Properties δ versus β

Helga Fetter and Berta Gamboa de Buen

CIMAT, Apdo. Postal 402, 36000 Guanajuato, Gto. México.

Abstract We compare Khamsi’s property β which implies weak normal structure with a weaker property called δ which also implies weak normal structure.

Introduction. The notions of normal and weak normal structures are intimately related to the fixed point theory of non expansive mappings in Banach spaces. In fact, it was proved by Kirk [4] that every Banach space with weak normal structure has the fixed point property

Khamsi in [3] gave a condition under which a Banach space has weak normal structure. Here we talk about a property which also implies weak normal structure and discuss its similarities and differences with Khamsi ’s condition.

1. Property δ

In this paper X will denote a real Banach space.

Definition 1.1. A sequence {x_{n}}_{n} in X is called diametral if c = diam {x_{n}}_{n} > 0 and

lim_{n→∞} d (x_{n}, conv {x_{i} : i < n}) = c,

where diamA denotes the diameter of A, convA the convex hull of A and d (x, A) the distance between x and A.

It follows that for every m, lim_{n→∞} ∥x_{n} − x_{m}∥ = c, that every subsequence of a diametral sequence is also diametral and if in addition x_{n} converges weakly to 0, (x_{n} →^{w} 0) then lim_{n→∞} ∥x_{n}∥ = c (see e.g. [2]).

The following definition of normal and weak normal structures is not the original one but is equivalent to it, thanks to a result by Brodskii and Milman [1].

Definition 1.2. 1. X has normal structure (NS) if no bounded convex subset K contains a diametral sequence.

2. X has weak normal structure (WNS) if no weakly compact convex subset K contains a diametral sequence.

Remark: In order to prove that X has WNS it suffices to show that it does not have a diametral sequence weakly convergent to zero.

Definition 1.3. Let X be a Banach space with a finite dimensional decomposition (F.D.D.) {X_{n}}_{n}. If x = ∑_{i} x_{i} with x_{i} ∈ X_{i} for every i ∈ ℕ, we define the support of x to be the set of integers i such that x_{i} ≠ 0. If y_{1}, y_{2}, y_{3}, ... ∈ X have finite support, we will say they have consecutive supports if the maximum of the support of y_{j} is less than the minimum of the support of y_{j+1}. We will say they have strictly consecutive supports if they have consecutive supports and have a hole between the supports.

Definition 1.4. Let K be a non-empty weakly compact convex subset of a Banach space X. We say that K has the fixed point property, if every T : K → K satisfying ∥T x − T y∥ ≤ ∥x − y∥ for all x, y ∈ K has a fixed point.

We say that X has the fixed point property (f.p.p.), if every weakly compact convex subset of X has the fixed point property.

Khamsi in [3] gave the following definition:

Definition 1.5. If X is a Banach space with a finite dimensional decomposition for which there exist p ∈ [0, ∞] and β < 2^{\frac{1}{p}} such that for every x, y ∈ X with consecutive supports

(∥x∥^{p} + ∥y∥^{p})^{\frac{1}{p}} ≤ β ∥x + y∥

then we say that X has property β.

We observe that for a space X with a F.D.D. if there exist p and β as in the previous theorem, this is equivalent to the following:

There exists δ > 0 such that for every x, y ∈ X with consecutive supports

∥x∥ + ∥y∥ ≤ (2 − δ) ∥x + y∥ .

This follows easily from the inequality for positive numbers a, b and p ∈ [1, ∞]

a + b ≤ 2^{\frac{p−1}{p}} (a^{p} + b^{p})^{\frac{1}{p}} ,

where for p = ∞ this is interpreted as a + b ≤ 2 max (a, b) .

Khamsi proved that a space having property β has weak normal structure; however in order to see that X has WNS, it is in fact enough to analyze the elements of norm one, as we state more precisely below.

Definition 1.6. Suppose X is such that there exists δ > 0 so that for every x, y ∈ X with consecutive supports and with ∥x∥ = ∥y∥

∥x∥ + ∥y∥ ≤ (2 − δ) ∥x + y∥ .

Then we say that X has property δ.

