A geometrical constant and normal normal structure in Banach Spaces

Abstract

Recently, we introduced a new coefficient as a generalization of the modulus of smoothness and Pythagorean modulus such as J _X , _p (t). In this paper, We can compute the constant J _X , _p (1) under the absolute normalized norms on ℝ² by means of their corresponding continuous convex functions on [0, 1]. Moreover, some sufficient conditions which imply uniform normal structure are presented.

2000 Mathematics Subject Classification: 46B20.

1. Introduction and preliminaries

We assume that X and X* stand for a Banach space and its dual space, respectively. By S _X and B _X we denote the unit sphere and the unit ball of a Banach space X, respectively. Let C be a non-empty bounded closed convex subset of a Banach space X. A mapping T : C → C is said to be non-expansive provided the inequality

holds for every x, y ∈ C. A Banach space X is said to have the fixed point property if every non-expansive mapping T : C → C has a fixed point, where C is a non-empty bounded closed convex subset of a Banach space X.

Recall that a Banach space X is called uniformly non-square if there exists δ > 0 such that ||x + y||/2 ≤ 1 - δ or ||x - y||/2 ≤ 1 - δ whenever x, y ∈ S _X . A bounded convex subset K of a Banach space X is said to have normal structure if for every convex subset H of K that contains more than one point, there exists a point x₀ ∈ H such that

A Banach space X is said to have uniform normal structure if there exists 0 < c < 1 such that for any closed bounded convex subset K of X that contains more than one point, there exists x₀ ∈ K such that

It was proved by Kirk that every reflexive Banach space with normal structure has the fixed point property.

There are several constants defined on Banach spaces such as the James [1] and von Neumann-Jordan constants [2]. It has been shown that these constants are very useful in geometric theory of Banach spaces, which enable us to classify several important concept of Banach spaces such as uniformly non-squareness and uniform normal structure [3–8]. On the other hand, calculation of the constant for some concrete spaces is also of some interest [2, 5, 6, 9].

Recently, we introduced a new coefficient as a generalization of the modulus of smoothness and Pythagorean modulus such as J _X , _p (t).

Definition 1.1. Let x ∈ S _X , y ∈ S _X . For any t > 0, 1 ≤ p < ∞ we set

Some basic properties of this new coefficient are investigated in [6]. In particular, we compute the new coefficient in the Banach spaces l _r , L _r , l₁, ∞ and give rough estimates of the constant in some concrete Banach spaces. In fact, the constant J_{X, p}(1) is also important from the below Corollary in [6].

Corollary 1.2. If . Then R(X) < 2, where R(X) and ω(X) stand for García-Falset constant and the coefficient of weak orthogonality, respectively (see [10, 11]). It is well known that a reflexive Banach space X with R(X) < 2 enjoys the fixed property (see [10]).

In this paper, we compute the constant J _X , _p (1) under the absolute normalized norms on ℝ², and give exact values of the constant J _X , _p (1) in some concrete Banach spaces. Moreover, some sufficient conditions which imply uniform normal structure are presented.

Recall that a norm on ℝ² is called absolute if ||(z, w)|| = ||(|z|, |w|)|| for all z, w ∈ ℝ and normalized if ||(1,0)|| = ||(0,1)||. Let N _α denote the family of all absolute normalized norms on ℝ², and let Ψ denote the family of all continuous convex functions on [0, 1] such that ψ (1) = ψ (0) = 1 and max{1 - s, s} ≤ ψ(s) ≤ 1(0 ≤ s ≤ 1). It has been shown that N _α and Ψ are a one-to-one correspondence in view of the following proposition in [12].

Proposition 1.3. If ||·||∈ N _α , then ψ(s) = ||(1 - s, s)|| ∈ Ψ. On the other hand, if ψ(s) ∈Ψ, defined a norm ||·|| _ψ as

then the norm ||·|| _ψ ∈ N _α .

A simple example of absolute normalized norm is usual l _r (1 ≤ r ≤ ∞) norm. From Proposition 1.3, one can easily get the corresponding function of the l _r norm:

Also, the above correspondence enable us to get many non-l _r norms on ℝ². One of the properties of these norms is stated in the following result.

