A 3-space problem related to the fixed point property

Helga Fetter and Berta Gamboa de Buen^{∗} CIMAT, Apdo. Postal 402, 36000 Guanajuato, Gto. México.

Abstract

We study some properties which imply weak normal satructure and thus the fixed point property, in order to know if they are inherited to spaces obtained by direct sum with a finite dimensional space. We exhibit a space X which satisfies Opial’s condition, X ⨁ ℝ does not have weak normal structure but X ⨁ ℝ has the fixed point property.

1. Introduction

In recent times several properties implying the fixed point property in Banach spaces have appeared in the literature. The object of this work is to study which of these properties are inherited to spaces obtained by direct sum with a finite dimensional space. In case the latter holds, we will say that the property is F DSP. This is different to the permanence properties under symmetric sums studied by Landes in [4] and [5] and by Sims and Smyth in [7], since besides that the sum is symmetric, they consider sums of the space with itself.

The main result is that there exists a space X which satisfies Opial’s condition and X ⨁ ℝ for a certain family of norms does not have weak normal structure (W NS), but X ⨁ ℝ still has the fixed point property.

2. Properties which are FDSP

Definition 2.1. Let P be a property of Banach spaces and suppose that each time that a Banach space Y has a finite codimensional subspace X with P, Y also has P. Then we say that the property is F DSP.

We will show that several of the known properties which are sufficient for weak normal structure and thus for the fixed point property, are F DSP. Next we will define those properties.

Definition 2.2. Let X be a Banach space.

∗Both authors supported partially by Conacyt Grant 32152-E 1991 Mathematics Subject Classification 46B20

1. X has the fixed point property (F P P) if for every weakly compact subset K of X, every non expansive mapping T : K → K has a fixed point.

2. X has weak normal structure (W NS) if X does not contain any weakly null sequence {x_{n}}_{n} such that for every x in the closed convex hull of {x_{n}}_{n}

diam {x_{n}} = lim_{n} ∥x_{n} − x∥ .

3. If there exists ϵ > 0 so that for every weakly null sequence {x_{n}}_{n} ⊂ X with

lim_{n} ∥x_{n}∥ = 1,

lim_{n} sup_{i,j≥n} ∥x_{i} − x_{j}∥ > 1 + ϵ,

we say that X satisfies Bynum’s condition (BC) .

4. If for every weakly null sequence {x_{n}}_{n} ⊂ X with lim inf_{n} ∥x_{n}∥ > 0 there exists 0 < r < 1 such that for every subsequence {y_{n}}_{n} ⊂ {x_{n}}_{n} ,

lim sup_{n} ∥y_{n}∥ ≤ rdiam {y_{n}} ,

we say that X has strong subsequential P (SSuP) .

5. If for every weakly null sequence {x_{n}}_{n} ⊂ X such that lim_{n} ∥x_{n}∥ = 1 exists,

1 < inf {diam {y_{n}} : {y_{n}}_{n} is a subsequence of {x_{n}}_{n}} = A^{∗} {x_{n}} ,

then X has subsequential P (SuP) .

6. If for every weakly null sequence {x_{n}}_{n} ⊂ X

lim_{n} sup ∥x_{n}∥ < lim_{n} sup_{i,j≥n} ∥x_{i} − x_{j}∥ ,

then X has generalized Gossez Lami-Dozo property (GGLD) .

Details about F P P and W NS can be found in [2]. The definition of BC is not the original given by Bynum in [1], but an equivalent found in [8]. Properties SSuP and SuP are defined by Sims and Smyth also in [8], and GGLD is defined by Jiménez-Melado in [6].

Theorem 2.3. Properties BC, SSuP, SuP and GGLD are F DSP.

Proof. We will only give the proof of the theorem for spaces with codimension one, however the generalization to spaces of any finite codimension is trivial.

