Comparison between Certain Equivalent Norms Regarding Some Familiar Properties Implying WFPP

Abstract

In a Banach space with a basis we define a similar norm to the norm shown by Lin to make 𝑙₁ into a space with FPP and make a comparative study of certain geometric properties such as the Opial property, WNS, and uniform nonsquareness of the original space and the space with the new norm.

1. Introduction

Dowling et al. in [1] defined a norm in 𝑙₁ which was used by Lin [2] to exhibit an equivalent norm which makes 𝑙₁ into a space with the fixed point property (FPP). A similar norm can be defined in every Banach space 𝑋 with a basis. Since 𝑙₁ with its usual norm does not have FPP, we asked ourselves if this norm in these spaces would also improve properties that imply the weak fixed point property (WFPP). We found out that in some instances it does, in some cases the original norm has better properties, and in some cases you cannot compare them. We give several examples to illustrate our assertions.

2. The Γ Norm in a Banach Space

We start by giving the definition of the generalization of the norm used by Lin in a Banach space with a basis.

Definition 2.1. Let (𝑋,‖⋅‖) be a Banach space with a basis {𝑒_𝑛}. Let 𝑥=∑^∞_𝑖=1𝑥_𝑖𝑒_𝑖∈𝑋 and 𝑄_𝑛∶𝑋→𝑋 be the projection 𝑄_𝑛(∑^∞_𝑖=1𝑥_𝑖𝑒_𝑖)=∑^∞_𝑖=𝑛𝑥_𝑖𝑒_𝑖. The basis {𝑒_𝑛} is called premonotone if ‖𝑄_𝑛𝑥‖≥‖𝑄_𝑛+1𝑥‖ and monotone if ‖𝑃_𝑛𝑥‖≤‖𝑃_𝑛+1𝑥‖ where 𝑃_𝑛𝑥=(𝐼−𝑄_𝑛+1)𝑥 for every 𝑥∈𝑋 and for every 𝑛∈ℕ.

Definition 2.2. Let (𝑋,‖⋅‖) be a Banach space with a basis {𝑒_𝑛} and Γ={𝛾_𝑛}⊂ℝ with 0<𝛾_𝑛<𝛾_𝑛+1 and lim_𝑛𝛾_𝑛=1. Let 𝑥=∑^∞_𝑖=1𝑥_𝑖𝑒_𝑖∈𝑋. Then if |‖𝑥‖|=sup_𝑛𝛾_𝑛𝑄_𝑛𝑥, (2.1)|‖⋅‖| is a norm in 𝑋 which we will call Γ-norm.
Clearly 𝛾₁‖𝑥‖≤|‖𝑥‖|≤sup_𝑛𝑄_𝑛‖𝑥‖. (2.2)
Observe that, if {𝑒_𝑛} is a basis in (𝑋,‖⋅‖), then it is always premonotone in (𝑋,|‖⋅‖|) and, if {𝑒_𝑛} is monotone in (𝑋,‖⋅‖), then it is also monotone in (𝑋,|‖⋅‖|). Also observe that since for every 𝑥∈𝑋 we have that lim_𝑛𝛾_𝑛‖𝑄_𝑛𝑥‖=0, there exists 𝑛₀ such that |‖𝑥‖|=𝛾_𝑛0‖𝑄_𝑛0𝑥‖.
Next we define the properties related to wfpp we are going to analyze. The definition of GGLD is not the original one found in [3], but an equivalent one found in [4].

Definition 2.3. Let 𝑌 be a Banach space.(1)𝑌 has the Opial property if for every weakly null sequence {𝑥_𝑛}⊂𝑌 and for every 𝑥∈𝑌, 𝑥≠0,limsup_𝑛𝑥_𝑛<limsup_𝑛𝑥_𝑛−𝑥. (2.3)(2)If 𝑌 has a basis, it has the generalized Gossez-Lami Dozo property (GGLD) [4] if, for every weakly null normalized block basic sequence {𝑦_𝑛}, we have that lim_𝑛sup_{𝑖,𝑗≥𝑛}𝑦_𝑖−𝑦_𝑗>1. (2.4)(3)A bounded sequence {𝑦_𝑛}⊂𝑌 is called diametral if lim_𝑛𝑑𝑦_𝑛+1,conv𝑦_𝑖^𝑛_𝑖=1=diam𝑦_𝑛^∞_𝑛=1. (2.5)𝑌 has weak normal structure (WNS) if there is no weakly null diametral nonzero sequence in 𝑌.(4)The coefficient 𝐽(𝑌), related to uniform nonsquareness, since 𝐽(𝑌)<2 if and only if 𝑌 is uniformly nonsquare, is given by 𝐽(𝑌)=supmin𝑥+𝑦,𝑥−𝑦∶𝑥,𝑦∈𝐵_𝑌. (2.6)(5)The coefficient 𝑅(𝑌) [5] is defined by 𝑅(𝑌)=supliminf_𝑛𝑥_𝑛+𝑥∶{𝑥},𝑥_𝑛⊂𝐵_𝑌,𝑥_𝑛⟶_𝑤0. (2.7)(6)Coefficients RW(𝑎,𝑌) and MW(𝑌) [6] are defined as follows: for each 𝑎>0RW(𝑎,𝑌)=supminliminf_𝑛𝑥_𝑛+𝑥,liminf_𝑛𝑥_𝑛−𝑥∶‖𝑥‖≤𝑎,𝑥_𝑛⊂𝐵_𝑌,𝑥_𝑛⟶_𝑤0,MW(𝑌)=sup1+𝑎().RW𝑎,𝑌∶𝑎>0 (2.8)It is known that GGLD⇒WNS⇒wfpp and the Opial property implies wfpp. Also 𝑅(𝑌)(𝑌)𝐽(𝑌)(𝑌)<2⟹MW>1⟹wfpp,<2⟹MW>1⟹wfpp. (2.9)(2.10)
First we will show that the Opial property is inherited from (𝑋,|‖⋅‖|) to (𝑋,‖⋅‖) and that (𝑋,‖⋅‖) has GGLD if and only if (𝑋,|‖⋅‖|) has GGLD. In order to achieve this, we need the following result shown in [7].

