Nonlinear Analysis 70 (2009) 2270–2276

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Nonlinear Analysis

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Criteria for complex strongly extreme points of Musielak–Orliczfunction spacesLili Chen a, Yunan Cui a, Henryk Hudzik b,∗

a Department of Mathematics, Harbin University of Science and Technology, Harbin 150080, PR Chinab Faculty of Mathematics and Computer Science, Adam Mickiewicz University, Umultowska 87 61-614, Poznań, Poland

a r t i c l e i n f o

Article history:Received 26 October 2007Accepted 5 March 2008

MSC:46B2046E30

Keywords:Musielak–Orlicz function spacesComplex strongly extreme pointsComplex midpoint local uniform rotundity

a b s t r a c t

The concepts of complex strongly extreme point and complex midpoint local uniformrotundity in complex Banach spaces are introduced. It is proved that every complexstrongly extreme point is a complex extreme point and every complex locally uniformlyrotund point is a complex strongly extreme point. Moreover, criteria for complex stronglyextreme points of the unit ball and, as a corollary, criteria for complex midpoint localuniform rotundity in Musielak–Orlicz function spaces are given.

© 2008 Elsevier Ltd. All rights reserved.

1. Introduction

Let (X, ‖ · ‖) be a Banach space over the complex field C and let B(X)and S(X) be the closed unit ball and the unit sphereof X, respectively. The same notation is used for real Banach spaces defined below. In the sequel N and R denote the set ofnatural numbers and the set of real numbers, respectively.

In the early 1980’s, many articles in the area of the geometry of Banach spaces were directed at the complex geometry ofcomplex Banach spaces. It is well known that the complex geometric properties of complex Banach spaces have applicationsin various branches, among others in Harmonic Analysis, Operator Theory, Banach Algebras, C∗-Algebras, DifferentialEquations, Quantum Mechanics and Hydrodynamics. It is also known that the strongly extreme points which are connectedwith the midpoint local uniform rotundity of the whole space, are the most basic and important geometric points in thegeometric theory of Banach spaces.

The notion of complex rotundity was introduced by Thorp and Whitley in [12], where they showed that the complexspace L1[0, 1] is complex rotund. They also showed that the maximum principle for analytic functions remains also validfor vector-valued analytic functions with values in X, if X is complex rotund. Next Globevnik [7] introduced the notion ofcomplex uniform rotundity and showed that the complex space L1[0, 1] is complex uniformly rotund. Davis, Garling, andTomczak-Jaegermann [4] introduced the notion of uniform PL-convexity of complex Banach spaces as a complex analogue ofuniform rotundity in real Banach spaces. Wu and Sun [13,14] gave criteria for complex extreme points, complex rotundity,complex local uniform rotundity and complex uniform rotundity in the class complex Musielak–Orlicz spaces. Wang andTeng [16] introduced the concepts of complex locally uniformly rotund points and complex local uniform rotundity and theygave criteria for them in complex Musielak–Orlicz spaces. Convexities and complex convexities of Musielak–Orlicz sequencespaces were also considered in [3,15] and in some other spaces in [1,5] and [9]. They continued the problems together with

∗ Corresponding author. Tel.: +48 61 8295356; fax: +48 61 8295315.E-mail addresses: chenlili0819@yahoo.com.cn (L. Chen), cuiya@hrbust.edu.cn (Y. Cui), hudzik@amu.edu.pl (H. Hudzik).

0362-546X/$ – see front matter © 2008 Elsevier Ltd. All rights reserved.doi:10.1016/j.na.2008.03.005

L. Chen et al. / Nonlinear Analysis 70 (2009) 2270–2276 2271

Bian in [17]. Hudzik and Narloch [8] have observed that a real Köthe function space is strictly monotone (resp. uniformlymonotone) if and only if its complexification is complex strictly rotund (resp. complex uniformly rotund). For more detailson the development of the complex geometry of complex Banach spaces, we refer to [4,10].

The aim of this paper is to introduce the concepts of complex strongly extreme points and complex midpoint localuniform rotundity for complex Banach spaces, and to give criteria for them in Musielak–Orlicz function spaces equipped withthe Luxemburg norm. Moreover, it will be proved that for any complex Musielak–Orlicz space LM complex strongly extremepoints of the unit ball B(LM) and complex midpoint local uniform rotundity of LM coincide, respectively, with complex locallyuniformly rotund points of the unit ball B(LM) and complex local uniform rotundity of LM . We will also prove that everycomplex strongly extreme point is a complex extreme point and every complex locally uniformly rotund point is a complexstrongly extreme point for arbitrary complex Banach spaces.

