CCCLXIII (ROZPRAWYMATEMATYCZNE) DISSERTATIONESMATHEMATICAE

D I S S E R T A T I O N E S M A T H E M A T I C A E

(ROZPRAWY MATEMATYCZNE)

K O M I T E T R E D A K C Y J N Y

A N D R Z E J B I A L Y N I C K I - B I R U L A, B O G D A N B O J A R S K I, Z B I G N I E W C I E S I E L S K I, J E R Z Y L O ´S,

Z B I G N I E W S E M A D E N I, J E R Z Y Z A B C Z Y K redaktor, W I E S L A W ˙Z E L A Z K O zast¸epca redaktora

CCCLXIII

S T A N I S L A W P R U S

Banach spaces and operators which are nearly uniformly convex

W A R S Z A W A 1997

Department of Mathematics Maria Curie-Sk lodowska University 20-031 Lublin, Poland

E-mail: bsprus@golem.umcs.lublin.pl

Published by the Institute of Mathematics, Polish Academy of Sciences Typeset in TEX at the Institute

Printed and bound by

P R I N T E D I N P O L A N D

c Copyright by Instytut Matematyczny PAN, Warszawa 1997

ISSN 0012-3862

Introduction . . . . 5

I. Basic definitions and notation . . . . 6

M-bases and finite-dimensional decompositions . . . . 6

Some geometric properties of Banach spaces . . . . 9

II. Constructions of equivalent norms . . . 12

III. (p, q)-estimates in interpolation spaces . . . 21

IV. Geometric properties of operators . . . 26

Nearly uniformly convex operators . . . 27

Nearly uniformly smooth operators . . . 30

V. Factoring operators through nearly uniformly convex spaces . . . 35

Factorizations and geometric properties of operators . . . 35

The case of spaces with finite-dimensional decompositions . . . 40

References . . . 45

1991 Mathematics Subject Classification: 46B03, 46B10, 46B70, 47A05.

Received 26.9.1994; revised version 8.11.1996.

One of the basic concepts of geometry of Banach spaces is that of uniform convexity.

It was introduced by J. A. Clarkson [9] in 1936. Five years later V. L. Shmulyan [41] cha- racterized the property dual to uniform convexity. It is called uniform smoothness. Since then these two notions have been extensively studied. They turned out to be useful for instance in the metric fixed point theory (see [15]). We point out, however, that uniform convexity and uniform smoothness describe geometric properties of finite-dimensional subspaces of a normed space. In contrast, most of the problems in the fixed point the- ory have global character. This was a motivation for considering infinite-dimensional counterparts of the classical geometric notions. One of them is nearly uniform convexity introduced by R. Huff in [19]. Independently a similar idea appeared in [16], where the modulus corresponding to nearly uniform convexity was defined.

In this paper we study some aspects of nearly uniform convexity and the dual property which is called nearly uniform smoothness. Their definitions can be found in the second part of Chapter I. The first part of that chapter contains basic facts concerning M-bases, bases and finite-dimensional decompositions of a Banach space. These notions play a fundamental role in this paper. For example, nearly uniform smoothness is defined in terms of basic sequences. We also recall some results from [34]. They give isomorphic characterizations of nearly uniform convexity and nearly uniform smoothness for Banach spaces with finite-dimensional decompositions. Those characterizations base on a concept of (p, q)-estimates.

In Chapter II we generalize an idea from [34] to M-bases. It allows us to prove coun- terparts of classical theorems due to G. Pisier and P. Enflo on equivalent norms. Namely we show that every nearly uniformly convex space has an equivalent norm for which the modulus satisfies a power type estimate. The second result shows that if the modulus is not identically equal to zero in the interior of its domain, then the space can be renormed in a nearly uniformly convex manner.

Chapter III is devoted to the study of the Lions–Peetre interpolation spaces. It is shown that this interpolation method preserves (p, q)-estimates. In the last chapter we use this result for constructing factorizations of operators through nearly uniformly convex spaces.

B. Beauzamy [5] carried over the idea of uniform convexity to bounded linear ope- rators. This kind of investigations was continued by V. I. Istr˘at¸escu [20]. He introduced the notion of a nearly uniformly convex operator. In Chapter IV we establish some pro- perties of such operators. We also define a class of nearly uniformly smooth operators.

It contains all operators adjoint to those which are nearly uniformly convex. Applying

one of the results of Beauzamy, we prove that nearly uniformly smooth operators factor through Banach spaces with the Banach–Saks property.