We will show that if X has property δ, the it has WNS. In order to do that we need the following lemma:

Lemma 1.7. Let X be a Banach space with a basis {x_{n}}_{n} . If there exists a diametral sequence {y_{n}}_{n} weakly convergent to zero with diameter 1, then there is a block basis {v_{n}}_{n} of {x_{n}}_{n} equivalent to a subsequence of {y_{n}}_{n} , so that

a) {v_{n}}_{n} is weakly null

b) ∥v_{n}∥ = 1

c) lim_{n→∞} lim_{m→∞} ∥v_{n} − v_{m}∥ = 1

Proof. Since {y_{n}}_{n} is weakly convergent to zero and is a diametral sequence it satisfies ∥y_{n} − y_{m}∥ ≤ 1, ∥y_{n}∥ ≤ 1, lim_{m→∞} ∥y_{n} − y_{m}∥ = 1 for every n, and lim_{m→∞} ∥y_{m}∥ = 1. Thus there exists a subsequence of {y_{n}}_{n} , which we will call {y_{n}}_{n} again, and a block basic sequence {u_{n}}_{n} of {x_{n}}_{n} so that ∥y_{n} − u_{n}∥ < \frac{1}{n}, lim_{n→∞} u_{n} = 1. But

− ∥y_{n} − u_{n}∥ − |1 − ∥u_{n}∥| + ∥y_{n} − y_{m}∥ − ∥y_{m} − u_{m}∥ − |1 − ∥u_{m}∥|

≤ ∥\frac{u_{n}}{∥u_{n}∥} − \frac{u_{m}}{∥u_{m}∥}∥ ≤ ∥y_{n} − \frac{u_{n}}{∥u_{n}∥}∥ + ∥y_{n} − y_{m}∥ + ∥y_{m} − \frac{u_{m}}{∥u_{m}∥}∥ ≤
∥y_{n} − u_{n}∥ + |1 − ∥u_{n}∥| + ∥y_{n} − y_{m}∥ + ∥y_{m} − u_{m}∥ + |1 − ∥u_{m}∥| .

Hence lim_{n→∞} lim_{m→∞} ∥\frac{u_{n}}{∥u_{n}∥} − \frac{u_{m}}{∥u_{m}∥} ∥ = 1. If v_{n} = \frac{u_{n}}{∥u_{n}∥}, this finishes the proof.

Theorem 1.8. Let X be a Banach space with F.D.D. and with property δ. Then X has WNS.

Proof. Suppose that X has property δ and there exists a weakly null diametral sequence {y_{n}}_{n=1}^{∞} in X. Let {v_{n}}_{n} be as in lemma 1.7. Then for every n and m

∥v_{n} − v_{m}∥ ≥ (2 − δ)

and this contradicts c) of said lemma.

However properties δ and β are not equivalent in general as the following example shows:

Example. Let X be the completion of the space of real sequences with finite support and

∥(a_{1}, a_{2}, ..., a_{n}, 0, ...)∥ = \{ \begin{matrix}max (|\frac{1}{8}a_{1} + a_{2}| , |a_{1} − \frac{1}{6}a_{2}|) + ∑_{i=3}^{n} |a_{i}| if a_{1}a_{2} ≥ 0 \\ ∑_{i=1}^{n} |a_{i}| if a_{1}a_{2} < 0 \end{matrix}

Clearly the sequence of unitary vectors {x_{n}}_{n} is a normalized basis for X. Let x, y ∈ X with consecutive support and ∥x∥ = ∥y∥ . For studying property δ the interesting cases are when x = ax_{1}, y = ax_{2}, or when x = a_{1}x_{1} + a_{2}x_{2} + ∑_{i=3}^{k} a_{i}x_{i}, y = ∑_{i=k+1}^{r} a_{i}x_{i} and a_{1}a_{2} > 0 or when x = a_{1}x_{1}, y = a_{2}x_{2} + ∑_{i=3}^{k} a_{i}x_{i} and a_{1}a_{2} > 0, because in the other cases the norm is the norm of l_{1} in which property δ holds with δ = 1.

In the first case

∥x + y∥ = \frac{9}{8} |a| = \frac{2 |a|}{2 − \frac{2}{9}} = \frac{∥x∥ + ∥y∥}{2 − \frac{2}{9}}.

In the second case, since ∥x∥ = ∥y∥ ,

max (|\frac{1}{8}a_{1} + a_{2}| , |a_{1} − \frac{1}{6}a_{2}|) + ∑_{i=3}^{k} |a_{i}| = ∑_{i=k+1}^{r} |a_{i}|

and by definition ∥x + y∥ = ∥x∥ + ∥y∥ .