Lemma 2.4. Let (𝑋,‖⋅‖) be a Banach space with a premonotone basis {𝑒_𝑛}. Then(1) if {𝑥_𝑛} converges weakly to 𝑥, lim_𝑛‖𝑥_𝑛−𝑥‖=𝑎 if and only if lim_𝑛|‖𝑥_𝑛−𝑥‖|=𝑎,(2) if {𝑥_𝑛} converges weakly to 0 and lim_𝑛lim_𝑟‖𝑥_𝑛−𝑥_𝑟‖=𝑎, there exists a subsequence {𝑦_𝑛} of {𝑥_𝑛} such that lim_𝑛lim_𝑟|‖𝑦_𝑛−𝑦_𝑟‖|=𝑎.

Lemma 2.5. Let (𝑋,‖⋅‖) be a Banach space with a premonotone basis {𝑒_𝑛}. If (𝑋,|‖⋅‖|) has the Opial property, then (𝑋,‖⋅‖) also has the Opial property, but the converse is false.

Proof. Let {𝑥_𝑛} be weakly null in 𝑋 and 𝑥∈𝑋,𝑥≠0. Then, by Lemma 2.4 and by (2.2), limsup_𝑛𝑥_𝑛=limsup_𝑛𝑥_𝑛<limsup_𝑛𝑥_𝑛−𝑥≤limsup_𝑛𝑥_𝑛−𝑥. (2.11)
It is known that, for 1<𝑝<∞, (𝑙_𝑝,‖⋅‖) has the Opial property. Consider any Γ-norm |‖⋅‖| in 𝑙_𝑝 with the canonical basis {𝑒_𝑛}, and let 𝛿>0 be such that𝛿<((𝛾^𝑝₂−𝛾^𝑝₁)/𝛾^𝑝₁)^1/𝑝. Then, for 𝑛≥2, 𝛿𝑒₁+𝑒_𝑛=max𝛾₁𝛿^𝑝+1^1/𝑝,𝛾_𝑛=𝛾_𝑛,lim_𝑛𝑒_𝑛=lim_𝑛𝛾_𝑛=lim_𝑛𝛿𝑒₁+𝑒_𝑛=1. (2.12) Thus, (𝑙_𝑝,|‖⋅‖|) does not have the Opial property.

Lemma 2.6. Let (𝑋,‖⋅‖) be a Banach space with a premonotone basis {𝑒_𝑛}. Then, (𝑋,‖⋅‖) has GGLD if and only if (𝑋,|‖⋅‖|) has GGLD.

Proof. Let {𝑦_𝑛} be a weakly null normalized block basic sequence. By Lemma 2.4, lim_𝑛‖𝑦_𝑛‖ exists if and only if lim_𝑛|‖𝑦_𝑛‖| exists and in this case lim_𝑛‖𝑦_𝑛‖=lim_𝑛|‖𝑦_𝑛‖|. Also lim_𝑛sup_{𝑖,𝑗≥𝑛}𝑦_𝑖−𝑦_𝑗=lim_𝑛sup_{𝑖,𝑗≥𝑛}𝑦_𝑖−𝑦_𝑗. (2.13) The above equality follows immediately from the following inequality, for 𝑖, 𝑗≥𝑛∶𝑦_𝑖−𝑦_𝑗≤𝑦_𝑖−𝑦_𝑗≤1𝛾_𝑛𝑦_𝑖−𝑦_𝑗. (2.14) This proves the lemma.

Now we will show that there exists a space with WNS such that with the Γ-norm it does not have WNS.

Lemma 2.7. Let 𝑋 be the space 𝑐₀ with the norm ‖𝑥‖=sup|𝑏_𝑖|+∑^∞_𝑖=1𝜀_𝑖|𝑏_𝑖|, where 𝑥=∑^∞_𝑖=1𝑏_𝑖𝑒_𝑖, 𝜀_𝑖>0 and ∑^∞_𝑖=1𝜀_𝑖<∞.Then 𝑋 has WNS.