Operators which factor through Banach spaces with some special properties were considered for instance by W. J. Davis, T. Figiel, W. B. Johnson and A. Pe lczy´nski [10], B. Beauzamy [6] and S. Byrd [8]. In Chapter V we show that the properties considered in the previous chapter are in a sense necessary for operators to factor through nearly uniformly convex (or smooth) spaces. In case of spaces with finite-dimensional decompo- sitions we also find sufficient conditions for the existence of such factorizations.

This work is an English version of the paper [36], which was written in Polish. However, it is not just a translation. Some parts of the material were considerably changed. Let us also point out that some results of this paper can be found in [27]. Their proofs are included here for the sake of completeness.

The author wishes to express his gratitude to Prof. K. Goebel for his encouragement and Dr. D. Kutzarova for collaboration in preparing part of the material.

I. Basic definitions and notation

M-bases and finite-dimensional decompositions. In this paper we consider real Banach spaces. Let Z be a nonempty subset of such a space X. By sp Z we denote the linear subspace of X spanned by Z. Moreover, co Z stands for the closed convex hull of Z.

We will deal with various kinds of systems of finite-dimensional subspaces of Banach spaces. Let {Xt}_t∈T be such a system, where Xt are finite-dimensional subspaces of a space X. We say that the system {X_t}_t∈T satisfies condition (I) if each nonzero element x of spS

t∈TXt has a unique decomposition of the form x = P

t∈Axt, where A is a finite subset of T and x_t is a nonzero element of X_t for every t ∈ A. In this case x is called a block of the system {Xt}_t∈T and the set A is called the support of x. We write A = supp x. To make this definition complete we treat 0 as a block with supp 0 = ∅.

Let now B ⊂ T be a finite set. Having a block x = P

t∈Axt, where xt ∈ Xt for t ∈ A = supp x, we put

SBx = X

t∈A∩B

xt.

Here a sum over an empty set of indices is considered to be equal to zero. The linear mapping SB is called the projection associated with the system {Xt}_t∈T. If A and B are finite subsets of T , then

SA◦ SB = SA∩B. Moreover,

SA+ SB= SA∪B

whenever A ∩ B = ∅.

Usually we will impose some additional conditions which guarantee continuity of the projections S_A. In this case S_A can be extended in the natural way to the closure of the subspace spS

t∈TXt. The extension will be denoted by the same symbol SA.

This terminology can be used also for a system {vt}_t∈T of linearly independent vectors in X. We simply identify {vt}_t∈T with a system consisting of one-dimensional spaces X_t= sp{v_t}, t ∈ T . Clearly, this system satisfies condition (I). For instance, this is the case if {vt}_t∈T is an M-basis. Recall that a system {vt}_t∈T of vectors of a Banach space X is an M-basis of X if sp {v_t}_t∈T is dense in X and there is a family {v_t^∗}_t∈T ⊂ X^∗such that

v^∗_t(vs) =n0 if s 6= t, 1 if s = t.

The M-basis is said to be shrinking provided that sp {v_t^∗}_t∈T is dense in X^∗. For each finite set B ⊂ T the projection SB associated with an M-basis is continuous. Moreover, if an M-basis {vt}_t∈T of a space X is shrinking, then

SBx =X

t∈B

v^∗_t(x)vt and S_B^∗x^∗=X

t∈B

x^∗(vt)v_t^∗

for all x ∈ X, x^∗∈ X^∗. It is also easy to see that if (x_n) is a bounded sequence of blocks of a shrinking M-basis such that the supports of the elements xn are pairwise disjoint, then (xn) converges weakly to zero. The basic result on the existence of shrinking M-bases was proved in [23]. In this paper we will use only the following partial case of that theorem.

Theorem 1.1. If X is a reflexive Banach space, then there exists a shrinking M-basis of X.

Up to now we have considered general families of subspaces indexed by elements of sets without any natural order. In the special case when a family is countable it is more convenient to treat it as a sequence.

So let (Xn) be a sequence of finite-dimensional subspaces of a Banach space X which satisfies condition (I). We say that a sequence of subspaces (Y_n) is a blocking of (X_n) if there is an increasing sequence of integers (nk) such that n1= 0 and

Y_k= X_nk+1⊕ . . . ⊕ Xnk+1

for every k. The sequence (Y_k) satisfies condition (I), too.

By F we denote the family of all finite intervals of natural numbers, i.e. sets of the form A = {n ∈ N : a ≤ n ≤ b} for some a, b ∈ N. In this paper we will frequently deal with finite-dimensional decompositions (FDD for short). Recall therefore that a sequence (X_n) of finite-dimensional subspaces of a Banach space X is an FDD of X if each element x ∈ X has a unique decomposition of the form x =P∞

n=1xn, where xn ∈ Xn for every n.