Let Y be a Banach space and X be a codimension one subspace. Then there is z ∈ Y such that every element of Y is of the form y = x + λz with x ∈ X and λ ∈ ℝ. Let {y_{n}}_{n} ⊂ Y be a weakly null sequence. Then

y_{n} = x_{n} + λ_{n}z

where {x_{n}}_{n} ⊂ X is weakly null in X and λ_{n} →_{n} 0. Also it is easy to see that:

(a) lim sup_{n} ∥y_{n}∥ = lim sup_{n} ∥x_{n}∥ and lim inf_{n} ∥y_{n}∥ = lim inf_{n} ∥x_{n}∥.

(b) lim_{n} sup_{i,j≥n} ∥y_{i} − y_{j}∥ = lim_{n} sup_{i,j≥n} ∥x_{i} − x_{j}∥ .

(c) A^{∗} {y_{n}} = A^{∗} {x_{n}} .

From (a), (b) and (c) it follows immediately that BC , SuP and GGLD are F DSP. With respect to SSuP, suppose that lim inf_{n} ∥y_{n}∥ > 0 and that there is 0 < r < 1 such that for every subsequence {u_{n}}_{n} of {x_{n}}_{n}

lim sup_{n} ∥u_{n}∥ ≤ rdiam {u_{n}} .

Then by (a), lim inf_{n} ∥x_{n}∥ > 0. Now let {v_{n}}_{n} be a subsequence of {y_{n}}_{n} with

v_{n} = u_{n} +λ_{m_{n}}z. Let ϵ > 0 and N ∈ ℕ such that if i, j > N, |λ_{m_{i}} − λ_{m_{j}} | < \frac{ϵ}{∥z∥}. Then

diam {u_{n}}_{n>N} ≤ diam {v_{n}} + ϵ

Thus, since lim sup_{n>N} ∥v_{n}∥ = lim sup_{n} ∥v_{n}∥ = lim sup_{n} ∥u_{n}, ∥ we get

\frac{1}{r} lim sup_{n} ∥v_{n}∥ = \frac{1}{r} lim sup_{n} ∥u_{n}∥ ≤ diam \{u_{n}\}_{n>N} ≤ diam \{v_{n}\} + ϵ

and thus lim sup_{n} ∥v_{n}∥ ≤ rdiam {v_{n}} . Hence SSuP is also F DSP.

The fact that GGLD is F DSP was already noted by Sims and Smyth in [7].

3. Properties which are not FDSP

There are several properties implying weak normal structure which are found in the literature and which are not F DSP. Examples of these are the Gossez Lami-Dozo (GLD)[3], the weakly uniformly Kadec-Klee (W UKK) and the W UKK′ [8] properties. In fact, if Y = l_{2} ⨁ ℝ with the norm ∥(x, λ)∥ = max (∥x∥_{l_{2}} , |λ|) , it is easy to see that l_{2} has all the above mentioned properties, but Y does not. However if X has one of the above properties and is of finite codimension in Y, then Y has weak normal structure and hence the fixed point property, since all of those properties imply Bynum’s condition.

In this section we show that other known conditions related to the fixed point property are not F DSP. We start by recalling the definitions of these properties.

Definition 3.1. Let X be a Banach space.

1. X satisfies Opial’s condition (OC) if whenever a sequence {x_{n}}_{n} in X is weakly null, then for x ≠ 0

lim inf_{n} ∥x_{n}∥ < lim inf_{n} ∥x_{n} − x∥ .

2. X satisfies Tingley’s condition (T C) if for every weakly null sequence {x_{n}}_{n} in X,

lim inf_{n} ∥x_{n}∥ < sup_{m} (lim sup_{n} ∥x_{m} − x_{n}∥) .

3. X has the weak sum property (W SP) if X ⨁_{l_{1}} ℝ has W NS.

Some of these definitions are not the original ones, but are equivalent. The definition of OC can be found in [2]. T C was defined in [9] and W SP in [5]. It is known that

OC ⇒ T C ⇒ W SP ⇒ W NS ⇒ F P P.  (3.1)

Definition 3.2. Let Y = c be the space of converging real sequences y = {y (n)} with the norm ∥·∥_{α} for α ≥ 0 :

∥y∥_{α} = ∥y∥_{∞} + α ∑_{i=1}^{∞} \frac{|y (i)|}{2^{i}},

where ∥y∥_{∞} = sup_{n} |y (n)| . Let X = (c_{0}, ∥·∥_{α}) be the subspace of Y consisting of the subsequences convergent to zero.