Proof. Let {𝑥_𝑛}⊂𝑋 be a weakly null nonzero sequence. We may assume that 𝑥₁≠0 and that there exists a block basic sequence {𝑢_𝑛}⊂𝑋 with ‖𝑢_𝑛−𝑥_𝑛‖→_𝑛→∞0. Suppose that 𝑢_𝑛=∑^𝑞𝑛_{𝑖=𝑝𝑛}𝑎_𝑖𝑒_𝑖 with 𝑝_𝑛≤𝑞_𝑛<𝑝_𝑛+1 for 𝑛∈ℕ and that 𝑥₁=∑^∞_𝑖=1𝑏_𝑖𝑒_𝑖. Let 𝑘 be such that ∑^𝑘_𝑖=1𝜀_𝑖|𝑏_𝑖|=𝛿≠0. Let 𝜀<𝛿/2, 𝑠>𝑘 with ‖𝑄_𝑠𝑥₁‖<𝜀 and 𝑛>𝑠. Then, 𝑥₁−𝑢_𝑛+𝜀≥𝑃_𝑠𝑥₁−𝑢_𝑛=max𝑃_𝑠𝑥_1∞,𝑢_𝑛∞+^𝑠_𝑖=1𝜀_𝑖𝑏_𝑖+^𝑞𝑛_{𝑖=𝑝𝑛}𝜀_𝑖𝑎_𝑖≥𝑢_𝑛+𝛿. (2.15) Hence, limsup_𝑛‖𝑥₁−𝑥_𝑛‖≥limsup_𝑛‖𝑥_𝑛‖+𝛿/2 and {𝑥_𝑛} cannot be a diametral sequence, since for a diametral sequence {𝑥_𝑛} it is true that lim_𝑛‖𝑥−𝑥_𝑛‖=diam{𝑥_𝑛} for every 𝑥∈conv{𝑥_𝑛}.

Lemma 2.8. Let Γ={𝛾_𝑛}⊂(0,1) be an increasing sequence with lim_𝑛𝛾_𝑛=1. Then there is a space (𝑋,‖⋅‖) with WNS, such that 𝑋 with the Γ-norm |‖⋅‖| does not have WNS.

Proof. Let {𝛾_𝑛𝑗}_𝑗 be a subsequence of {𝛾_𝑛} such that, if 𝜀_𝑛𝑗=(1/3)(1/𝛾_𝑛𝑗−1), then ∑^∞_𝑖=1𝜀_𝑛𝑖<∞. Observe that 𝜀_𝑛𝑗<𝜀_𝑛𝑗+1. Let 𝑋 be the space 𝑐₀ with the norm ‖𝑥‖=sup|𝑎_𝑖|+∑^∞_𝑖=1𝜀_𝑛𝑖|𝑎_𝑛𝑖|+∑^∞_{𝑖=1,𝑖≠𝑛𝑗}(|𝑎_𝑖|/2^𝑖), where 𝑥=∑^∞_𝑖=1𝑎_𝑖𝑒_𝑖. By Lemma 2.7, since ∑^∞_𝑖=1(𝜀_𝑛𝑖+1/2^𝑖)<∞, we know that 𝑋 has WNS. Now let |‖⋅‖| be the Γ-norm in 𝑋 with respect to {𝛾_𝑖}. Let 𝑢_𝑗=𝑒_𝑛𝑗; then 𝛾_𝑛𝑗=1/(1+3𝜀_𝑛𝑗), and thus 𝑢_𝑗=𝛾_𝑛𝑗1+𝜀_𝑛𝑗<1, (2.16) and, if 𝑗<𝑚, 𝑢_𝑗−𝑢_𝑚=max𝛾_𝑛𝑗1+𝜀_𝑛𝑗+𝜀_𝑛𝑚,𝛾_𝑛𝑚1+𝜀_𝑛𝑚≤max𝛾_𝑛𝑗1+2𝜀_𝑛𝑗,𝛾_𝑛𝑚1+𝜀_𝑛𝑚<1. (2.17) Therefore, since 0∈conv{𝑢_𝑛} and lim_𝑛|‖𝑢_𝑛‖|=1, we have that diam_|‖⋅‖|{𝑢_𝑛}=1. Also, if 0≤𝜆_𝑖, ∑^𝑛−1_𝑖=1𝜆_𝑖=1, 𝑢_𝑛≤^𝑛−1_𝑖=1𝜆_𝑖𝑢_𝑖−𝑢_𝑛≤1. (2.18) Hence, {𝑢_𝑛} is diametral in (𝑋,|‖⋅‖|).

The above example is another proof of the fact that for every 𝜀>0 there are Banach spaces 𝑋 and 𝑌 with 𝑑(𝑋,𝑌)<1+𝜀 so that 𝑋 has WNS but 𝑌 does not.

With regard to the coefficient MW(𝑋), we will see that, if 𝑋 has a premonotone basis and MW(𝑋,‖⋅‖)>1, then MW(𝑋,|‖⋅‖|)>1 and we will show a sufficient condition for the reverse implication. For this we need the following lemma.

Lemma 2.9. Let 𝑋 be a Banach space with a basis. If RW₁(𝑎,𝑋)=supminliminf𝑢_𝑛+𝑦,liminf𝑢_𝑛−𝑦∶𝑢_𝑛⟶_𝑤𝑢0,_𝑛⊂𝐵_𝑋𝑦,isablockbasicsequence,≤𝑎andsupportof𝑦isfinite (2.19) then RW(𝑎,𝑋)=RW₁(𝑎,𝑋).