Clearly, an FDD (X_n) satisfies condition (I) and the projections S_A associated with it are continuous. Moreover, there exists a constant C such that if A ∈ F , then kSAk ≤ C (see [30]). Put An= {k ∈ N : 1 ≤ k ≤ n} for n ∈ N. The number sup_n∈NkSA_nk is called the decomposition constant of (X_n).

We now consider a sequence (Pn) of finite-dimensional continuous projections on a Banach space X such that spS∞

n=1Xn is dense in X, Pm◦ Pn = Pmin{m,n} for all m, n and sup_n∈NkPnk < ∞. Then the subspaces X1 = P1(X), Xn = (Pn− Pn−1)(X), n = 2, 3, . . . form an FDD of the space X (see [30]).

The definition of a shrinking M-basis has a counterpart for FDD. Namely, an FDD (X_n) of a Banach space X is said to be shrinking if the spaces S_{n}^∗ (X^∗) form an FDD of X^∗. If the space X is reflexive, then every FDD of X is shrinking.

We also recall that a sequence (en) of nonzero vectors of a Banach space X is a basis of this space provided that the corresponding sequence of one-dimensional subspaces sp{en} is an FDD of X. Therefore each element x ∈ X has a unique decomposition of the form x =P∞

n=1αnenfor some sequence (αn) of scalars. The decomposition constant C of this FDD is called the basic constant of (e_n). It is well-known that the functionals e^∗_n given by the formula

e^∗_nX^∞

k=1

αkek



= αn

are uniformly bounded. In fact, ke^∗_nk ≤ 2C for every n. This shows, in particular, that a basis is an M-basis. The functionals e^∗_n are called the coefficient functionals of the basis (e_n).

If a sequence (xn) is a basis only of the closure of the subspace Y = sp {xn}_n∈N, then (x_n) is said to be a basic sequence. Clearly, the coefficient functionals of the basic sequence (xn) can be extended from the subspace Y to the whole space X without changing their norms.

We now state a version of the well-known theorem on the existence of basic sequen- ces [30].

Theorem 1.2. Let a sequence (xⁿ) converge weakly to zero in a Banach space X. If lim inf_n→∞kxnk > 0, then for every ε > 0 there exists a subsequence (xnk) which is a basic sequence with the basic constant not exceeding 1 + ε. Moreover , if x16= 0, then the subsequence can be chosen in such a way that n₁= 1.

Let a sequence (X_n) of finite-dimensional subspaces of a Banach space X satisfy condition (I). We say that blocks x1, . . . , xmof (Xn) are disjoint if the following condition holds with α = 0:

max{min supp x_i− max supp xj, min supp x_j− max supp xi} > α

whenever i 6= j. If this condition is satisfied with α = 1, then the blocks x1, . . . , xm are said to be strongly disjoint .

Fix now p, q ∈ [1, ∞) so that q ≤ p. We say that (X_n) satisfies (p, q)-estimates if there are positive constants c, C such that

(1.1) cX^∞

i=1

kSA_ixk^p^1/p

≤ kxk ≤ CX^∞

i=1

kSA_ixk^q^1/q

for every system of pairwise disjoint intervals A₁, . . . , A_n ∈ F and every block x with supp x ⊂Sn

i=1Ai. This definition can be extended to the case when p = ∞ by replacing the left hand side expression in (1.1) by c max_1≤i≤nkS_Aixk.

We point out that for q = 1 the right inequality in (1.1) is just a consequence of the triangle inequality. So (p, 1)-estimates reduce in fact to the left inequality. If p = ∞, then the left inequality in (1.1) means in turn that the norms of the projections S_A, where A ∈ F , are uniformly bounded. This is equivalent to (Xn) being an FDD.

Some geometric properties of Banach spaces. By BX we denote the closed unit ball of a Banach space X. Let (xn) be a sequence in X. We put

sep(xn) = inf{kxm− xnk : m 6= n}.

In this paper we deal with geometric properties of infinite-dimensional character. We start by recalling the classical notions. The modulus of convexity δX of a Banach space X is a function defined on the interval [0, 2] by

δX(ε) = inf

 1 − 1

2kx + yk : x, y ∈ BX, kx − yk ≥ ε

 .

The space X is uniformly convex if δ_X(ε) > 0 for every ε ∈ (0, 2]. It is well-known that uniform convexity implies reflexivity (see for instance [11], p. 37).

Another function which describes the shape of the unit ball of X is the modulus of smoothness %X. It is defined by

%X(τ ) = sup 1

2(kx + τ yk + kx − τ yk) − 1 : x, y ∈ BX

 ,

where τ ∈ [0, ∞). The space X is uniformly smooth if limτ →0%X(τ )/τ = 0. These two properties are dual to each other. So a Banach space X is uniformly convex if and only if X^∗ is uniformly smooth (see [41]).

