Weak nearly uniform smoothness of the ψ-direct sums (X

Mikio KATO^∗, Takayuki TAMURA^∗

Weak nearly uniform smoothness of the ψ-direct sums (X

⊕ · · · ⊕ X

)

Abstract. We shall characterize the weak nearly uniform smoothness of the ψ- direct sum (X1⊕ · · · ⊕ XN)ψ of N Banach spaces X1, . . . , XN,where ψ is a convex function satisfying certain conditions on the convex set ∆N = {(s1, . . . , sN−1) ∈ R^N₊⁻¹ : PN−1

i=1 s_i¬ 1}. To do this a class of convex functions which yield `1-like norms will be introduced. We shall apply our result to the fixed point property for nonexpansive mappings (FPP). In particular an example will be presented which indicates that there are plenty of Banach spaces with FPP failing to be uniformly non-square.

2010 Mathematics Subject Classification: 46B20, 52A21, 26A51, 47H10, 46B10.

Key words and phrases: absolute norm, convex function, ψ-direct sum of Banach spaces, weak nearly uniform smoothness, Garc´ıa-Falset coefficient, Schur property, fixed point property.

1. Introduction. Since it was introduced in Takahashi, Kato and Saito [27], the ψ-direct sum X ⊕^ψY of Banach spaces X and Y has attracted a good deal of attention and many investigations have been done, where ψ is a convex function on the unit interval satisfying some conditions (e.g., [3, 4, 5, 6, 7, 11, 12, 13, 14, 15, 16, 20, 21, 22, 24, 28]). The finitely many Banach spaces case is treated in Kato, Saito and Tamura [11]. The present authors [14] showed that under the condition that X or Y is of infinite dimension, X⊕^ψY is weakly nearly uniformly smooth if and only if X and Y are weakly nearly uniformly smooth and ψ 6= ψ1, where ψ₁(t) = 1 is the convex function corresponding to the `1-norm.

The aim of this paper is to extend the above result for the ψ-direct sum (X1⊕

· · · ⊕ X^N)ψ of N Banach spaces X1, . . . , XN. The situation is much more compli-

∗The authors are Supported in part by a Grant-in-Aid for Scientific Research, Japan Society for the Promotion of Science (23540216).

cated than expected in comparison with the two Banach spaces case, that is, the weak nearly uniform smoothness of X1, . . . , XN and the condition ψ 6= ψ1 do not imply that of (X1⊕· · ·⊕X^N)ψ. The point is to find out a subclass of convex functions in ΨN including ψ₁ which should be excluded. As such one we shall introduce a subclass Ψ⁽¹⁾_N consisting of the convex functions which yield ”`₁-like norms”(Section 5; cf. [17]). In the case of N = 2 the subclass Ψ⁽¹⁾_N contains only the function ψ₁.

As the main results we shall have the following: (i) Let X1, . . . , XN be all infinite dimensional spaces. Then (X₁⊕ · · · ⊕ X^N)ψ is weakly nearly uniformly smooth if and only if X1, . . . , XN are weakly nearly uniformly smooth and ψ 6∈ Ψ⁽¹⁾N (Theorem 6.14). (ii) In the case where some of X1, . . . , XN are infinite dimensional spaces and the rest finite dimensional we shall obtain a more precise result depending on which spaces Xjare of infinite dimension (Theorem 6.12). This extends the previous result for the two Banach spaces case stated above (Corollary 6.13). (iii) The condition that all X₁, . . . , XN are finite dimensional is equivalent to that (X₁⊕ · · · ⊕ X^N)_ψis weakly nearly uniformly smooth for all ψ ∈ Ψ⁽¹⁾N , or equivalently for ψ₁ (Theorem 6.17).