Proof. It is clear that RW₁(𝑎,𝑋)≤RW(𝑎,𝑋).
Now let 𝜀>0, 𝑥∈𝑋, ‖𝑥‖≤𝑎, and {𝑥_𝑛}⊂𝐵_𝑋, with 𝑥_𝑛→_𝑤0such that min(lim_𝑛‖𝑥_𝑛+𝑥‖,lim_𝑛‖𝑥_𝑛−𝑥‖)>𝑅𝑊(𝑎,𝑋)−𝜀. By passing to a subsequence, we may assume that there exist a block basic sequence {𝑢_𝑛}⊂𝐵_𝑋 with ‖𝑥_𝑛−𝑢_𝑛‖<𝜀 and 𝑚∈ℕ such that ‖𝑥−𝑃_𝑚𝑥‖<𝜀 and ‖𝑃_𝑚𝑥‖<(1+𝜀)‖𝑥‖≤𝑎(1+𝜀). Then, 𝑃_𝑚𝑥1+𝜀+𝑢_𝑛≥𝑥_𝑛+𝑥−𝑥_𝑛−𝑢_𝑛−𝑥−𝑃_𝑚𝑥−𝑃_𝑚𝑥𝜀,𝑃1+𝜀_𝑚𝑥1+𝜀−𝑢_𝑛≥𝑥_𝑛−𝑥−𝑥_𝑛−𝑢_𝑛−𝑥−𝑃_𝑚𝑥−𝑃_𝑚𝑥𝜀.1+𝜀 (2.20) Let 𝑦=𝑃_𝑚𝑥/(1+𝜀); then ‖𝑦‖≤𝑎, and we conclude that RW₁(𝑎,𝑋)≥RW(𝑎,𝑋)−(3+𝑎)𝜀, thus proving the assertion.

Similarly one can prove that, if 𝑋 is a space with a basis, 𝑅(𝑌)=supliminf_𝑛𝑢_𝑛+𝑥∶{𝑥},𝑢_𝑛⊂𝐵_𝑌,𝑢_𝑛⟶_𝑤𝑢0,_𝑛isablockbasicsequenceandsupportof𝑥isfinite. (2.21)

It is known (see [6, 8]) that, if 𝑋, 𝑌 are Banach spaces, thenMW(𝑋)≤MW(𝑌)𝑑(𝑋,𝑌),𝐽(𝑋)≤𝐽(𝑌)𝑑(𝑋,𝑌). (2.22) So, if 𝑋 is a Banach space with a basis, Γ={𝛾_𝑛} with 0≤𝛾_𝑛≤𝛾_𝑛+1≤1 and 𝛾₁>1/MW(𝑋), and 𝑌 is 𝑋 with the Γ-norm, then MW(𝑌)≥MW(𝑋)/𝑑(𝑋,𝑌)≥𝛾₁MW(𝑋)>1, and if 𝛾₁>1/MW(𝑌), MW(𝑋)>1. Similarly, if 𝛾₁>𝐽(𝑋)/2, then 𝐽(𝑌)<2 and, if 𝛾₁>𝐽(𝑌)/2, then 𝐽(𝑋)<2. But the next proposition shows that in fact MW(𝑋)>1 always implies MW(𝑌)>1. For the coefficient 𝐽, in general neither 𝐽(𝑋)<2 implies 𝐽(𝑌)<2 nor the other way round, as we will see in Examples 2.16 and 2.17.

Proposition 2.10. Suppose that 𝑋 is a Banach space with a premonotone basis {𝑒_𝑛}. Then MW=MW(𝑋,‖⋅‖)>1 implies that MW₁=MW(𝑋,|‖⋅‖|)>1.

