengen einer Menge Ω und X ein Banachraum. Wir werden sagen, daß X die Lyapunoveigenschaft hat, falls der Abschluß des Wertebereichs jedes nicht-atomaren X-wertigen Maßes eine konvexe Menge ist.

Im ersten Kapitel wird allgemeinere Information über Vektormaße dargestellt und das Dreiraumproblem in bezug auf diese Eigenschaft betrachtet. Nämlich wird folgendes bewiesen: wenn ein Unterraum eines Banachraums und sein Faktorraum die Lyapunoveigenschaft haben, besitzt dieser Raum selbst die Eigenschaft.

Im zweiten Kapitel werden die Verallgemeinerungen der Begriffe des Typs und Cotyps, der B- und C-Konvexität bezüglich der Lyapunoveigenschaft eingeführt. Es wird untersucht, unter welchen Voraussetzungen “Lyapunov” B- und C-konvexe Räume die Lyapunoveigenschaft haben.

Das letzte Kapitel beschäftigt sich mit einigen Beispielen von klassischen Banachräumen, die die Lyapunoveigenschaft haben oder nicht haben. Es wird bewiesen, daß die Orlicz-, Lorentz-, Baernsteinfolgenräume, die asymptotischen l_{p}-Räume, die keine isomorphe Kopie von l_{2} besitzen, der Schreierraum, der Tsirelsonraum, der Schlumprechtraum, der Gowers-Maurey-Raum sowie der Gowersraum die Lyapunoveigenschaft haben und daß der Tokarevraum die Lyapunoveigenschaft nicht hat.