According to Garc´ıa-Falset [8] a Banach space X is weakly nearly uniformly smooth if and only if X is reflexive and R(X) < 2, where R(X) is the Garc´ıa- Falset coefficient ([8]). Thus to obtain the above-mentioned results we shall prove that for the spaces X₁, . . . , XN, some of which do not have the Schur property, R((X1⊕· · ·⊕X^N)ψ) < 2 if and only if R(Xj) < 2 for all 1 ¬ j ¬ N and ψ 6∈ Ψ⁽¹⁾N (S) for all nonempty subsets S of {1, . . . , N − 1} such that X^j does not have the Schur property either for all j ∈ S +1 or for all j ∈ (S +1)^c, where Ψ⁽¹⁾_N (S) is a subclass of Ψ⁽¹⁾_N depending on the set S (Theorem 6.2). In particular, if all Xj do not have the Schur property, R((X1⊕ · · · ⊕ X^N)ψ) < 2 if and only if R(Xj) < 2 for all 1 ¬ j ¬ N and ψ 6∈ Ψ⁽¹⁾N (Theorem 6.4). This yields a previous result by Dhompongsa et al. [3]: If R(Xj) < 2 for all 1 ¬ j ¬ N and ψ ∈ Ψ^N is strictly convex, then R((X1⊕ · · · ⊕ X^N)ψ) < 2 (Corollary 6.6). Also their another result ([3]) which states that, if ψ ∈ Ψ^N is strictly convex, then (X1⊕ · · · ⊕ X^N)ψ is weakly nearly uniformly smooth if and only if X1, . . . , XN are so, is a corollary of Theorem 6.14 (Corollary 6.16).

In the course of doing the above we shall present a characterization for the strict monotonicity of an absolute norm on Cⁿ, from which a previous result in Dowling and Turett [7] will be immediately derived (Section 3) and also we shall discuss generalized H¨older’s inequality and the dual space of (X1⊕ · · · ⊕ X^N)ψ(Section 4).

As an application we shall obtain that if X₁, . . . , XN are weakly nearly uni- formly smooth and ψ 6∈ Ψ⁽¹⁾N , then (X₁⊕ · · · ⊕ X^N)ψ has the fixed point property for nonexpansive mappings, FPP in short (Theorem 7.2); in particular the `_∞-sum (X1⊕ · · · ⊕ X^N)_∞, which is not uniformly non-square, has FPP (recall that all uniformly non-square spaces have FPP ([10])). Also another example will be con- structed, which indicates that there are plenty of Banach spaces with FPP failing to be uniformly non-square (Corollary 7.6).

2. Preliminaries. A norm k · k on C^N is calledabsolute if

k(z¹,· · · , z^N)k = k(|z¹|, · · · , |z^N|)k for all (z¹,· · · , z^N) ∈ C^N and normalized if

k(1, 0, · · · , 0)k = · · · = k(0, · · · , 0, 1)k = 1.

The collection of all such norms on C^N is denoted by ANN. In case of N = 2 for every absolute normalized norm k · k on C² there corresponds a unique convex (continuous) function ψ on the unit interval [0, 1] satisfying max{1−t, t} ¬ ψ(t) ¬ 1 under the equation ψ(t) = k(1−t, t)k ([5]; see also [25]). The N-dimensional version of this fact was presented by K.-S. Saito et al. [26] as follows: Let k · k ∈ AN^N. Let

ψ(s) =k(1 −

NX−1 i=1

si, s1,· · · , s^N−1)k for s = (s¹,· · · , s^N−1) ∈ ∆^N, (1)

where

∆N = {s = (s1,· · · , s^N−1) ∈ R^N−1:

N−1X

i=1

si¬ 1, sⁱ­ 0}. (2)

Then ψ is convex (continuous) on the convex set ∆N and satisfies the following:

(A0) ψ(0,· · · , 0) = ψ(1, 0, · · · , 0) = · · · = ψ(0, · · · , 0, 1) = 1, (A1) ψ(s1,· · · , s^N−1) ­

^NX−1 i=1

si

 ψ

 s1

PN−1 i=1 si

,· · · , sN−1

PN−1 i=1 si



if 0 <

NX−1 i=1

si¬ 1,

(A2) ψ(s1,· · · , s^N−1) ­ (1 − s¹)ψ

 0, s2

1 − s¹,· · · , sN−1

1 − s¹



if 0 ¬ s¹< 1,

· · · ·

(AN) ψ(s1,· · · , s^N−1) ­ (1 − s^N−1)ψ

 s1

1 − s^N−1,· · · , sN−2

1 − s^N−1, 0



if 0 ¬ sN−1 < 1 The converse holds true: Denote by ΨN the family of all convex functions ψ on ∆N

satisfying (A0) − (A^N). For any ψ ∈ Ψ^N define

k(z¹,· · · , z^N)k^ψ =













 PN j=1|zⁱ|

ψ

 |z²| PN

j=1|zⁱ|,· · · , |z^N| PN

j=1|zⁱ|



if (z1,· · · , z^N) 6= (0, · · · , 0), 0 if (z1,· · · , z^N) = (0, · · · , 0)