Recall that a sequence {y_{n}}_{n} in Y is weakly null if and only if lim_{n} y_{n} (i) = 0 for every i ∈ ℕ and lim_{n} [lim_{i} y_{n} (i)] = 0.

Lemma 3.3. For α > 0, X = (c_{0}, ∥·∥_{α}) satisfies Opial’s condition.

Proof. Let {x_{n}}_{n} ⊂ X be a weakly null sequence with lim_{n} ∥x_{n}∥_{α} = 1. Then

lim_{n} ∥x_{n}∥_{∞} = 1, lim_{n} ∑_{i=1}^{∞} \frac{x_{n} (i)}{2^{i}} = 0 and by passing to a subsequence we can assume

that there exist 0 < i_{1} < i_{2} < ... such that ∥x_{n}∥_{∞} = |x_{n} (i_{n})| .

Let x ∈ X, x ≠ 0; then

∥x_{n} − x∥_{α} ≥ ∥x_{n} − x∥_{∞} + α ∑_{i=1}^{∞} \frac{|x (i)|}{2^{i}} − α ∑_{i=1}^{∞} \frac{|x_{n} (i)|}{2^{i}}.

Hence

lim inf_{n} ∥x_{n} − x∥_{α} ≥ lim inf_{n} ∥x_{n} − x∥_{∞} + α ∑_{i=1}^{∞} \frac{|x (i)|}{2^{i}}.

Now let ϵ > 0 and J be such that for j ≥ J, |x (j)| < ϵ and let N be so that for n > N, i_{n} > J. Then for n > N

∥x_{n} − x∥_{∞} ≥ |x_{n} (i_{n}) − x (i_{n})| ≥ |x_{n} (i_{n})| − ϵ.

Thus

lim inf_{n} ∥x_{n} − x∥_{∞} ≥ 1

and

lim inf_{n} ∥x_{n} − x∥_{α} ≥ 1 + α ∑_{i=1}^{∞} \frac{|x (i)|}{2^{i}} > 1.

This finishes the proof.

Jiménez-Melado in [6] mentioned that (c_{0}, ∥·∥_{α}) for α = 1 fails to have property GGLD. Thus we obtain the following: Corollary 3.4. OC does not imply GGLD. Theorem 3.5. For α ∈ (0, 1] , X = (c, ∥·∥_{α}) does not have W NS.

Proof. Let \{x_{n}\}_{n} ⊂ X be the sequence defined by x_{n} (i) = \{ \begin{matrix}0 if i < n \\ 1 − \frac{α}{2^{n}} for i = n \\ \frac{α}{2^{n}} for i > n \end{matrix} .

Then {x_{n}}_{n} is weakly null,

∥x_{n}∥_{α} = 1 − \frac{α}{2^{n}} + \frac{α}{2^{n}} (1 − \frac{α}{2^{n}} + ∑_{i=n+1}^{∞} \frac{α}{2^{i}} ) →_{n→∞} 1  (3.2)

and

∥x_{n} − x_{m}∥_{α} →_{n→∞} 1.  (3.3)

Also, for n > 1

∑_{j=1}^{n} |x_{j} (i) − x_{n} (i)| = |∑_{j=1}^{n} (x_{j} (i) − x_{n} (i))| = \{ \begin{matrix}1 − \frac{α}{2} for i = 1 \\ (∑_{j=1}^{i−1} \frac{α}{2^{j}} ) + 1 − \frac{α}{2^{i}} for 1 < i < n \\ ∑_{j=1}^{n−1} (1 − \frac{α}{2^{n}} − \frac{α}{2^{j}} ) for i = n \\ ∑_{j=1}^{n−1} (\frac{α}{2^{j}} − \frac{α}{2^{n}} ) for i > n \end{matrix}

(3.4)

On the other hand, if n > 3, ∑_{j=1}^{n} ∥x_{j} − x_{n}∥_{∞} = ∑_{j=1}^{n−1} (1 − \frac{α}{2^{j}} ) and

∥∑_{j=1}^{n} (x_{j} − x_{n})∥_{∞} = ∑_{j=1}^{n−1} (1 − \frac{α}{2^{n}} − \frac{α}{2^{j}} ) .