Proof. Let RW(𝑎,(𝑋,‖⋅‖))=𝑅(𝑎) and RW(𝑎,(𝑋,|‖⋅‖|))=𝑅₁(𝑎). Suppose that MW₁=1. Then, 𝑅₁(𝑎)=1+𝑎 for every 𝑎>0. Let 𝑎>0 and 0<𝜀<𝑎,𝑦∈𝑋 with finite support, |‖𝑦‖|≤𝑎, and let {𝑢_𝑛} be a weakly null block basic sequence with |‖𝑢_𝑛‖|≤1 such that lim_𝑛|‖𝑢_𝑛+𝑦‖|>1+𝑎−𝜀 and lim_𝑛|‖𝑢_𝑛−𝑦‖|>1+𝑎−𝜀. Then we may suppose that for every 𝑛,|‖𝑢_𝑛+𝑦‖|>1+𝑎−𝜀 and |‖𝑢_𝑛−𝑦‖|>1+𝑎−𝜀. Hence, 𝑢_𝑛≥1−𝜀,𝑦≥𝑎−𝜀. (2.23) We may also assume that the supports of 𝑦 and 𝑢_𝑛 are disjoint. Let 𝑢_𝑛=∑^𝑟𝑛_{𝑖=𝑙𝑛}𝑎_𝑖𝑒_𝑖 and 𝑦=∑^𝑟_𝑖=1𝑏_𝑖𝑒_𝑖.
Suppose that |‖𝑢_𝑛+𝑦‖|=𝛾_𝑚𝑛‖∑^𝑟𝑛_{𝑖=𝑚𝑛}(𝑎_𝑖+𝑏_𝑖)𝑒_𝑖‖ for some 𝑚_𝑛≤𝑟_𝑛. It is not possible that 𝑚_𝑛>𝑟, because this would mean that 1+𝑎−𝜀≤𝑢_𝑛+𝑦=𝛾_𝑚𝑛^𝑟𝑛_𝑚𝑛𝑎_𝑖𝑒_𝑖≤𝑢_𝑛≤1. (2.24) So, by passing to a subsequence if necessary, we may assume that for every 𝑛 we have that 𝑚_𝑛=𝑖₀≤𝑟. Thus, 1+𝑎−𝜀≤𝑢_𝑛+𝑦=𝛾_𝑖0^𝑟𝑛_𝑖=𝑖0𝑎_𝑖+𝑏_𝑖𝑒_𝑖≤𝛾_𝑖0^𝑟𝑛_{𝑖=𝑙𝑛}𝑎_𝑖𝑒_𝑖+^𝑟_𝑖=𝑖0𝑏_𝑖𝑒_𝑖≤𝛾_𝑖0𝛾_𝑙𝑛+𝑎. (2.25) Since lim_𝑛𝛾_𝑙𝑛=1, by passing to the limit, we obtain that 𝛾_𝑖0≥1−𝜀. (2.26) Similarly, there exists 𝛾_𝑖1≥1−𝜀 with 1+𝑎−𝜀≤𝑢_𝑛≤𝛾−𝑦^𝑖1𝛾_𝑙𝑛+𝑎. (2.27) Suppose that 𝑖₁≥𝑖₀, and let 𝑦₀=∑^𝑟_𝑖=𝑖0𝑏_𝑖𝑒_𝑖 and 𝑦₁=∑^𝑟_𝑖=𝑖1𝑏_𝑖𝑒_𝑖. Then, since the basis is premonotone, 1+𝑎−𝜀≤|‖𝑢_𝑛+𝑦‖|=|‖𝑢_𝑛+𝑦₀‖|≤‖𝑢_𝑛+𝑦₀‖ and 1+𝑎−𝜀≤|‖𝑢_𝑛−𝑦‖|=|‖𝑢_𝑛−𝑦₁‖|≤‖𝑢_𝑛−𝑦₁‖≤‖𝑢_𝑛−𝑦₀‖; thus, 𝑎−𝜀≤𝑦₀≤𝑦≤𝑎. (2.28) Therefore, 𝑎−𝜀≤𝑦₀≤𝑦₀≤1𝛾_𝑖0𝑦₀≤𝑎𝛾_𝑖0≤𝑎1−𝜀 (2.29) and |‖𝑦₀‖−𝑎|≤max{𝜀,𝑎𝜀/(1−𝜀)}. Further, since 𝛾_𝑖0≤𝛾_𝑙𝑛, 1−𝜀≤𝑢_𝑛≤𝑢_𝑛≤1𝛾_𝑙𝑛𝑢_𝑛≤1𝛾_𝑙𝑛≤11−𝜀 (2.30) and |‖𝑢_𝑛‖−1|≤𝜀/(1−𝜀).
Hence, 𝑢_𝑛𝑢_𝑛+𝑦₀𝑦₀𝑎≥𝑢_𝑛+𝑦₀−𝑢_𝑛−1−𝑦₀𝜀−𝑎≥1+𝑎−𝜀−1−𝜀−max𝜀,𝑎𝜀.1−𝜀 (2.31) Similarly, 𝑢_𝑛𝑢_𝑛−𝑦₀𝑦₀𝑎𝜀≥1+𝑎−𝜀−1−𝜀−max𝜀,𝑎𝜀.1−𝜀 (2.32) We deduce that min(liminf_𝑛‖(𝑢_𝑛/‖𝑢_𝑛‖)+(𝑦₀/‖𝑦₀‖)𝑎‖,liminf_𝑛‖(𝑢_𝑛/‖𝑢_𝑛‖)−(𝑦₀/‖𝑦₀‖)𝑎‖)≥1+𝑎−𝜀−𝜀/(1−𝜀)−max{𝜀,𝑎𝜀/(1−𝜀)}, and letting 𝜀 tend to zero we obtain 𝑅(𝑎)=1+𝑎 and MW=1.

Examples 2.14 and 2.17 exhibit spaces in which MW(𝑋,‖⋅‖)=1 and MW(𝑋,|‖⋅‖|)>1. There is however a special case in which MW(𝑋,‖⋅‖)≥MW(𝑋,|‖⋅‖|).

Recall that a basis {𝑒_𝑛} of a Banach space 𝑋 is 1-spreading if, whenever 𝑥=∑^∞_𝑖=1𝑎_𝑖𝑒_𝑖∈𝑋 and {𝑒_𝑛𝑖}_𝑖 is a subsequence of {𝑒_𝑛},∑^∞_𝑖=1𝑎_𝑖𝑒_𝑛𝑖∈𝑋 and ‖∑^∞_𝑖=1𝑎_𝑖𝑒_𝑖‖=‖∑^∞_𝑖=1𝑎_𝑖𝑒_𝑛𝑖‖.

Proposition 2.11. If 𝑋 is a Banach space with a premonotone 1-spreading basis, then 𝑀𝑊(𝑋,‖⋅‖)≥𝑀𝑊(𝑋,|‖⋅‖|).