(3)

Then k·k^ψ∈ AN^N and k·k^ψsatisfies (1). Thus we have a one-to-one correspondence between ANN and ΨN with the equation (1). The `p-norms

k(z¹,· · · , z^N)k^p=

( {|z¹|^p+ · · · + |z^N|^p}^1/p if 1 ¬ p < ∞, max{|z1|, · · · , |z^N|} if p = ∞

are typical examples of absolute normalized norms and for any k·k ∈ AN^N we have

k · k∞¬ k · k ¬ k · k¹ (4)

([26, Lemma 3.1], cf. [5]). The convex function corresponding to the `p-norm, which is denoted by ψ_p, is

ψ_p(s1,· · · , s^N−1) =







n1 −PN−1 i=1 si

p

+ s^p₁+ · · · + s^pN−1

o1/p

if 1 ¬ p < ∞, max{1 −PN−1

i=1 si, s1,· · · , s^N−1} if p = ∞ and for any ψ ∈ Ψ^N we have

1

N ¬ ψ∞(·) ¬ ψ(·) ¬ ψ1(·) (5)

([26, Lemma 3.2]). A function ψ ∈ Ψ^N is called strictly convex provided, if s, t ∈

∆N, s6= t, one has ψ((1 − α)s + αt) < (1 − α)ψ(s) + αψ(t) for all 0 < α < 1.

Let X1, ..., XN be Banach spaces and let ψ ∈ Ψ^N. Let (X1⊕ · · · ⊕ X^N)ψ be the direct sum of X1, ..., XN equipped with the norm

k(x1,· · · , x^N)k^ψ := k(kx1k, · · · , kx^Nk)k^ψ for (x₁, . . . , xN) ∈ X1⊕ · · · ⊕ X^N (6) ([11, 27]). Then (X1⊕ · · · ⊕ X^N)ψ is a Banach space, which extends the notion of

`p-sum.

As usual BX and SX stand for the closed unit ball and unit sphere of a Banach space X, respectively. A sequence {xⁿ} in X is called a basic sequence if it is a Schauder basis for its closed linear span, that is, if every x in the span of {xⁿ} has a unique representation of the form x =P∞

n=1αnxn. A Banach space X is said to be weakly nearly uniformly smooth ([18]) provided there exist 0 <  < 1 and ν > 0 such that for any basic sequence {xⁿ} in B^X and any 0 < t < ν there is k > 0 so that kx¹+ txkk ¬ 1 + t. The constant R(X) ([8]), which is referred to as Garc´ıa-Falset coefficient in [2], is defined by

R(X) = sup{lim inf_n

→∞ kxⁿ+ xk}, (7)

where the supremum is taken over all weakly null sequences {xⁿ} in B^Xand all x ∈ BX. Clearly the formula (7) can be rewritten as R(X) = sup {lim infⁿ→∞kxⁿ− xk}.

As is readily seen, 1 ¬ R(X) ¬ 2 and it is known that R(c⁰) = R(`1) = 1, R(`p) = 2^1/p (1 < p < ∞), R(c) = R(`∞) = 2 (cf. [19, p. 165]). It is known

that uniformly convex, resp., uniformly smooth spaces are weakly nearly uniformly smooth (cf. [19, p. 165], [23, p. 508]), and X is weakly nearly uniformly smooth if and only if X is reflexive and R(X) < 2 (Garc´ıa-Falset [8]). A Banach space X is said to have theSchur property if every weakly convergent sequence in X converges strongly.

It is clear that R(X) = 1 if X has the Schur property. X is called strictly convex provided, whenever x, y ∈ S^X, x6= y, one has k(x+y)/2k < 1. X is said to have the fixed point property (resp. weak fixed point property) for nonexpansive mappings if every nonexpansive self-mapping T of any nonempty bounded closed (resp. weakly compact) convex subset C of X has a fixed point (T is called nonexpansive if kT x − T yk ¬ kx − yk for all x, y ∈ C). We say the former as FPP in short. It is well known that, if R(X) < 2, X has the weak fixed point property (Garc´ıa-Falset [9]) and hence weakly nearly uniformly smooth spaces have FPP.

3. Strict monotonicity of absolute norms. We begin with recalling a fun- damental fact.

Lemma 3.1 ([26]; cf. [5]) Let k · k ∈ AN^N.