Hence

∑_{j=1}^{n} ∥x_{j} − x_{n}∥_{∞} − ∥∑_{j=1}^{n} (x_{j} − x_{n})∥_{∞} = \frac{(n − 1) α}{2^{n}} →_{n→∞} 0.  (3.5)

Thus by (3.4) and (3.5)

lim_{n→∞} (∑_{j=1}^{n} ∥x_{j} − x_{n}∥_{α} − ∥∑_{j=1}^{n} (x_{j} − x_{n})∥_{α}) = 0  (3.6)

and by (3.2), (3.3) and (3.6), using a result of Landes found in [4] we get that X does not have W NS.

The following result is an immediate consequence of the previous theorem, lemma 3.3 and (3.1). Corollary 3.6. Properties OC, T C, W SP and W NS are not F DSP.

Next we will see that the spaces (c, ∥·∥_{α}) in spite of not having W NS still have F P P. The proof follows along the lines of Lin’s proof that spaces with a 1-unconditional basis have F P P (see e.g. [2]). Theorem 3.7. For α ∈ [0, 1) , X = (c, ∥·∥_{α}) has the fixed point property. Proof. Let K be a weakly compact convex set in X and suppose T : K → K is a non expansive mapping without a fixed point. We may suppose that K is a minimal T − invariant weakly compact convex set with diamK = 1 and that there is a weakly null approximate fixed point sequence {x_{n}}_{n} ⊂ K for T so that:

(1) lim_{n} ∥x_{n}∥_{α} = 1

(2) lim_{n} ∥x − x_{n}∥_{α} = 1 for every x ∈ K

(3) lim_{n} ∥x_{n} − x_{n+1}∥_{α} = 1.

Since {x_{n}}_{n} is weakly null:

(4) lim_{n} x_{n} (i) = 0 for every i ∈ ℕ and if k_{n} = lim_{i} x_{n} (i) , then lim_{n} k_{n} = 0.

Hence we have also, by passing to a subsequence and taking −K if it is necessary:

(5) lim_{n} α ∑_{i=1}^{∞} \frac{|x_{n} (i)|}{2^{i}} = 0.

(6) lim_{n} ∥x_{n}∥_{∞} = 1

(7) There exist i_{1} < i_{2} < ... such that ∥x_{n}∥_{∞} = x_{n} (i_{n}) .

Then, if for x ∈ K, lim_{i} x (i) = k_{x}, we have

|k_{x}| ≤ 1 and ∥x − x_{n}∥_{α} ≥ |x (i_{n}) − x_{n} (i_{n})| + α ∑_{i=1}^{∞} \frac{|x (i) − x_{n} (i)|}{2^{i}}.

Thus, by (2), (4) (5), (6) and (7), 1 ≥ 1 − k_{x} + α ∑_{i=1}^{∞} \frac{|x (i)|}{2^{i}} and

k_{x} ≥ α ∑_{i=1}^{∞} \frac{|x (i)|}{2^{i}} ≥ 0.  (3.7)

On the other hand, suppose that there exist r_{1} < r_{2} < ... such that x_{n} (r_{n}) < 0.

If x ∈ K, then 1 ≥ ∥x − x_{n}∥_{α} ≥ |x (r_{n}) − x_{n} (r_{n})| + α ∑_{i=1}^{∞} \frac{|x (i) − x_{n} (i)|}{2^{i}}. Hence

1 ≥ k_{x} + lim sup_{n} |x_{n} (r_{n})| + α ∑_{i=1}^{∞} \frac{|x (i)|}{2^{i}} and thus

lim sup_{n} |x_{n} (r_{n})| ≤ 1 − k_{x} − α ∑_{i=1}^{∞} \frac{|x (i)|}{2^{i}}.  (3.8)

By a standard procedure, using (4), we may extract a subsequence of {x_{n}}_{n} which we will keep calling {x_{n}}_{n} , and construct finite sets F_{1}, F_{2}, ... with sup F_{i} < inf F_{i+1} and projections P_{n} : X → X such that

1 ≥ ∥P_{n}x_{n}∥_{α} > 1 − \frac{1}{2^{n}} and ∥(I − P_{n}) x_{n}∥_{α} < \frac{1}{2^{n}} ,  (3.9)

where for x ∈ X, (P_{n}x) (j) = \{ \begin{matrix}x (j) if j ∈ F_{n} \\ 0 otherwise \end{matrix} .