In [19] R. Huff introduced the following infinite-dimensional counterpart of uniform convexity. A Banach space X is nearly uniformly convex (NUC for short) provided that for every ε > 0 there is δ > 0 such that if (xn) is a sequence in BX with sep(xn) ≥ ε, then

inf{kxk : x ∈ co{xn}} < 1 − δ.

Also the next definition was formulated in [19]. A Banach space X is uniformly Kadec–

Klee (UKK for short) whenever for every ε > 0 there is δ > 0 such that if (xn) is a sequence in B_X which converges weakly to x and sep(x_n) ≥ ε, then kxk < 1 − δ. If X is finite-dimensional, then it satisfies these conditions in a trivial way.

Theorem 1.3 [19]. A Banach space X is NUC if and only if X is reflexive and UKK.

Consider a sequence (x_n) converging weakly to x and such that a = lim inf

n→∞ kxn− xk > 0.

Having γ > 0 we can assume that kxn− xk ≥ a(1 − γ/2) for every n. Moreover, The- orem 1.2 shows that there exists a subsequence (xn_k− x) which is a basic sequence with the basic constant not exceeding 1 + γ/2. Therefore

a(1 − γ/2) ≤ kxn_k− xk ≤ (1 + γ/2)kxn_k− xnik, whenever k < i. Consequently,

a(1 − γ) ≤ sep(xn_j).

On the other hand,

sep(xn) ≤ kxj− xk + kxm− xk

if j 6= m. It follows that the inequality kxm− xk < ¹₂sep(xn) may hold for at most one index m. Hence

1

2sep(xn) ≤ lim inf

m→∞ kxm− xk.

The above argument shows that in the definition of UKK one can replace the condition sep(xn) ≥ ε by lim infn→∞kxn− xk ≥ ε.

A condition equivalent to NUC was independently formulated in [16] (see also [15]).

The authors used the notion of a measure of noncompactness of a set (see [3]). Here we recall only the definition of so-called Hausdorff measure. Let A be a bounded subset of a Banach space X. The Hausdorff measure of noncompactness χ(A) is the infimum of all numbers r > 0 such that A can be covered by finitely many balls with radii r.

Let now X be an infinite-dimensional Banach space. Having ε ∈ [0, 1] we put

∆_X(ε) = inf{1 − inf{kxk : x ∈ A}},

where the infimum is taken over all convex closed sets A ⊂ BXwith χ(A) ≥ ε. Following [16] the function ∆X is called the modulus of noncompact convexity of the space X. It is easy to see that X is NUC if and only if ∆X(ε) > 0 for every ε ∈ (0, 1]. Other properties of ∆X were investigated in [1], [2], [35] and [37].

Consider an infinite-dimensional reflexive space X. For ε ∈ [0, 1] we set KX(ε) = inf{1 − kxk},

where the infimum is taken over all x ∈ X which are weak limits of sequences (xn) in B_X such that lim inf_n→∞kxn− xk ≥ ε (cf. [37]).

Remark 1.4. If X is an infinite-dimensional reflexive space, then KX(ε) ≤ ∆X(ε) for every ε ∈ [0, 1].

P r o o f. Let A be a closed convex subset of B_X with χ(A) ≥ ε. An easy induction procedure gives us a sequence (xn) in A such that

(1.2) dist(xn+1, co{xi}ⁿ_i=1) ≥ ε − 1/n

for every n ∈ N. Since X is reflexive, we can assume that (xn) converges weakly to an x ∈ A. From (1.2) it follows that

lim inf

n→∞ kxn− xk ≥ ε.

Therefore

K_X(ε) ≤ 1 − kxk ≤ 1 − inf{kyk : y ∈ A}.

This gives us the desired inequality.

Examples. 1. If p ∈ (1, ∞), then the space l^p is NUC. Moreover, in [16] it was proved that

∆l_p(ε) = 1 − (1 − ε^p)^1/p

for every ε ∈ [0, 1]. The same argument shows that if (Xn) is an arbitrary sequence of finite dimensional Banach spaces, then the modulus of noncompact convexity of the space (P∞

n=1Xn)l_p is equal to ∆l_p.

2. Take p ∈ (1, ∞), p 6= 2, and put X = Lp(0, 1). The exact value of ∆X(ε) for ε > 0 is not known. Some estimates of that value were established in [2]. To obtain an upper estimate it suffices to notice that X contains a subspace Y such that Y is isometrically isomorphic with lp and there is a norm-one projection of X onto Y . It follows that

∆_X(ε) ≤ ∆_lp(ε) for ε ∈ [0, 1].