(i)If |p^j| ¬ |q^j| for all 1 ¬ j ¬ N, then k(p¹, . . . , pN)k ¬ k(q1, . . . , qN)k.

(ii)If |pj| < |q^j| for all 1 ¬ j ¬ N, then k(p¹, . . . , pN)k < k(q1, . . . , qN)k.

A norm k · k in AN^N is calledstrictly monotone if |p^j| ¬ |q^j| for all 1 ¬ j ¬ N and |p^j0| < |q^j0| for some 1 ¬ j⁰ ¬ N, then k(p¹, . . . , pN)k < k(q¹, . . . , qN)k. As is readily seen, the `_∞-norm does not have this property. The next result is useful in our later discussion.

Proposition 3.2 Let ψ ∈ Ψ^N. Let (p1, . . . , pN) ∈ C^N and 0 < |p^j0| < |q^j0| for some 1 ¬ j⁰¬ N. Then the following are equivalent.

(i) k(p¹, . . . , pj₀, . . . , pN)k^ψ<k(p¹, . . . , qj₀, . . . , pN)k^ψ

(ii) k(p1, . . . ,

j0

^0 , . . . , pN)k^ψ<k(p1, . . . , qj₀, . . . , pN)k^ψ

Proof The implication (i) ⇒ (ii) is clear. Assume (ii) to be true. Let |p^j0| = α|q^j0| with some 0 < α < 1. Then we have

k(p¹, . . . , pj0, . . . , pN)k^ψ

= k(p¹, . . . , αqj0, . . . , pN)k^ψ

= kα(p¹, . . . , qj₀, . . . , pN) + (1 − α)(p¹, . . . ,

j₀

^0 , . . . , pN)k^ψ

¬ αk(p1, . . . , qj₀, . . . , pN)k^ψ+ (1 − α)k(p1, . . . ,

j0

^0 , . . . , pN)k^ψ

<k(p¹, . . . , qj0, . . . , pN)k^ψ,

or (i). _

Now, we note that according to the formulae (1) and (3) the properties (A0) − (AN) for a function ψ ∈ Ψ^N are rewritten in words of the corresponding norm k·k^ψ

as follows:

(A⁰₀) k(1, 0, · · · , 0)k^ψ= · · · = k(0, · · · , 0, 1)k^ψ= 1,

(A⁰₁) k(1 −

N−1X

i=1

si, s1,· · · , s^N−1)k^ψ­ k(0, s¹,· · · , s^N−1)k^ψ,

(A⁰₂) k(1 −

NX−1 i=1

si, s1,· · · , s^N−1)k^ψ ­ k(1 −

NX−1 i=1

si, 0, s2,· · · , sN−1)k^ψ,

· · · · (A⁰_N) k(1 −

NX−1 i=1

si, s1,· · · , s^N−1)k^ψ ­ k(1 −

NX−1 i=1

si, s1,· · · , s^N−2, 0)k^ψ

As is readily seen from the above observation, the next result in Dowling and Turett [7] is an immediate consequence of Proposition 3.2.

Corollary 3.3 ([7]) Let ψ ∈ Ψ^N. Then k · k^ψ is strictly monotone if and only if ψ satisfies the following conditions:

(sA₁) ψ(s₁,· · · , s^N−1) >

^NX−1 i=1

si

 ψ

 s1

PN−1 i=1 si

,· · · , sN−1

PN−1 i=1 si



if 0 <

NX−1 i=1

si< 1,

(sA2) ψ(s1,· · · , s^N−1) > (1 − s¹)ψ

 0, s2

1 − s¹,· · · , sN−1

1 − s¹



if 0 < s1< 1,

· · · ·

(sAN) ψ(s1,· · · , s^N−1) > (1 − s^N−1)ψ

 s1

1 − sN−1,· · · , sN−2

1 − s^N−1, 0



if 0 < sN−1 < 1

4. Generalized H¨older’s inequality and dual space. In this section we shall discuss generalized H¨older’s inequality for the ψ-direct sum (X1⊕ · · · ⊕ X^N)ψ

and its dual space (cf. [4, 20, 14]); for completeness we shall present our proofs. In the following X^∗ stands for the dual space of X.