Observe that

P_{n} ∘ P_{m} = 0 if n ≠ m and ∥P_{n}∥ = ∥P_{n} + P_{n+1}∥ = ∥I − P_{n}∥ = 1.  (3.10)

and for x ∈ X,

lim_{n} ∥P_{n}x∥_{α} = k_{x}  (3.11)

Now define for x ∈ X

(R_{n}x) (j) = \{ \begin{matrix}0 if j ∉ F_{n} \\ x (j) − k_{x} if j ∈ F_{n} \end{matrix} .

Hence

R_{n} ∘ R_{m} = 0 if n ≠ m, ∥R_{n}∥ = ∥R_{n} + R_{n+1}∥ = 2,  (3.12)
∥R_{n}x∥_{α} = max_{i∈F_{n}} |x (i) − k_{x}| + α ∑_{i∈F_{n}} \frac{|x (i) − k_{x}|}{2^{i}} →_{n→∞} 0  (3.13)

and

∥(I − R_{n}) x∥_{α} = sup \{|k_{x}| , |x (i)| : i ∉ F_{n}\} + α ( ∑_{i∉F_{n}} \frac{|x (i)|}{2^{i}} + ∑_{i∈F_{n}} \frac{|k_{x}|}{2^{i}} ) →_{n→∞} ∥x∥_{α} .

Thus

lim_{n} ∥I − R_{n}∥ = 1.  (3.14)

Also

∥(I − R_{n}) x_{n}∥_{α} = ∥x_{n} − P_{n}x_{n} + k_{n}χ_{F_{n}}∥_{α} →_{n→∞} 0,  (3.15)

where χ_{F} (j) = \{ \begin{matrix}1 for j ∈ F \\ 0 otherwise \end{matrix} .

By (3.7) there are two cases to consider:

I. For every x ∈ K, k_{x} = α ∑_{i=1}^{∞} \frac{|x (i)|}{2^{i}}.

II. There exists u ∈ K with k_{u} > α ∑_{i=1}^{∞} \frac{|u (i)|}{2^{i}}.

We now translate some of the above into the terminology of ultrapowers. For details see e.g. [2].

Let U be a free ultrafilter on ℕ. K̃ will denote the set

K̃ = {K}_{U} = {ũ = {u_{i}}_{U} ∈ {X}_{U} : u_{i} ∈ K, i ∈ ℕ} ,

where {X}_{U} = X̃ is the ultrapower of X. If {S_{i}} is an equibounded family of operators S_{i} : X → X, we define the operator S̃ = {S_{i}}_{U} : X̃ → X̃ by

S̃ũ = {S_{i}u_{i}}_{U} .

In particular T̃ : K̃ → K̃ is the non expansive mapping given by

T̃ũ = T̃ {u_{i}}_{U} = {T u_{i}}_{U} .

We identify K with the subset of K̃ represented by the constant sequences {x}_{U} with x ∈ K.

Let P̃ = {P_{n}}_{U} , Q̃ = {P_{n+1}}_{U} , R̃ = {R_{n}}_{U} and S̃ = {R_{n+1}}_{U} . Then by (3.10) and (3.14)

∥P̃∥ = ∥I − P̃∥ = ∥Q̃∥ = ∥I − Q̃∥ = ∥P̃ + Q̃∥ = ∥I − R̃∥ = ∥I − S̃∥ = 1 (3.16)

and by (3.12)

∥R̃∥ = ∥S̃∥ = ∥R̃ + S̃∥ = 2.  (3.17)

Now let x̃ = {x_{n}}_{U} , ỹ = {x_{n+1}}_{U} . Then since {x_{n}}_{n} is an approximate fixed point sequence for T,