Proof. Let 𝑇_𝑚∶𝑋→𝑋 be the translation given by 𝑇_𝑚∑^∞_𝑖=1𝑎_𝑖𝑒_𝑖=∑^∞_𝑖=1𝑎_𝑖𝑒_𝑖+𝑚, and let 𝑎>0,𝑦∈𝑋, ‖𝑦‖≤𝑎, {𝑢_𝑛}⊂𝐵_𝑋, where 𝑦 has finite support and {𝑢_𝑛} is a weakly null block basic sequence such that min(‖𝑦+𝑢_𝑛‖,‖𝑦−𝑢_𝑛‖)>𝑅𝑊(𝑎,(𝑋,‖⋅‖))−𝜀 for every 𝑛. Let 𝑚∈ℕ; then there exists 𝑁∈ℕ such that for 𝑛>𝑁 the supports of 𝑇_𝑚𝑦 and 𝑢_𝑛 are disjoint, and thus, since the basis is 1-spreading, ‖𝑦−𝑢_𝑛‖=‖𝑇_𝑚𝑦−𝑢_𝑛‖ and ‖𝑦+𝑢_𝑛‖=‖𝑇_𝑚𝑦+𝑢_𝑛‖. Then, |‖𝑇_𝑚𝑦‖|≤‖𝑇_𝑚𝑦‖≤𝑎, |‖𝑢_𝑛‖|≤‖𝑢_𝑛‖≤1 and for 𝑛>𝑁𝑇_𝑚𝑦−𝑢_𝑛≥𝛾_𝑚𝑇_𝑚𝑦−𝑢_𝑛=𝛾_𝑚𝑦−𝑢_𝑛,𝑇_𝑚𝑦+𝑢_𝑛≥𝛾_𝑚𝑇_𝑚𝑦+𝑢_𝑛=𝛾_𝑚𝑦+𝑢_𝑛. (2.33) Hence, RW(𝑎,(𝑋,|‖⋅‖|))≥𝛾_𝑚RW(𝑎,(𝑋,‖⋅‖)), and, by passing to the limit as 𝑚 tends to infinity, we get RW(𝑎,(𝑋,|‖⋅‖|))≥RW(𝑎,(𝑋,‖⋅‖)) and thus the desired result.

Similarly to Propositions 2.10 and 2.11 we can prove the following.

Proposition 2.12. Suppose that 𝑋 is a Banach space with a premonotone basis {𝑒_𝑛}. Then 𝑅=𝑅(𝑋,‖⋅‖)<2 implies that 𝑅₁=𝑅(𝑋,|‖⋅‖|)<2 and, if the basis is premonotone and 1-spreading, then 𝑅=𝑅(𝑋,‖⋅‖)≤𝑅(𝑋,|‖⋅‖|)=𝑅₁.

Corollary 2.13. If 𝑋 is a Banach space with a premonotone 1-spreading basis, then 𝑀𝑊₁>1 if and only if 𝑀𝑊>1; also 𝑅₁<2 if and only if 𝑅<2.

Next we will show an example of a space without a 1-spreading basis, such that 𝑅=2, MW=1 but 𝑅₁<2 and thus MW₁>1.

Example 2.14. Let 𝑋 be 𝑐₀ with the following norm: 𝑎_𝑛=𝑎₁+max_𝑖≥2𝑎_𝑖. (2.34) Let {𝑒_𝑛} denote the canonical basis of 𝑐₀. Then for every 𝑎>0, ‖𝑎𝑒₁+𝑒_𝑛‖=‖𝑎𝑒₁−𝑒_𝑛‖=1+𝑎, and thus 𝑅=2 and MW=1. On the other hand let 𝑥∈𝑋 with finite support and |‖𝑥‖|≤1 and suppose that {𝑢_𝑛} is a block basic sequence with |‖𝑢_𝑛‖|≤1 for 𝑛∈𝑁, 𝑢_𝑛=∑^𝑟𝑛_{𝑖=𝑚𝑛}𝑎_𝑖𝑒_𝑖 with 𝑚_𝑛≤𝑟_𝑛<𝑚_𝑛+1 and the support of 𝑥 does not intersect the support of 𝑢_𝑛. Then, for every 𝑚≥2 and 𝑛≥𝑚,𝛾_𝑚‖𝑄_𝑚𝑥+𝑢_𝑛‖=max(𝛾_𝑚‖𝑄_𝑚𝑥‖_𝑐0,𝛾_𝑚‖𝑢_𝑛‖_𝑐0)≤1, 𝛾₁𝑥+𝑢_𝑛≤1+𝛾₁𝑢_𝑛𝑐0𝛾≤1+¹𝛾_𝑚𝑛. (2.35) Thus, liminf|‖𝑥+𝑢_𝑛‖|≤1+𝛾₁. Hence 𝑅₁≤1+𝛾₁<2 and, by (2.9), MW₁>1.

With regard to the coefficient 𝐽 we have the following which is proved similarly to Proposition 2.11.

Proposition 2.15. Suppose that 𝑋 is a Banach space with a premonotone 1-spreading basis {𝑒_𝑛}.Then 𝐽(𝑋,‖⋅‖)≤𝐽(𝑋,|‖⋅‖|).

In general neither 𝐽(𝑋,‖⋅‖)<2 implies 𝐽(𝑋,|‖⋅‖|)<2 nor the other way round, as the following examples show.