Theorem 4.1 (Generalized H¨older’s inequality; cf. [20, 14]) Let X1,· · · , X^N be Banach spaces and let ψ ∈ Ψ^N. Let ψ^∗ be the function on ∆N defined by

ψ^∗(s1, . . . , sN−1) = sup

(t1,...,t_N−1)∈∆N

(1 −PN−1

i=1 si)(1 −PN−1

i=1 ti) +PN−1 i=1 siti

ψ(t1,· · · , t^N−1) (8)

Then ψ^∗∈ Ψ^N and   

XN j=1

x^∗_j(xj)  

¬ k(x^∗1,· · · , x^∗N)k^ψ∗k(x¹,· · · , x^N)k^ψ (9)

for all (x1,· · · , x^N) ∈ (X1⊕ · · · ⊕ X^N)ψ and (x^∗₁,· · · , x^∗N) ∈ (X1^∗⊕ · · · ⊕ XN^∗)ψ^∗.

Proof It is straightforward to see that ψ^∗is convex and the proof is omitted. Since ti¬ max{t¹, . . . , t_N−1} ¬ ψ(t¹,· · · , t^N−1) and ψ(0, . . . , 0,

^i

1 , 0, · · · , 0) = 1, we have

ψ^∗(0, . . . , 0,

^i

1 , 0, · · · , 0) = sup

(t1,...,tN−1)∈∆^N

ti

ψ(t1,· · · , tN−1) = 1, or (A₀). Let 0 <PN−1

i=1 si¬ 1. Then we have

^NX−1 i=1

si

 ψ^∗

 s1

PN−1 i=1 si

, . . . , sN−1

PN−1 i=1 si



= sup

(t₁,...,tN−1)∈∆N

PN−1 i=1 siti

ψ(t1,· · · , t^N−1)

¬ sup

(t1,...,tN−1)∈∆^N

(1 −PN−1

i=1 si)(1 −PN−1

i=1 ti) +PN−1 i=1 siti

ψ(t1,· · · , t^N−1)

= ψ^∗(s1, . . . , sN−1), or (A₁). Let 0 ¬ s < 1. Then

(1 − s¹)ψ^∗

 0, s2

1 − s¹, . . . , sN−1

1 − s¹



= sup

(t1,...,t_N−1)∈∆N

(1 −PN−1

i=1 si)(1 −PN−1

i=1 ti) +PN−1 i=2 siti

ψ(t1,· · · , t^N−1)

¬ sup

(t₁,...,tN−1)∈∆N

(1 −PN−1

i=1 si)(1 −PN−1

i=1 ti) +PN−1 i=1 siti

ψ(t1,· · · , t^N−1)

= ψ^∗(s1, . . . , s_N−1),

or we have (A2). The rest properties (A3)-(AN) are similarly shown. Thus we have ψ^∗ ∈ Ψ^N. Next, let (x1,· · · , x^N) and (x^∗₁,· · · , x^∗N) be arbitrary nonzero elements in (X1⊕ · · · ⊕ X^N)ψ and (X₁^∗⊕ · · · ⊕ XN^∗)ψ^∗, respectively. Let

sj= kx^∗j+1k PN

j=1kx^∗jk and tj = kx^j+1k PN

j=1kx^jk for 1 ¬ j ¬ N − 1

Then (s1, . . . , sN−1), (t1, . . . , t_N−1) ∈ ∆^N, and by (8) we have ψ^∗(s₁, . . . , sN−1)ψ(t₁,· · · , t^N−1)

­ 1 −

NX−1 i=1

si

! 1 −

NX−1 i=1

ti

! +

NX−1 i=1

siti

= kx^∗1k PN

j=1kx^∗jk· kx¹k PN

j=1kx^jk+

NX−1 j=1

kx^∗j+1k PN

j=1kx^∗jk· kx^j+1k PN

j=1kx^jk

= XN j=1

kx^∗jkkx^jk

PN

j=1kx^∗jk PN

j=1kx^jk

­ 1

PN

j=1kx^∗jk PN

j=1kx^jk   

XN j=1

x^∗_j(xj)   ,

from which it follows that x^∗₁, . . . , x^∗_N

ψ^∗

x1, . . . , xN

ψ ­PN

j=1x^∗_j(xj) , or

(9). This completes the proof. _

Theorem 4.2 (cf. [4, 20, 14]) Let X1, . . . , XN be Banach spaces and ψ ∈ Ψ^N. Then