In the third case ∥x∥ = |a_{1}| = ∑_{i=2}^{k} |a_{i}| = ∥y∥ and it is straightforward to see that

∥x + y∥ = max (|\frac{1}{8}a_{1} + a_{2}| , |a_{1} − \frac{1}{6}a_{2}|) + ∑_{i=3}^{k} |a_{i}| =
= max (|\frac{1}{8}a_{1} + a_{2}| , |a_{1} − \frac{1}{6}a_{2}|) + |a_{1}| − |a_{2}| ≥
≥ \frac{9}{8} |a_{1}| = \frac{∥x∥ + ∥y∥}{2 − \frac{2}{9}}.

Hence X has property δ for δ = \frac{2}{9}. However if x = x_{1} and y = \frac{3}{4}x_{2}, then

∥x + y∥ = \frac{7}{8} = \frac{1}{2} (∥x∥ + ∥y∥)

and therefore property β does not hold.

However there is an instance when properties β and δ are equivalent:

Theorem 1.9. Let X be a Banach space with a monotone basis {x_{n}}_{n} such that for every n and every k ≤ n, ∥∑_{i=k}^{n} a_{i}x_{i}∥ ≤ ∥∑_{i=1}^{n} a_{i}x_{i}∥ . Then properties β and δ are equivalent.

Proof. Property β always implies property δ.

Suppose now that property δ holds but property β does not. Then there exist sequences {y_{n}}_{n} and {z_{n}}_{n} so that for every n, y_{n} and z_{n} have consecutive supports and

∥y_{n}∥ + ∥z_{n}∥ ≥ (2 − \frac{1}{n}) ∥y_{n} + z_{n}∥ .

Without loss of generality we may suppose that 1 = ∥y_{n}∥ ≥ ∥z_{n}∥ = 1−ε_{n}. Hence, by hypothesis

2 − ε_{n} ≥ (2 − \frac{1}{n}) ∥y_{n} + z_{n}∥ ≥ (2 − \frac{1}{n}) ∥y_{n}∥ = 2 − \frac{1}{n}.

Therefore ε_{n} →_{n→∞} 0, ∥z_{n}∥ →_{n→∞} 1 and ∥y_{n} + z_{n}∥ →_{n→∞} 1. Since property δ holds we have:

2 ≤ (2 − δ) ∥y_{n} + \frac{z_{n}}{∥z_{n}∥}∥ ≤ (2 − δ) (∥y_{n} + z_{n}∥ + ∥z_{n} − \frac{z_{n}}{∥z_{n}∥}∥) =
= (2 − δ) (∥y_{n} + z_{n}∥ + ε_{n}) ,

and by passing to the limit as n → ∞ we get a contradiction.

Remark: One may restrict property δ even further without losing WNS. More precisely if X is a Banach space with a basis, so that there exists δ such that for every x and y with strict consecutive supports and ∥x∥ = ∥y∥ one has ∥x∥ + ∥y∥ ≤ (2 − δ) ∥x + y∥ , then X has WNS. This property is not necessarily equivalent to property δ if the basis of X is not symmetric as the following example shows.

Take X as the James space where if x = ∑_{i=1}^{∞} a_{i}x_{i} ∈ X

∥x∥ = sup (\frac{1}{2} ∑_{i=0}^{n} (a_{p_{i+1}} − a_{p_{i}})^{2})^{\frac{1}{2}} .

Here a_{0} = 0 and the sup is taken over all choices of n and all positive integers 0 = p_{0} < p_{1} < ⋯ < p_{n}. Then, ∥x_{1}∥ = ∥x_{2}∥ = ∥x_{1} + x_{2}∥ = 1 and thus X does not have property δ. However it is easy to see that if x and y have strictly consecutive supports and ∥x∥ = ∥y∥ ,

2 ∥x∥^{2} = ∥x∥^{2} + ∥y∥^{2} ≤ ∥x + y∥^{2}

and X has the weaker property.

References

[1] M. S. Brodskii and D. P. Milman, On the center of a convex set. Dokl. Akad. Nauk. SSSR 59 (1948), 837-840.

[2] J. Diestel, “Geometry of Banach spaces- Selected Topics ”. Springer Verlag, Berlin 1984.

[3] M. A. Khamsi, Normal structure for Banach spaces with Schauder decomposition. Can. Math. Bull. 32 (1989), 344-351.

[4] W. A. Kirk, A fixed point theorem for mappings which do not increase distance. Amer. Math. Monthly 72 (1965), 1004-1006.

[5] J. Lindenstrauss and L. Tzafriri, “Classical Banach spaces I ”. Springer Verlag, Berlin 1977.