Proposition 1.4. Let ψ, φ ∈ Ψ and φ ≤ ψ. Put , then

The Cesà ro sequence space was defined by Shue [13] in 1970. It is very useful in the theory of matrix operators and others. Let l be the space of real sequences.

For 1 < p < ∞, the Cesà ro sequence space ces _p is defined by

The geometry of Cesà ro sequence spaces have been extensively studied in [14–16]. Let us restrict ourselves to the two-dimensional Cesà ro sequence space which is just ℝ² equipped with the norm defined by

2. Geometrical constant J _{X, p} (1) and absolute normalized norm

In this section, we give a simple method to determine and estimate the constant J _{X, p} (1) of absolute normalized norms on ℝ². For a norm || · || on ℝ², we write J _{X, p} (1)(|| · ||) for J _{X, p} (1)(ℝ², || · ||). The following is a direct result of Proposition 2.4 in [6].

Proposition 2.1. Let X be a non-trivial Banach space. Then

Proposition 2.2. Let X be the space l _r or L _r [0, 1] with dimX ≥ 2 (see [6])

• (1) Let 1 < r ≤ 2 and 1/r + 1/r' = 1. Then for all t > 0

if 1 < p < r' then .

if r' ≤ p < ∞ then , for some K ≥ 1.

• (2) Let 2 ≤ r < ∞, 1 ≤ p < ∞ and h = max{r, p}. Then

  

Proposition 2.3. Let φ ∈ Ψ and ψ(s) = φ (1 - s). Then

Proof. For any x = (a, b) ∈ ℝ² and a ≠ 0, b ≠ 0, put . Then

Consequently, we have

We now consider the constant J _{X, p} (1) of a class of absolute normalized norms on ℝ². Now let us put

Theorem 2.4. Let ψ ∈ Ψ and ψ ≤ ψ _r (2 ≤ r < ∞). If the function attains its maximum at s = 1/ 2 and r ≥ p, then

Proof. By Proposition 1.4, we have || · || _ψ ≤ || · || _r ≤ M₁|| · || _ψ . Let x, y ∈ X, (x, y) ≠ (0, 0), where X = ℝ². Then

from the definition of J _{X, p} (t), implies that

Note that r ≥ p and the function attains its maximum at s = 1/ 2, i.e.,. From Proposition 2.2, implies that

(1)

On the other hand, let us put x = (a, a), y = (a, -a), where . Hence ||x|| _ψ = ||y|| _ψ = 1, and

(2)

From (1) and (2), we have

Theorem 2.5. Let ψ ∈ Ψ and ψ ≥ ψ _r (1 ≤ r ≤ 2). If the function attains its maximum at s = 1/ 2 and 1 ≤ p < r', then

Proof. By Proposition 1.4, we have || · || _r ≤ || · || _ψ ≤ M₂|| · || _r . Let x, y ∈ X, (x, y) ≠ (0, 0), where X = ℝ². Then

From the definition of J _{X, p} (t), it implies that

note that 1 ≤ p < r' and the function attains its maximum at s = 1/ 2, i.e.,. From Proposition 2.2, it implies that

(3)

On the other hand, let us put x = (1, 0), y = (0, 1). Then ||x|| _ψ = ||y|| _ψ = 1, and

(4)

From (3) and (4), we have

Lemma 2.6 (see [6]). Let || · || and |.| be two equivalent norms on a Banach space. If a|.| ≤ || · || ≤ b|.| (b ≥ a > 0), then

Example 2.7. Let X = ℝ² with the norm

Then

Proof. It is very easy to check that ||x|| = max{||x||₂, λ||x||₁} ∈ ℕ _α and its corresponding function is

Therefore,

Since ψ₂(s) attains minimum at s = 1/ 2 and hence attains maximum at s = 1/ 2. Therefore, from Theorem 2.5, we have

Example 2.8. Let X = ℝ² with the norm

Then

Proof. It is obvious to check that the norm ||x|| = max{||x||₂, λ||x||_∞} is absolute, but not normalized, since ||(1, 0)|| = ||(0, 1)|| = λ. Let us put