We point out that these functions are not equal. Indeed, X does not have the weak Opial property (see [32]). Therefore a result from [35] shows that

∆_X(1−) < 1.

The lower estimate given in [2] can be improved. For this purpose it suffices to apply the following result (see [29] and [40]). There is a constant c_p ∈ (0, 1) such that for all x, y ∈ X with x 6= 0 we have

kxk^q+ c_pky − xk^q+ τ_q(x, y − x) ≤ kyk^q,

where q = max{2, p} and τq(x, y − x) denotes the directional derivative of the function z 7→ kzk^q at x in the direction y − x. If 1 < p < 2 one can just take cp = p − 1 and if p > 2 it is known that c_p≥ 2^2−p.

Let (x_n) converge weakly to x in the space X. Then the above inequality shows that kxk^q+ cplim inf

n→∞ kxn− xk^q≤ lim inf

n→∞ kxnk^q. Therefore by Remark 1.4 we obtain

1 − (1 − c_pε^q)^1/q≤ ∆_X(ε) for ε ∈ [0, 1].

In the next chapter we will show that each NUC space can be renormed in such a way that its modulus of noncompact convexity satisfies an estimate of this kind.

We now turn to a condition dual to NUC (see [34]). A Banach space X is nearly uniformly smooth (NUS for short) provided that for every ε > 0 there exists η > 0 such that if 0 < τ < η and (xn) is a basic sequence in BX, then

kx1+ τ xkk ≤ 1 + ετ

for some k > 1. Clearly, this condition becomes trivial if (x_n) has a convergent sub- sequence. Therefore finite-dimensional spaces are NUS. Moreover, uniform smoothness implies NUS, which in turn implies reflexivity of the space (see [34]).

Theorem 1.5 [34]. A Banach space X is NUC if and only if the dual space X^∗ is NUS.

Obviously, in Theorem 1.5 we can replace X by X^∗ and vice versa. In the sequel we frequently consider a few norms in a given Banach space X. Then it is more convenient to denote the space X endowed with a norm k · k by (X, k · k). We say that the norm k · k has some property whenever the space (X, k · k) has this property.

Banach spaces with bases which admit equivalent NUC norms were characterized in [34]. The same method of proof can be used for spaces with FDD.

Theorem 1.6. Let X be a reflexive space with an FDD (Xn). Then the following conditions are equivalent :

(1) The space X admits an equivalent NUC norm.

(2) There exist a constant p > 1 and a blocking (Yk) of (Xn) such that (Yk) satisfies (p, 1)-estimates.

The paper [34] contains an analogous result on NUS spaces with bases. It can be easily generalized to spaces with FDD, too.

Theorem 1.7. Let X be a reflexive space with an FDD (Xn). Then the following conditions are equivalent :

(1) The space X admits an equivalent NUS norm.

(2) There exist a constant q > 1 and a blocking (Yk) of (Xn) such that (Yk) satisfies (∞, q)-estimates.

II. Constructions of equivalent norms

In this chapter we will establish counterparts of some well-known renorming results concerning uniformly convex spaces. One of them is the theorem of G. Pisier [33]. It states that for every uniformly convex space X there exist an equivalent norm k · k₀ and constants C > 0, p > 1 such that the modulus of convexity δ of the space (X, k · k₀) satisfies

δ(ε) ≥ Cε^p

for all ε ∈ [0, 2]. We will prove an analogous result for the modulus of noncompact convexity. Our proof extends the idea used in [34] for spaces with bases. This time we use shrinking M-bases.

In [28] a weak version of NUS was introduced. A Banach space X is weakly NUS (WNUS for short) whenever there exist constants c, η ∈ (0, 1) such that if 0 < τ < η and (xn) is a basic sequence in BX, then

kx1+ τ xkk ≤ 1 + cτ

for some k > 1. Clearly, if a space X is NUS, then it is WNUS. We also recall two observations from [28].

Remark 2.1. A Banach space X is WNUS if and only if there exists a constant r ∈ (0, 2) such that if (xn) is a basic sequence in BX, then

kx1+ x_kk ≤ 2 − r for some k > 1.

Remark 2.2. If a Banach space X is WNUS , then X is reflexive.

From Remark 2.2 and Theorem 1.1 it follows that every WNUS space has a shrinking M-basis.

Lemma 2.3. Let X be a WNUS Banach space with a shrinking M-basis {vt}_t∈T. There is a constant r ∈ (0, 2) such that for every finite set A ⊂ T there exists a finite set B ⊂ T for which A ⊂ B and

kx + yk ≤ 2 − r

whenever x, y ∈ B_X are blocks with supp x ⊂ A and supp y ⊂ T \ B.