(X1⊕ · · · ⊕ X^N)^∗_ψ= (X₁^∗⊕ · · · ⊕ XN^∗)ψ^∗ (10) Proof Let (x^∗₁, . . . , x^∗_N) ∈ (X1^∗⊕ · · · ⊕ XN^∗)ψ^∗. Let

f (x1, . . . , xN) = XN j=1

x^∗_j(xj) for (x₁, . . . , xN) ∈ (X1⊕ · · · ⊕ X^N)ψ (11)

Then by Theorem 4.1 we have |f(x¹, . . . , xN)| ¬ k(x^∗1, . . . , x^∗_N)k^ψ∗k(x¹, . . . , xN)k^ψ, from which it follows that f ∈ (X¹⊕ · · · ⊕ X^N)^∗_ψ and kfk ¬ k(x^∗1, . . . , x^∗_N)k^ψ∗. Conversely, take an arbitrary f ∈ (X¹⊕ · · · ⊕ X^N)^∗_ψ. For each 1 ¬ j ¬ N let

x^∗_j(xj) = f(0, . . . , 0,

^j

xj, 0 . . . , 0) for xj ∈ X^j

Then we have x^∗_j ∈ Xj^∗ and the formula (11). Next we see k(x^∗1, . . . , x^∗_N)k^ψ∗¬ kfk.

Assume that (x^∗₁, . . . , x^∗_N) 6= (0, . . . , 0). For any ε > 0 take u^j∈ S^Xj, the unit sphere of Xj, so that kx^∗jk ¬ x^∗j(uj) + ε. Let

sj = kx^∗j+1k PN

j=1kx^∗jk for 1 ¬ j ¬ N − 1 Then, noting that k(x^∗1, . . . , x^∗_N)k^ψ∗ = (PN

j=1kx^∗jk)ψ^∗(s1, . . . , sN−1), we have for

any (t1, . . . , tN−1) ∈ ∆^N



 XN j=1

kx^∗jk





1 −PN−1 j=1 sj

 1 −PN−1 j=1 tj

+PN−1 j=1 sjtj

ψ(t1,· · · , t^N−1)

= 1

ψ(t1,· · · , t^N−1)



kx^∗1k



1 −

NX−1 j=1

tj



 +

NX−1 j=1

kx^∗j+1kt^j





¬ 1

ψ(t1,· · · , t^N−1)



(x^∗₁(u1) + ε)



1 −

NX−1 j=1

tj



 +

NX−1 j=1

(x^∗_j+1(uj+1) + ε)tj





= 1

ψ(t1,· · · , t^N−1)



x^∗₁(u1)



1 −

NX−1 j=1

tj



 +

N−1X

j=1

x^∗_j+1(uj+1)tj



 + ε

ψ(t1,· · · , t^N−1)

¬ x^∗1 1 −PN−1 j=1 tj

ψ(t1, . . . , tN−1)u1

! +

XN j=2

x^∗_j

 tj−1

ψ(t1, . . . , tN−1)uj

 + Nε

(as ψ(t1, . . . , tN−1) ­ 1/N by (5))

= f 1 −PN−1 j=1 tj

ψ(t1, . . . , tN−1)u1, t1

ψ(t1, . . . , tN−1)u2, . . . , tN−1

ψ(t1, . . . , tN−1)uN

! + Nε

¬ kfk + Nε,

where one should note that

 1 − P^Nj=1⁻¹tj

ψ(t1, . . . , tN−1)u1, t1

ψ(t1, . . . , t_N−1)u2, . . . , tN−1

ψ(t1, . . . , tN−1)uN

 _ψ

= 1

ψ(t1, . . . , tN−1) k(1 −

N−1X

j=1

tj, t1, . . . , tN−1)k^ψ= 1

Therefore k(x^∗1, . . . , x^∗_N)k^ψ∗ ¬ kfk + Nε. Since ε > 0 is arbitrary, we have k(x^∗1, . . . , x^∗_N)k^ψ∗¬ kfk, which completes the proof. 

By Theorem 4.2 we have the next result.

Proposition 4.3 Let X1, . . . , XN be Banach spaces and ψ ∈ Ψ^N. Then the follo- wing are equivalent.

(i) {(x^(k)1 , . . . , x^(k)_N )}^k is a weakly null sequence in (X1⊕ · · · ⊕ X^N)ψ. (ii) {x^(k)j }^k is a weakly null sequence in Xj for each 1 ¬ j ¬ N.