Example 2.16. Let 1>𝜇≥1/√2 and 𝑋=ℝ²_𝜇⨁₂𝑙₂, where ℝ²_𝜇=(ℝ²,‖⋅‖_(𝜇)) and ‖(𝑥₁,𝑥₂)‖_(𝜇)=max(|𝑥₁|,|𝑥₂|,𝜇(|𝑥₁|+|𝑥₂|)). Then, if 𝜇≤𝛾₂/(𝛾₁+𝛾₂), 𝐽(𝑋)<2 but for every Γ, 𝐽(𝑋,|‖⋅‖|)=2.
Since 𝜇≥1/√2, it is easy to see that for 𝑥=(𝑥₁,𝑥₂)∈ℝ², 𝜇1−2𝜇+2𝜇²‖𝑥‖₂≤‖𝑥‖_(𝜇)≤𝜇√2‖𝑥‖₂. (2.36) Thus, 𝑑(𝑙₂,𝑋)≤𝑑(ℝ²,ℝ²_𝜇)=√21−2𝜇+2𝜇²<√2, and by (2.22), since 𝐽(𝑙₂)=√2, we obtain that𝐽(𝑋)<2.
Now let Γ={𝛾_𝑛}, 𝜇≤𝛾₂/(𝛾₁+𝛾₂), 𝑥=(1/𝛾₁,1/𝛾₂,0,0,…), and 𝑦=(−1/𝛾₁,1/𝛾₂,0,0,…). Then |‖𝑥‖|=|‖𝑦‖|=1 but |‖𝑥+𝑦‖|=|‖𝑥−𝑦‖|=2.

This last example is another proof of the known fact that for every 𝜀>0 there are Banach spaces 𝑋 and 𝑌 with 𝑑(𝑋,𝑌)<1+𝜀, 𝐽(𝑋)<2 but 𝐽(𝑌)=2. In the following example we exhibit a space 𝑋 with 𝐽(𝑋)=2 such that 𝐽(𝑋,|‖⋅‖|)<2.