T̃x̃ = x̃ and T̃ỹ = ỹ.  (3.18)

By (3.9), (3.10), (3.15) and (3.12)

P̃x̃ = x̃, Q̃ỹ = ỹ, P̃ỹ = 0 and Q̃x̃ = 0  (3.19)

and

R̃x̃ = x̃, S̃ỹ = ỹ, R̃ỹ = 0 and S̃x̃ = 0.  (3.20)

Also by (3.11) and (3.13), for x ∈ K

∥P̃x∥ = ∥(P̃ + Q̃) x∥ = k_{x} and ∥R̃x∥ = ∥(R̃ + S̃) x∥ = 0.  (3.21)

Finally, by (3), (3.9), and from the fact that ∥P_{n}x_{n} − P_{n}x_{n+1}∥_{α} = ∥P_{n}x_{n} + P_{n}x_{n+1}∥_{α} , it follows that

∥x̃ − ỹ∥ = lim_{U} ∥P_{n}x_{n} − P_{n+1}x_{n+1}∥ = lim_{U} ∥P_{n}x_{n} + P_{n+1}x_{n+1}∥ = ∥x̃ + ỹ∥ = 1.

(3.22)

Case I. In this case for every x ∈ K, 0 ≤ k_{x} ≤ \frac{α}{α + 1}. If this were not true

and k_{x} > \frac{α}{α + 1}, then ∥x∥_{∞} = ∥x∥_{α} − k_{x} < 1 − \frac{α}{α + 1} = \frac{1}{α + 1} and thus, since

k_{x} = α ∑_{i=1}^{∞} \frac{|x (i)|}{2^{i}} this would yield a contradiction. Let

W̃ = \{w̃ ∈ K̃ : ∥w̃ − x∥ ≤ \frac{1}{2} for some x ∈ K, ∥w̃ − x̃∥ ≤ \frac{1}{2}, ∥w̃ − ỹ∥ ≤ \frac{1}{2} \} .

Clearly W̃ is a closed and convex subset of K̃ and by (3.22) z̃ = \frac{x̃ + ỹ}{2} ∈ W̃ (taking x = 0 ∈ K). Also by (3.18) and since T is non expansive,

∥T̃w̃ − T x∥ ≤ ∥w̃ − x∥ , ∥T̃w̃ − x̃∥ ≤ ∥w̃ − x̃∥ and ∥T̃w̃ − ỹ∥ ≤ ∥w̃ − ỹ∥ ,

showing that W̃ is T̃ invariant.

Now, if w̃ ∈ W̃, then by (3.19), (3.16) and (3.21) we obtain, since α < 1,

\left∥{ \atop w}\right∥ = \frac{1}{2} ∥(P̃ + Q̃) w̃ + (I − P̃) w̃ + (I − Q̃) w̃∥ ≤
≤ \frac{1}{2} (∥(P̃ + Q̃) (w̃ − x)∥ + ∥(P̃ + Q̃) x∥ +
+ ∥(I − P̃) (w̃ − x̃)∥ + ∥(I − Q̃) (w̃ − ỹ)∥)
≤ \frac{1}{2} (\frac{1}{2} + k_{x} + \frac{1}{2} + \frac{1}{2}) ≤ \frac{3}{4} + \frac{α}{2 (α + 1)} < 1,

but by Karlovitz’s lemma

sup_{_{w}∈_{W}} ∥W̃∥ = 1.

Case II. There exists u ∈ K with k = k_{u} = lim_{i} |u (i)| > α ∑_{i=1}^{∞} \frac{|u (i)|}{2^{i}}.

In this case we will show that there is 0 < b < \frac{1}{2} such that the closed convex set W̃ given by

W̃ = \{w̃ ∈ K̃ : ∥w̃ − x∥ ≤ \frac{1}{2} − b for some x ∈ K, ∥w̃ − x̃∥ ≤ \frac{1}{2}, ∥w̃ − ỹ∥ ≤ \frac{1}{2} \} .

is not empty. Then, as above W̃ is T̃ invariant and if w̃ ∈ W̃, by (3.21), (3.20), (3.17) and (3.16).