Before starting with our results, we need to recall some notions.A point x ∈ S(X) is said to be a complex extreme point of B(X) if max|λ|=1 ‖x+λy‖ > 1 for every non-zero y ∈ X. A complex

Banach space X is said to be complex strictly rotund if every x ∈ S(X) is a complex extreme point of B(X).A point x ∈ S(X) is said to be a complex locally uniformly rotund point of B(X) if for any ε > 0, there exists δ = δ(x, ε) > 0

such that max|λ|=1 ‖x+ λy‖ ≥ 1+ δ for every non-zero y ∈ X with ‖y‖ ≥ ε. A complex Banach space X is said to be complexlocally uniformly rotund if every x ∈ S(X) is a complex locally uniformly rotund point of B(X).

Let (X, ‖ · ‖) be a real Banach space, x ∈ S(X) and ε ≥ 0. Then we consider the following modulus at x:

∆(x, ε) = inf{1− λ : ∃y ∈ X s.t. ‖λx± y‖ ≤ 1 and ‖y‖ ≥ ε}.

The infimum is taken here over all λ ∈ R such that there is y ∈ X such that ‖y‖ ≥ ε, ‖λx+ y‖ ≤ 1 and ‖λx− y‖ ≤ 1.Let us note that such y ∈ X satisfies the inequality ‖y‖ ≤ 1. Really, if ‖y‖ > 1, then

2 < ‖2y‖ = ‖y+ λx+ (y− λx)‖ ≤ ‖y+ λx‖ + ‖y− λx‖ ≤ 2,

a contradiction.It is easy to see that ∆(x, ε) is a nonnegative and non-decreasing function of ε ≥ 0 with ∆(x, 0) = 0 for any x ∈ S(X), and

it is known that x is a strongly extreme point of B(X) if and only if ∆(x, ε) > 0 for every ε > 0 (see [6]).Let (T,

∑,µ) be a non-atomic and complete measure space. By M we denote a Musielak–Orlicz function, i.e. M : T ×

[0,+∞)→ [0,+∞] satisfies the conditions:

(1) M(·, u) is a µ-measurable function of t ∈ T for any u from the interval [0,+∞);(2) for µ-a.e. t ∈ T we have that M(t, 0) = 0, limu−→∞M(t, u) = ∞ and there exists ut > 0 such that M(t, ut) <∞;(3) M(t, ·) is a convex function of u on the interval [0,∞) for µ-a.e. t ∈ T.

Let (X, ‖ ·‖) be a complex Banach space and XT be the set of all strongly∑

-measurable functions from T to X. Let us definefor x ∈ XT the modular of x by

ρM(x) =∫TM(t, ‖x(t)‖)dt.

Then the Musielak–Orlicz function space

LM = {x : ρM(λx) <∞ for some λ > 0}

equipped with the Luxemburg norm

‖x‖M = inf{λ > 0 : ρM

(x

λ

)≤ 1

}(∀x ∈ LM)

is a Banach space. Let us define

e(t) = sup{u ≥ 0 : M(t, u) = 0}, E(t) = sup{u ≥ 0 : M(t, u) <∞}.

Then both functions e and E are∑

-measurable (see [2]).We say that M satisfies condition ∆2(M ∈ ∆2 for short) if there exist K ≥ 2 and a Σ-measurable nonnegative function δ

on T such that∫T M(t, δ(t))dt < ∞ and M(t, 2u) ≤ KM(t, u) for µ-almost all t ∈ T and all u ≥ δ(t). Obviously condition ∆2

for M implies that the functions M(t, ·) have finite values for µ-a.e. t ∈ T (see [11]).

2. Results

We begin this section with formulating some definitions.

Definition 1. Let (X, ‖ · ‖) be a complex Banach space over the complex field C, i be the complex number satisfying i2= −1.

A point x ∈ S(X) is said to be a complex strongly extreme point of B(X) if ∆c(x, ε) > 0 for every ε > 0, where

∆c(x, ε) = inf{1− |λ| : λ ∈ C s.t. ∃y ∈ X s.t. ‖λx± y‖ ≤ 1, ‖λix± y‖ ≤ 1 and ‖y‖ ≥ ε}.