Proof (i) ⇒ (ii). Let {(x^(k)1 , . . . , x^(k)_N )}ktend weakly to 0 in (X₁⊕ · · · ⊕ X^N)_ψ. For each 1 ¬ j ¬ N take arbitrary x^∗j ∈ Xj^∗. Since (0, . . . ,

^j

x^∗_j, 0 . . . , 0)∈ (X¹⊕· · ·⊕X^N)^∗_ψ,

we have

klim→∞x^∗_j(x^(k)_j ) = lim

k→∞h(0, . . . ,

^j

x^∗_j, 0 . . . , 0), (x^(k)₁ , . . . , x^(k)_N )i = 0

(ii) ⇒ (i). Let {x^(k)1 }^k, . . . ,{x^(k)N }^k be weakly null sequences in X₁, . . . , XN re- spectively. By Theorem 4.2 for any f ∈ (X1⊕ · · · ⊕ X^N)^∗_ψ there exists a unique (x^∗₁, . . . , x^∗_N) ∈ (X1^∗⊕ · · · ⊕ XN^∗)ψ^∗ such that

f (x1, . . . , xN) = XN j=1

x^∗_j(xj) for all (x1, . . . , xN) in (X1⊕ · · · ⊕ X^N)ψ

Therefore we have limk→∞f (x^(k)₁ , . . . , x^(k)_N ) = limk→∞PN

j=1x^∗_j(x^(k)_j ) = 0, which

completes the proof. _

By Proposition 4.3 we have the following.

Proposition 4.4 Let X1, . . . , XN be Banach spaces and ψ ∈ Ψ^N. Then the follo- wing are equivalent.

(i) (X1⊕ · · · ⊕ X^N)ψ has the Schur property.

(ii)All X1, . . . , XN have the Schur property.

5. A class of convex functions Ψ⁽¹⁾_N . We shall introduce a subclass Ψ⁽¹⁾_N of ΨN.

Definition 5.1 Let ψ ∈ Ψ^N. Let S be a nonempty subset of {1, . . . , N − 1} and χ_S its characteristic function.

(i) We say ψ ∈ Ψ⁽¹⁾N (S) if there exists an element (s1, . . . , s_N−1) ∈ ∆^N with

0 <

N−1X

i=1

χ_S(i)si< 1 (12)

such that, letting M =PN−1

i=1 χ_S(i)si, ψ(s1, . . . , sN−1) = Mψ

χ_S(1)s₁

M , . . . ,χ_S(N − 1)s^N−1

M



(13) + (1 − M) ψ

χ_S^c(1)s1

1 − M , . . . ,χ_S^c(N − 1)s^N−1

1 − M



(ii) We say ψ ∈ Ψ⁽¹⁾N when ψ ∈ Ψ⁽¹⁾N (S) for some nonempty subset S of {1, . . . , N − 1}.

According to (1) and (3) the equation (14) is interpreted as

k(1 −

NX−1 i=1

si, s1, . . . , sN−1)k^ψ (14)

= k(0, χS(1)s1, . . . , χ_S(N − 1)s^N−1)k^ψ +k(1 −

N−1X

i=1

si, χ_S^c(1)s1, . . . , χ_S^c(N − 1)sN−1)k^ψ

This indicates that the norm k · k^ψfor ψ ∈ Ψ⁽¹⁾N has an `1-norm like property, which will be reformulated in Theorems 5.5 and 5.8 below.

Example 5.2 We see that ψ₁∈ Ψ⁽¹⁾N . In the case N = 2 the class Ψ⁽¹⁾₂ consists of only the function ψ₁. Indeed, according to the formula (15), ψ ∈ Ψ⁽¹⁾2 if and only if there exists s ∈ (0, 1) such that ψ(s) = k(1 − s, s)k^ψ= k(0, s)k^ψ+ k(1 − s, 0)k^ψ= 1, which is possible only when ψ = ψ₁. If N ­ 3, let S = {1} and (s¹, . . . , sN−1) = (1/N, . . . , 1/N). Then

k(1 −

NX−1 i=1

si, s1, . . . , sN−1)k^ψ1= k(1/N, 1/N, . . . , 1/N)k1

= k(0, 1/N, 0, . . . , 0)k¹+ k(1/N, 0, 1/N, . . . , 1/N)k¹

= k(0, χS(1)s1, . . . , χ_S(N − 1)s^N−1)k^ψ1

+k(1 −

N−1X

i=1

si, χ_S^c(1)s₁, . . . , χ_S^c(N − 1)s^N−1)k^ψ1,

which implies ψ₁∈ Ψ⁽¹⁾N .