Example 2.17. Let 𝑋=(𝑙₂,‖⋅‖), where, for 𝑥=(𝑎_𝑛)∈𝑙₂, ‖𝑥‖=|𝑎₁|+(∑^∞_𝑖=2𝑎²_𝑖)^1/2. Then 𝐽(𝑋)=2,MW(𝑋)=1, and, if Γ is such that 𝛾₂>1/√2 and 𝛾₁/𝛾₂<1/√2, 𝐽(𝑋,|‖⋅‖|)<2 and thus MW(𝑋,|‖⋅‖|)>1.
Obviously if {𝑒_𝑛} is the canonical basis of 𝑙₂, and 𝑎>0, then ‖𝑎𝑒₁+𝑒_𝑛‖=‖𝑎𝑒₁−𝑒_𝑛‖=1+𝑎 for 𝑛>1 and thus 𝐽(𝑋)=2 and MW(𝑋)=1.
Suppose now that 𝐽(𝑌)=𝐽(𝑋,|‖⋅‖|)=2.Then there exist sequences {𝑥_𝑛},{𝑦_𝑛}⊂𝐵_𝑌 such that lim_𝑛|‖𝑥_𝑛+𝑦_𝑛‖|=lim_𝑛|‖𝑥_𝑛−𝑦_𝑛‖|=2and {𝑚_𝑛},{𝑙_𝑛}⊂ℕ so that 𝑥_𝑛+𝑦_𝑛=𝛾_𝑚𝑛𝑄_𝑚𝑛𝑥_𝑛+𝑦_𝑛,𝑥_𝑛−𝑦_𝑛=𝛾_𝑙𝑛𝑄_𝑙𝑛𝑥_𝑛−𝑦_𝑛. (2.37)
By passing to a subsequence, we may assume that, for every 𝑛∈ℕ, 𝑒^∗₁(𝑥_𝑛)≥0, 𝑒^∗₁(𝑦_𝑛)≥0 and 𝑒^∗₁(𝑥_𝑛)≥𝑒^∗₁(𝑦_𝑛), and that the subsequence satisfies one of the next three cases:(1)𝑚_𝑛>1 and 𝑙_𝑛>1 for every 𝑛,(2)𝑚_𝑛=𝑙_𝑛=1 for every 𝑛,(3)𝑚_𝑛=1 and 𝑙_𝑛>1 for every 𝑛.
Observe that 𝛾_𝑚𝑄_𝑚𝑥+𝑦≥2−𝜀implies𝛾_𝑚𝑄_𝑚(𝑥)≥1−𝜀and𝛾_𝑚𝑄_𝑚𝑦≥1−𝜀. (2.38)
Case 1. Let 𝑍=(𝑙₂,|‖⋅‖|_Γ1), where |‖⋅‖|_Γ1 is the Γ₁={𝛾_𝑖}^∞_𝑖=2-norm. Then 𝑄₂𝑥_𝑛Γ1≤1,𝑄₂𝑦_𝑛Γ1𝑄≤1,₂𝑥_𝑛+𝑦_𝑛Γ1≥2−𝜀,𝑄₂𝑥_𝑛−𝑦_𝑛Γ1≥2−𝜀. (2.39) Since 1/𝛾₂<√2, then, by (2.2), 𝑑(𝑍,𝑙₂)<√2 and, by (2.22), since 𝐽(𝑙₂)=√2,𝐽(𝑍)<2 and this is a contradiction.Case 2. Let 𝜀>0. Suppose that 𝑥=∑^∞_𝑖=1𝑎_𝑖𝑒_𝑖, 𝑦=∑^∞_𝑖=1𝑏_𝑖𝑒_𝑖∈𝐵_𝑌, 𝑎₁>0,𝑏₁>0, 𝑎₁>𝑏₁ and 𝛾₁^∞_𝑖=2𝑎_𝑖+𝑏_𝑖^21/2≥2−𝜀−𝛾₁𝑎₁+𝑏₁,𝛾₁^∞_𝑖=2𝑎_𝑖−𝑏_𝑖^21/2≥2−𝜀−𝛾₁𝑎₁−𝑏₁. (2.40) Squaring both inequalities and adding them, since 𝛾₂(∑^∞_𝑖=2𝑎²_𝑖)^1/2≤|‖𝑥‖|, we get 2𝛾²₁𝛾²₂≥𝛾²₁^∞_𝑖=2𝑎²_𝑖+𝑏²_𝑖≥2−𝜀−𝛾₁𝑎₁²+𝛾²₁𝑏²₁≥2−𝜀−𝛾₁𝑎₁². (2.41) By passing to the limit as 𝜀→0, since 𝛾₁𝑎₁≤1, we obtain that √2(𝛾₁/𝛾₂)≥2−𝛾₁𝑎₁≥1, and this contradicts 𝛾₁/𝛾₂<1/√2.Case 3. Let 2>𝜀>0. Suppose that 𝑥=∑^∞_𝑖=1𝑎_𝑖𝑒_𝑖,𝑦=∑^∞_𝑖=1𝑏_𝑖𝑒_𝑖∈𝐵_𝑌,𝑎₁>0,𝑏₁≥0,𝑎₁>𝑏₁ and |‖𝑥+𝑦‖|=𝛾₁(|𝑎₁+𝑏₁|+(∑^∞_𝑖=2(𝑎_𝑖+𝑏_𝑖)²)^1/2)≥2−𝜀 and |‖𝑥−𝑦‖|=𝛾_𝑚(∑^∞_𝑖=𝑚(𝑎_𝑖−𝑏_𝑖)²)^1/2≥2−𝜀.Then, by (2.38), 𝛾_𝑚^∞_𝑖=𝑚𝑎²_𝑖^1/2≥1−𝜀. (2.42)
Since in 𝑙₂ if 𝑢,𝑣∈𝐵_𝑙2, one has that ‖𝑢−𝑣‖≥𝛿 implies ‖𝑢+𝑣‖≤21−(𝛿/2)², then 𝛾_𝑚(∑^∞_𝑖=𝑚(𝑎_𝑖+𝑏_𝑖)²)^1/2≤√4𝜀−𝜀².Also 2−𝜀≤𝛾₁𝑎₁+𝑏₁+^𝑚−1_𝑖=2𝑎_𝑖+𝑏_𝑖^21/2+^∞_𝑖=𝑚𝑎_𝑖+𝑏_𝑖^21/2≤𝛾₁𝑎₁+𝑏₁+𝛾₁^𝑚−1_𝑖=2𝑎_𝑖+𝑏_𝑖^21/2+𝛾₁𝛾_𝑚√4𝜀−𝜀²≤𝛾₁𝑎₁+𝑏₁+𝛾₁^𝑚−1_𝑖=2𝑎_𝑖+𝑏_𝑖^21/2+√4𝜀−𝜀². (2.43) Let 𝜙=𝜀+√4𝜀−𝜀²; then 2−𝜙≤𝛾₁𝑎₁+𝑏₁+𝛾₁^𝑚−1_𝑖=2𝑎_𝑖+𝑏_𝑖^21/2. (2.44) By (2.38),𝛾₁|𝑎₁|+𝛾₁(∑^𝑚−1_𝑖=2𝑎²_𝑖)^1/2≥1−𝜙, and since, for 𝐴,𝐵>0,(𝐴+𝐵)^1/2−𝐴^1/2=𝐵(𝐴+𝐵)^1/2+𝐴^1/2, (2.45) we have, using (2.42), that 1≥𝛾₁𝑎₁+𝛾₁^∞_𝑖=2𝑎²_𝑖^1/2≥𝛾₁𝑎₁+𝛾₁^𝑚−1_𝑖=2𝑎²_𝑖+(1−𝜀)²𝛾²_𝑚^1/2=𝛾₁𝑎₁+𝛾₁^𝑚−1_𝑖=2𝑎²_𝑖^1/2+𝛾₁(1−𝜀)²𝛾²_𝑚1(𝐴+𝐵)^1/2+𝐴^1/2𝛾≥1−𝜙+¹(1−𝜀)²𝛾²_𝑚1(𝐴+𝐵)^1/2+𝐴^1/2, (2.46) where 𝐴=(∑^𝑚−1_𝑖=2𝑎²_𝑖)^1/2 and 𝐵=(1−𝜀)²/𝛾²_𝑚.But (𝐴+𝐵)^1/2+𝐴^1/2≤1𝛾²₂+(1−𝜀)²𝛾²_𝑚^1/2+1𝛾₂≤3𝛾₂; (2.47) therefore, 𝛾₁<𝛾₁𝛾²_𝑚≤𝜙(1−𝜀)²3𝛾₂, (2.48) and, taking the limit as 𝜀→0, we get 𝛾₁=0 which is a contradiction.
Hence, 𝐽(𝑋,|‖⋅‖|)<2, and, by (2.10), we have that MW(𝑋,|‖⋅‖|)>1.

Acknowledgments

The authors thank the referees for their detailed review and valuable observations. This work was partially funded by Grant SEP CONACYT 102380.