It is not difficult to see that ∆c(x, ε) is also a nonnegative and non-decreasing function of ε ≥ 0 with ∆c(x, 0) = 0 for anyfixed x ∈ S(X).

2272 L. Chen et al. / Nonlinear Analysis 70 (2009) 2270–2276

Definition 2. A complex Banach space X is said to be complex midpoint locally uniformly rotund if every element of S(X) isa complex strongly extreme point of B(X).

Remark 1. If X is a real Banach space and x ∈ S(X), then it is known that if x is a locally uniformly rotund point of B(X) thenx is a strongly extreme point of B(X) which next implies that x is an extreme point of B(X).

We will show in the first two theorems that similar relationships hold also among complex analogues of these points.

Theorem 1. Let (X, ‖ · ‖) be a complex Banach space and suppose that x ∈ S(X) is a complex strongly extreme point of B(X). Thenx is a complex extreme point.

Proof. Assume that x is not a complex extreme point. Then there exists y ∈ X with y 6= 0 such that max|λ|=1 ‖x + λy‖ = 1.Take ε0 = ‖y‖ > 0. Since max|λ|=1 ‖x + λy‖ = 1 implies that ‖x ± y‖ ≤ 1 and ‖x ± iy‖ ≤ 1, we get ∆c(x, ε0) = 0, whichcontradicts the assumption that x is a complex strongly extreme point. �

Theorem 2. Let (X, ‖ · ‖) be a complex Banach space and suppose that x ∈ S(X) is a complex locally uniformly rotund point ofB(X). Then x is a complex strongly extreme point.

Proof. If the assertion is not true, then there exists ε0 > 0 such that ∆c(x, ε0) = 0, i.e. for any η > 0 there exist λ0 ∈ C with1− |λ0| <

η1+η and y0 ∈ X with ‖y0‖ ≥ ε0 such that ‖λ0x± y0‖ ≤ 1 and ‖λ0ix± y0‖ ≤ 1. Hence it follows that∥∥∥∥x± 1λ0

y0

∥∥∥∥ ≤ ∣∣∣∣ 1λ0

∣∣∣∣ < 1+ η,∥∥∥∥x± 1

λ0iy0

∥∥∥∥ ≤ ∣∣∣∣ 1λ0

∣∣∣∣ < 1+ η.

Defining τ1 =1λ0

y0, we have ‖τ1‖ = ‖1λ0

y0‖ ≥ε0|λ0|

, ‖x ± τ1‖ < 1 + η and ‖x ± iτ1‖ < 1 + η, whence it follows thatmax|λ|=1 ‖x+ λτ1‖ < 1+ η, which means that x is not a complex locally uniformly rotund point. �

Before presentation of criteria for complex strongly extreme points and complex midpoint local uniform rotundity inMusielak–Orlicz spaces, let us recall two useful lemmas. For the convenience, from now on, we write LM for (LM, ‖ · ‖M).

Lemma 1 (See [16]). Let (X, ‖ · ‖) be a complex Banach space and x, y ∈ X. If

‖x+ y‖ + ‖x− y‖ + ‖x+ iy‖ + ‖x− iy‖ =∑

k=±1,±i‖x+ ky‖ ≤ 4(1+ δ)‖x‖,

then max|λ|≤1 ‖x+λ2 y‖ ≤ (1+ 13

√δ)‖x‖.

Lemma 2 (See [16]). If M ∈ ∆2, then for any sequence {xn} in LM , we have that ρM(xn)→ 1 if and only if ‖xn‖ → 1.

We are now ready to prove the main theorem of this paper which gives criteria for complex strongly extreme points ofB(LM).

Theorem 3. Let LM be a complex Musielak–Orlicz space. Then x ∈ S(LM) is a complex strongly extreme point of B(LM) iff:

(i) ρM(x) = 1 or ‖x(t)‖ = E(t) for µ-a.e. t ∈ T;(ii) ‖x(t)‖ ≥ e(t) for µ-a.e. t ∈ T;

(iii) for any ε > 0 there exists δ > 0 such that for every y ∈ LM(X), we have ‖y|A(x,y,δ)‖M < ε3 for the set

A(x, y, δ) =

t ∈ T :∑

k=±1,±i‖x(t)+ ky(t)‖ ≤ 4(1+ δ)‖x(t)‖

.