Example 5.3 Let α + β = 1, α, β > 0 and let

ψ(s1, s2) = max{1 − αs¹, 1− αs², αs1+ β, αs2+ β} for (s¹, s2) ∈ ∆³ Then ψ ∈ Ψ⁽¹⁾3 . Indeed it is obvious that ψ is convex on ∆3and ψ(0, 0) = ψ(1, 0) = ψ(0, 1) = 1. Also we have

(s1+ s2)ψ

 s₁ s1+ s2, s₂

s1+ s2



= maxn

(s1+ s2) − αs1,(s1+ s2) − αs2, αs1+ β(s1+ s2), αs2+ β(s1+ s2)o

¬ ψ(s1, s2),

(1 − s1)ψ



0, s₂ 1 − s1



= maxn

1 − s1,(1 − s1) − αs2, β(1 − s1), αs2+ β(1 − s1)o

¬ ψ(s1, s2),

and in the same way, (1 − s²)ψ (s1/(1− s²), 0) ¬ ψ(s¹, s2). Thus we have ψ ∈ Ψ³. Let next (s1, s2) = (α/2, β/2) and S = {1, 2}. Then, letting M = χS(1)s1+χ_S(2)s2, we have

M ψ

χ_S(1)s1

M ,χ_S(2)s2

M



+ (1 − M)ψ

χ_S^c(1)s1

1 − M ,χ_S^c(2)s2

1 − M



= 1

2ψ (α, β) +1

2ψ(0, 0) = max

1 − α²

2 , 1 − αβ

2 , α²+ β

2 , αβ + β 2

 +1

2

= max



1 − α ·α

2, 1− α ·β 2



= ψ(s1, s2),

where one should note that αs₁+ β ¬ (α²+ β + 1)/2 = 1 − αβ/2 and αs2+ β ¬ (αβ + β + 1)/2 = 1 − α²/2. Therefore we have ψ∈ Ψ⁽¹⁾3 .

The above function ψ yields the following norm onC³: k(z1, z₂, z₃)k^ψ =

max{|z1| + β|z2| + |z3|, |z1| + |z2| + β|z3|, β|z1| + |z2| + β|z3|, β|z1| + β|z2| + |z3|}

The next lemma is useful in Theorem 5.5.

Lemma 5.4 ([12]) For nonzero elements x and y in a Banach space X the following are equivalent.

(i) kx + yk = kxk + kyk (ii)

x

kxk + y kyk

= 2

For a nonempty subset S of {1, . . . , N − 1} let S + 1 = {i + 1 : i ∈ S}.

Theorem 5.5 Let ψ ∈ Ψ^N and let S be a nonempty subset of {1, . . . , N −1}. Then the following are equivalent.

(i) ψ ∈ Ψ⁽¹⁾N (S)

(ii)There exists (a1, . . . , aN) ∈ R^N+ (that is, a1, . . . , an­ 0) such that

k(a¹, . . . , aN)k^ψ= k(0, χS+1(2)a2, . . . , χ_S+1(N)aN)k^ψ (15) +k(a¹, χ_(S+1)c(2)a2, . . . , χ_(S+1)c(N)aN)k^ψ,

where (0, χ_S+1(2)a2, . . . , χ_S+1(N)aN)and (a1, χ_(S+1)c(2)a2, . . . , χ_(S+1)c(N)aN)are nonzero.

(iii)There exists (a1, . . . , aN) ∈ R^N+ such that the formula (16) holds true with k(0, χS+1(2)a2, . . . , χ_S+1(N)aN)k^ψ (16)

= k(a¹, χ_(S+1)^c(2)a2, . . . , χ_(S+1)^c(N)aN)k^ψ= 1 (iv) There exists (a₁, . . . , aN) ∈ R^N+ such that for all α, β ∈ R

kα(0, χS+1(2)a₂, . . . , χ_S+1(N)a_N) + β(a₁, χ_(S+1)c(2)a₂, . . . , χ_(S+1)c(N)a_N)kψ (17)

= |α|k(0, χS+1(2)a2, . . . , χ_S+1(N)aN)kψ

+|β|k(a1, χ_(S+1)c(2)a2, . . . , χ_(S+1)c(N)aN)kψ,