Then |.| ∈ ℕ _α and its corresponding function is

Then

Consider the increasing continuous function . Because g(0) = 1 and , there exists a unique 0 ≤ a ≤ 1 such that g(a) = λ. In fact g(s) is symmetric with respect to s = 1/ 2. Then we have

Obviously, g(s) attains its maximum at s = 1/ 2. Hence, from Theorem 2.4 and Lemma 2.6, we have

Example 2.9. Let X = ℝ² with the norm

Then

Proof. It is obvious to check that the norm is absolute, but not normalized, since ||(1, 0)|| = ||(0, 1)|| = (1 + λ)^{1/ 2}. Let us put

Therefore, |.| ∈ ℕ _α and its corresponding function is

Obvious ψ(s) ≤ ψ₂(s). Since is symmetric with respect to s = 1/ 2, it suffices to consider for s ∈ [0, 1/ 2]. Note that, for any s ∈ [0, 1/ 2], put . Taking derivative of the function g(s), we have

We always have g'(s) ≥ 0 for 0 ≤ s ≤ 1/ 2. This implies that the function g(s) is increased for 0 ≤ s ≤ 1/2. Therefore, the function attains its maximum at s = 1/2. By Theorem 2.4 and Lemma 2.6, we have

Example 2.10. (Lorentz sequence spaces). Let ω₁ ≥ ω₂> 0, 2 ≤ r < ∞. Two-dimensional Lorentz sequence space, i.e. ℝ² with the norm

where is the rearrangement of (|z|, |ω|) satisfying , then

Proof. It is obvious that , and the corresponding convex function is given by

Obviously ψ(s) ≤ ψ _r (s) and . It suffices to consider Φ(s) for s ∈ [0, 1/2] since Φ(s) is symmetric with respect to s = 1/2. Note that for s ∈ [0, 1/2]

Some elementary computation shows that u(s) - v(s) = (1-(ω₂/ω₁))s^r attains its maximum and v(s) attains its minimum at s = 1/2. Hence,

attains its maximum at s = 1/2 and so does Φ(s). Then by Theorem 2.4 and Lemma 2.6, we have

Example 2.11. Let X be two-dimensional Cesà ro space , then

Proof. We first define

for (x, y) ∈ ℝ². It follows that is isometrically isomorphic to (ℝ², |.|) and |.| is an absolute and normalized norm, and the corresponding convex function is given by

Indeed, defined by is an isometric isomorphism. We prove that ψ(s) ≥ ψ₂(s). Note that

Consequently,

Some elementary computation shows that attains its maximum at s = 1/2. Therefore, from Theorem 2.5, we have

3. Constant and uniform normal structure

First, we recall some basic facts about ultrapowers. Let l_∞(X) denote the subspace of the product space II_n∈ℕX equipped with the norm ||(x _n )|| := sup_n∈ℕ||x _n || < ∞. Let be an ultrafilter on ℕ and let

The ultrapower of X, denoted by , is the quotient space equipped with the quotient norm. Write to denote the elements of the ultrapower. Note that if is non-trivial, then X can be embedded into isometrically. We also note that if X is super-reflexive, that is , then X has uniform normal structure if and only if has normal structure (see [17]).

Theorem 3.1. Let X be a Banach space with

for some t ∈ (0, 1]. Then X has uniform normal structure.

Proof. Observe that X is uniform non-square (see [6]) and then X is super-reflexive, it is enough to show that X has normal structure. Suppose that X lacks normal structure, then by Saejung [18, Lemma 2], there exist and satisfying:

• (1)
  

  and for all i ≠ j.

• (2)
  

  for i = 1, 2, 3.

• (3)
  

  .

Let and consider three possible cases.

First, if . In this case, let us put and . It follows that , and

Secondly, if and . In this case, let us put and . It follows that , and

Thirdly, and . In this case, let us put and . It follows that , and

Then, by definition of J _{X, p} (t) and the fact ,

This is a contradiction and thus the proof is complete.