P r o o f. Assume that a WNUS space X does not satisfy the conclusion of the lemma.

By Remark 2.1 there is a constant r ∈ (0, 2) such that if (zn) is a basic sequence in BX, then there is a subsequence (zn_k) with

kz1+ zn_kk ≤ 2 − r

for every k. Now, from our assumption it follows that there exist a finite set A ⊂ T and two sequences of blocks (x_n), (y_n) in B_Xsuch that the supports of blocks y_nare pairwise disjoint, supp xn⊂ A and

(2.1) kxn+ ynk > 2 −1

3r

for every n. Since (xn) is a bounded sequence in a finite-dimensional space sp {vt}_t∈A, we can assume that it converges in norm to an element x. Clearly, x ∈ B_Xand inequality (2.1) shows that x 6= 0. The sequence (yn) converges weakly to zero and from (2.1) we see that

lim inf

n→∞ kynk ≥ 1 3.

In view of Theorem 1.2 we can therefore assume that the elements x, y1, y2, . . . form a basic sequence. Thus there exists a subsequence (y_nk) such that

kx + yn_kk ≤ 2 − r for every k. Consequently,

lim inf

n→∞ kxn+ ynk = lim inf

n→∞ kx + ynk ≤ 2 − r which contradicts (2.1).

Now consider a WNUS Banach space X with a shrinking M-basis {vt}_t∈T. We fix a constant r > 0 which satisfies the conclusion of Lemma 2.3.

We say that finite sets A1, . . . , An⊂ T (in a given order) form an admissible system if for every k = 1, . . . , n there is a finite set B ⊂ T such that

(1)Sk

i=1A_i⊂ B, (2)Sn

i=k+1A_i⊂ T \ B,

(3) kx + yk ≤ 2 − r whenever x, y ∈ BX are blocks with supp x ⊂Sk

i=1Ai, supp y ⊂ T \ B.

Sets forming an admissible system are clearly pairwise disjoint. Moreover, any sub- sequence of an admissible system is an admissible system. Let us also formulate the following obvious remark.

Remark 2.4. Let A¹, . . . , An be an admissible system and 0 = k1< k2< . . . < km= n. Then the sets B_j=S^kj+1

i=k_j+1A_i, j = 1, . . . , m − 1, form an admissible system.

Theorem 2.5. Let X be a WNUS Banach space with a shrinking M-basis {vt}_t∈T. There exist constants q > 1, C > 0 such that

n

X

i=1

xi

≤ CXⁿ

i=1

kxik^q1/q

whenever x1, . . . , xn are blocks for which the sets supp x1, . . . , supp xnform an admissible system.

P r o o f. Let r > 0 satisfy the conclusion of Lemma 2.3. We take n ∈ N so large that (2.2)

 1 −r

2

n

<1 2.

Assume that the conclusion of the theorem is false. Following the idea of the proof of Theorem 2 [22], we obtain blocks z₁, . . . , z₂n ∈ B_Xsuch that the sets supp z₁, . . . , supp z₂n

form an admissible system and

(2.3) 2ⁿ⁻¹≤

2ⁿ

X

i=1

zi

. Then kz2j−1+ z2jk ≤ 2 − r for every j = 1, . . . , 2ⁿ⁻¹.

Remark 2.4 shows that the supports of the blocks z₁+ z₂, . . . , z₂n−1+ z₂n also form an admissible system. Consequently,

kz4j−3+ z4j−2+ z4j−1+ z4jk ≤ (2 − r)² for every j = 1, . . . , 2ⁿ⁻².

After n steps, this induction procedure leads us to the inequality

2ⁿ

X

i=1

z_i

≤ (2 − r)ⁿ.

From this and (2.3) we see that 2ⁿ⁻¹≤ (2 − r)ⁿ, which contradicts (2.2).

A different proof of Theorem 2.5 can be obtained as a modification of the proof of the well-known theorem of V. I. and N. I. Gurari˘ı on lp-type estimates in uniformly convex spaces (see [17] or [12]).

Consider the special case when the space X is separable. Then its M-basis is countable.

Remark 2.6. Let X be a separable WNUS Banach space with a shrinking M-basis (v_n). There exist a blocking (X_k) of (v_n) and a constant q > 1 such that (X_k) satisfies (∞, q)-estimates.

P r o o f. We construct an increasing sequence of integers (n_k). First we put n₁ = 1.

Next, having numbers n1, . . . , nm, we apply Lemma 2.3 with A = {1, . . . , nm}. It gives us a finite set of integers B such that A ⊂ B. Let n_m+1= max B + 1.