\left∥{ \atop w}\right∥ ≤ \frac{1}{2} (∥(R̃ + S̃) w̃∥ + ∥(I − R̃) w̃∥ + ∥(I − S̃) w̃∥) =
= \frac{1}{2} (∥(R̃ + S̃) (w̃ − x)∥ + ∥(I − R̃) (w̃ − x̃)∥ + ∥(I − S̃) (w̃ − ỹ)∥)
≤ \frac{1}{2} (2 (\frac{1}{2} − b) + \frac{1}{2} + \frac{1}{2}) = 1 − b,

and this again contradicts Karlovitz’s lemma.

We will prove now that \frac{1}{2} (x̃ + ỹ) ∈ W̃ by exhibiting an x ∈ K and a b < \frac{1}{2} satisfying ∥\frac{1}{2} (x̃ + ỹ) − x∥ ≤ \frac{1}{2} − b. To this end let u ∈ K as above and 0 < λ < 1. Since K is convex and 0 ∈ K, λu ∈ K. Now we will estimate ∥\frac{1}{2} (P_{n}x_{n} + P_{n+1}x_{n+1}) − λu∥_{α} .

|\frac{1}{2} (P_{n}x_{n} + P_{n+1}x_{n+1}) (i) − λu (i)| = \{ \begin{matrix}λ |u (i)| if i ∉ F_{n} ∪ F_{n+1} \\ |\frac{x_{r} (i)}{2} − λu (i)| if r ∈ \{n, n + 1\} and i ∈ F_{r} \end{matrix}

Let 0 < ϵ < k − α ∑_{j=1}^{∞} \frac{|u(j)|}{2^{j}} and suppose that n is sufficiently large in order that u (i) > 0 , |u (i) − k| < ϵ for i ∈ F_{r} and if x_{r} (i) < 0, then by (3.8)

|x_{r} (i)| < 1 − k − α ∑_{j=1}^{∞} \frac{|u (j)|}{2^{j}} + ϵ.

But |\frac{x_{r} (i)}{2} − λu (i)| =

= \{ \begin{matrix}\frac{x_{r} (i)}{2} − λu (i) < \frac{1}{2} − λ (k − ϵ) if λu (i) < \frac{x_{r} (i)}{2} \\ 0 ≤ λu (i) − \frac{x_{r} (i)}{2} ≤ λu (i) ≤ λ (k + ϵ) if 0 ≤ \frac{x_{r} (i)}{2} ≤ λu (i) \\ λu (i) + \frac{|x_{r} (i)|}{2} ≤ λ (k + ϵ) + \frac{1}{2} (1 − k − α ∑_{j=1}^{∞} |\frac{u (j)}{2^{j}} | + ϵ) if x_{r} (i) < 0 \end{matrix} .

Let λ be sufficiently small so that λ |u (i)| < \frac{1}{2} − λk for every i and so that 2λk <

\frac{1}{2} (k + α ∑_{j=1}^{∞} |\frac{u(j)}{2^{j}} | − ϵ) . Then

∥\frac{1}{2} (P_{n}x_{n} + P_{n+1}x_{n+1}) − λu∥_{α} ≤ (\frac{1}{2} − λk + λϵ)+λα ∑_{j=1}^{∞} |\frac{u (j)}{2^{j}} |+α ∑_{j∈F_{n}∪F_{n+1}} |\frac{x_{r} (j)}{2^{j}} |

and finally

lim_{n} ∥\frac{1}{2} (P_{n}x_{n} + P_{n+1}x_{n+1}) − λu∥_{α} ≤ \frac{1}{2} − λk + λϵ + λα ∑_{j=1}^{∞} |\frac{u (j)}{2^{j}} | < \frac{1}{2}.

Therefore, if x = λu, ∥\frac{1}{2} (x̃ + ỹ) − x∥ = \frac{1}{2} − b for some b > 0 and this concludes the proof.

We do not know of an example of a space having F P P and codimension one in another space without F P P. We also do not know if the previous theorem is still valid for α = 1.

References

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