Now consider intervals A1, . . . , Am ∈ F such that max Ai < min Ai+1 for every i = 1, . . . , m − 1. We put ki = min{nj : j ∈ Ai}, li = max{nj : j ∈ Ai} and Fi = {n ∈ N : k_i ≤ n ≤ li}. Then the sets F1, F₃, . . . form an admissible system and the same is true for the sets F2, F4, . . .

Therefore Theorem 2.5 gives us constants q > 1, C > 0 such that if x ∈ X is a block of (vn) with supp x ⊂Sm

i=1Fi, then kxk ≤

XS_F2i−1x +

XS_F2ix

≤ C X

kSF2i−1xk^q^1/q

+ X

kSF2ixk^q^1/q

≤ C2^1/pX^m

i=1

kSFixk^q^1/q , where 1/p + 1/q = 1 and S_Fj are projections associated with (v_n).

We take a blocking (X_k) of (v_n) given by

Xk = sp{vn_k, . . . , vn_k+1−1},

for every k. Then each projection SF_j is also associated with (Xk), but this time it corresponds to the interval Aj. Thus our last inequality means that (Xk) satisfies (∞, q)- estimates.

We can apply Theorem 2.5 to construct a special norm in a WNUS space.

Theorem 2.7. Let X be a WNUS Banach space. There exist constants q > 1, M > 0 and an equivalent norm k · k₁ such that if (x_n) is a weakly null sequence in X, then

lim inf

n→∞ kx1+ xnk^q₁≤ kx1k^q₁+ M lim inf

n→∞ kxnk^q₁.

P r o o f. Let (X, k · k) be a WNUS space. By Theorem 1.1 there exists a shrinking M-basis {vt}_t∈T of X. In this proof SA will denote a projection associated with this M-basis. Let q > 1, C > 0 be as in Theorem 2.5. Having a block x ∈ X, we put

|x| = infnXⁿ

i=1

kS_Aixk^q1/qo ,

where the infimum is taken over all admissible systems A₁, . . . , A_n such that supp x ⊂ Sn

i=1Ai. Clearly, in this case x =Pn

i=1SA_ix. So Theorem 2.5 shows that kxk ≤ C|x|.

On the other hand, considering an admissible system consisting of the single set supp x, we see that |x| ≤ kxk. Therefore C⁻¹kxk ≤ |x| ≤ kxk.

The desired norm for a block x ∈ X is given by kxk1= infnXⁿ

i=1

|zi|o ,

where the infimum is taken over all finite systems of blocks z₁, . . . , z_n such that x = Pn

i=1zi. It is easy to see that

(2.4) C⁻¹kxk ≤ kxk₁≤ kxk

for every block x. Since the subspace of all blocks is dense in X, the norm k · k₁ can be extended in a natural way to the whole space X. Inequalities (2.4) hold also for the extension (this time for all elements x ∈ X).

Fix γ > 0 and take a sequence of blocks (u_n) whose supports are pairwise disjoint.

We consider the case when u16= 0.

From the definition of the norm k · k₁ it follows that there exist nonzero blocks z1, . . . , zmsuch that u1=Pm

i=1zi and (2.5)

m

X

i=1

|zi| ≤ (1 + γ0)^1/(2q)ku1k1

where γ0= γku1k^−q₁ . Then for a fixed index i the definition of |zi| gives us an admissible system Aⁱ₁, . . . , Aⁱ_n for which supp zi⊂Sk

j=1Aⁱ_j and

n

X

j=1

kS_Aⁱ

jz_ik^q ≤ (1 + γ0)^1/2|zi|^q.

For simplicity of notation we can assume that n does not depend on i.

We put

A =

m

[

i=1 n

[

j=1

Aⁱ_j.

From Lemma 2.3 it follows that there exists a finite set B ⊂ T such that A ⊂ B and if D ⊂ T \ B is finite, then for every i = 1, . . . , m the system Aⁱ₁, . . . , Aⁱ_n, D is admissible.

Therefore for each element y with supp y ⊂ D we have

|zi+ y| ≤Xⁿ

j=1

kS_Ai

j(zi+ y)k^q+ kSD(zi+ y)k^q1/q

(2.6)

=Xⁿ

j=1

kS_Ai

jzik^q+ kyk^q1/q

≤ ((1 + γ0)^1/2|zi|^q+ kyk^q)^1/q

for i = 1, . . . , m. Since the supports of the blocks ui are pairwise disjoint, we can find k0> 1 so that supp uk ⊂ T \ B for all k ≥ k0.

Fix such k and consider the decomposition u1+ uk =

m

X

i=1



zi+X^m

j=1

|zj|−1

|zi|uk

 . By the definition of the norm k · k₁ and inequality (2.6) we obtain

ku1+ ukk1≤

m

X

i=1

zi+X^m

j=1

|zj|−1

|zi|uk

≤

m

X

i=1

(1 + γ₀)^1/2|zi|^q+X^m

j=1

|zj|^−q

(|z_i|kukk)^q^1/q

=

(1 + γ₀)^1/2X^m

i=1

|z_i|q

+ ku_kk^q1/q

. In view of (2.5) and (2.4) this shows that

ku1+ ukk^q₁≤ (1 + γ0)ku1k^q₁+ kukk^q ≤ ku1k^q₁+ C^qkukk^q₁+ γ.

Since C ≥ 1, this inequality holds also in the case when u1 = 0. Taking lim inf of both

sides and letting γ tend to 0, we get lim inf

k→∞ ku1+ u_kk^q₁≤ ku1k^q₁+ C^qlim inf

k→∞ kukk^q₁.

Now consider an arbitrary weakly null sequence (xn) in X. There is a sequence of blocks (zn) with kx1− znk1 ≤ 1/n for every n. Next, take a subsequence (xn_k) for which limk→∞kxn_kk₁= lim infn→∞kxnk₁. Passing to a subsequence once more, we can assume that there exists a sequence of blocks (u_k) such that supports of u_k are pair- wise disjoint and kxn_k− ukk₁ ≤ 1/k for all k. Then we also have |kzn+ ukk1− kx1+ x_nkk₁| ≤ 1/n+1/k for every k and n. It follows that lim inf_n→∞lim inf_k→∞kz_n+ u_kk₁= lim infk→∞kx1+ xn_kk₁. But the first part of the proof shows that

lim inf

k→∞ kzn+ ukk^q₁≤ kznk^q₁+ C^qlim inf

k→∞ kukk^q₁ for each n. Therefore

lim inf

k→∞ kx1+ xkk^q₁≤ lim inf

n→∞ lim inf

k→∞ kzn+ ukk^q₁

≤ lim

n→∞kznk^q₁+ C^qlim inf

k→∞ kukk^q₁= kx1k^q₁+ C^qlim inf

k→∞ kxkk^q₁. By duality we can now obtain some special norms in NUC spaces.

Theorem 2.8. Let X be a NUC Banach space. There exist constants p ∈ (1, ∞), m > 0 and an equivalent norm k · k₀ such that if (x_n) is a weakly null sequence in X, then

kx1k^p₀+ m lim inf

n→∞ kxnk^p₀≤ lim inf

n→∞ kx1+ x_nk^p₀.

P r o o f. If a space X is NUC, then its dual X^∗ is WNUS. From Theorem 2.7 we obtain constants q, M and an equivalent norm k · k₁ in X^∗ which satisfy its conclusion.

By Remark 2.2 the space X^∗is reflexive. So there exists an equivalent norm k·k0in X for which k · k₁ is the dual norm. We take a weakly null sequence (x_n) in X. We can choose a subsequence (xn_k) so that limk→∞kx1+ xn_kk0= a, where a = lim infn→∞kx1+ xnk0

and the limit b = lim_k→∞kx_nkk₀ exists. Then there are norm-one functionals y^∗_k ∈ X^∗ such that y^∗₁(x1) = kx1k0 and y_k^∗(xn_k) = kxn_kk0 for every k > 1. We can assume that the sequence (y^∗_k) converges weakly to some y^∗. We put x^∗₁ = kx1k^p−1₀ y₁^∗ and x^∗_k = kxnkk^p−1₀ 2^−pM^1−p(y^∗_k− y^∗) for k > 1, where 1/p + 1/q = 1. Theorem 2.7 shows that (2.7) lim inf

k→∞ kx^∗₁+ x^∗_kk^q₁≤ kx^∗₁k^q₁+ M lim inf

k→∞ kx^∗_kk^q₁≤ kx₁k^p₀+ M^1−p2^−pb^p.

Consider functionals z_k^∗ = (kx1k^p₀+ M^1−p2^−pkxn_kk^p₀)^−1/q(x^∗₁+ x^∗_k). From (2.7) we see that lim inf_k→∞kz_k^∗k₁≤ 1. Hence

a ≥ lim inf

k→∞ z_k^∗(x1+ xn_k)

= (kx₁k^p₀+ M^1−p2^−pb^p)^−1/q(x^∗₁(x₁) + lim

k→∞x^∗_k(x_nk))

= (kx1k^p₀+ M^1−p2^−pb^p)^1/p.

Theorem 2.8 gives us a norm with a nice estimate for the modulus of noncompact convexity. Namely we have the following result.