Proof. Necessity. If condition (i) is not true, then ρM(x) < 1 and µ{t ∈ T : ‖x(t)‖ < E(t)} > 0. Choose b > 0 such thatT0 = {t ∈ T : ‖x(t)‖ + b < E(t)} is not a null set. We may assume, passing to a subset of T0 if necessary, that∫

T\T0

M(t, ‖x(t)‖)dt +∫T0

M(t, ‖x(t)‖ + b)dt ≤ 1.

Take θ ∈ S(X) and define

y(t) ={bθ, t ∈ T0;

0, t ∈ T \ T0.(2.1)

It is obvious that y 6= 0. However, for any λ ∈ C with |λ| ≤ 1, we have ρM(x + λy) ≤ 1, whence ‖x + λy‖M ≤ 1. This showsthat x is not a complex extreme point of B(LM).

L. Chen et al. / Nonlinear Analysis 70 (2009) 2270–2276 2273

If condition (ii) does not hold, then µ{t ∈ T : ‖x(t)‖ < e(t)} > 0. Pick d > 0 such that T1 = {t ∈ T : ‖x(t)‖ + d < e(t)} isnot a null set. Take θ ∈ S(X) and define

y(t) ={

dθ, t ∈ T1;

0, t ∈ T \ T1.(2.2)

It is obvious that y 6= 0. However, for any λwith |λ| ≤ 1, we have

ρM(x+ λy) ≤ ρM(x|T\T1)+

∫T1

M(t, ‖x(t)‖ + d)dt

= ρM(x|T\T1) ≤ 1,

which implies that ‖x+ λy‖M ≤ 1. In such a way we proved that x is not a complex extreme point of B(LM).Finally, if condition (iii) is not satisfied, then there exists ε0 > 0 such that for any n ∈ N, there exists yn ∈ LM(X) with

‖yn|A(x,yn, 1n )‖M ≥

ε03 , where

An = A(x, yn,

1n

)=

t ∈ T :∑

k=±1,±i‖x(t)+ kyn(t)‖ ≤ 4

(1+

1n

)‖x(t)‖

.

By Lemma 1, max|λ|≤1 ‖x(t)+λ2 yn(t)‖ ≤ (1+ 13

√1n)‖x(t)‖ for t ∈ An.

Let us set

zn(t) ={yn(t), t ∈ An;

0, t ∈ T \ An(2.3)

for any n ∈ N. Then

ρM

x+ λ2 zn

1+ 13√

1n

= ∫T\An

M

t,‖x(t)‖

1+ 13√

1n

dt +∫An

M

t,‖x(t)+ λ

2 yn(t)‖

1+ 13√

1n

dt

≤

∫TM(t, ‖x(t)‖)dt ≤ 1.

Hence it follows that ‖x+ λ2 zn‖M ≤ 1+ 13

√1n

. Setting λn = 1+ 13√

1n

, it is clear that |λn| → 1. Noticing that ‖ zn2λn‖M ≥

ε012 for

all n large enough and considering λ = ±1,±i, we have4c(x,ε012 ) = 0. This shows that x is not a complex strongly extreme

point of B(LM).Sufficiency. First we assume that ‖x(t)‖ = E(t) for µ-a.e. t ∈ T. If x is not a complex strongly extreme point, then there

exists ε0 > 0 such that

∆c(x, ε0) = inf{1− |λ| : λ ∈ C is s.t. ∃y ∈ X s.t. ‖λx± y‖ ≤ 1, ‖λix± y‖ ≤ 1 and ‖y‖ ≥ ε0} = 0.

As in the proof of Theorem 2, we can deduce that there exists ε1 > 0 such that for any η > 0 there exists τ ∈ LM(X) with‖τ‖M ≥ ε1 and

max|λ|=1‖x+ λτ‖M < 1+ η.

Given any ε1 > 0, by (iii), there exists δ > 0 such that for any y ∈ LM(X), we have ‖y|A(x,y,δ)‖M < ε13 , where

A(x, y, δ) =

t ∈ T :∑

k=±1,±i‖x(t)+ ky(t)‖ ≤ 4(1+ δ)‖x(t)‖

.

Taking η = δ > 0, we have max|λ|=1 ‖x + λτ‖M < 1+ δ and ‖τ|A(x,τ,δ)‖M < ε13 . The last condition gives ‖τ|T\A(x,τ,δ)‖M ≥ 2

3ε1.For µ-a.e. t ∈ T \ A(x, τ, δ), we have

14(1+ δ)

∑k=±1,±i

‖x(t)+ kτ(t)‖ > ‖x(t)‖ = E(t) for µ-a.e. t ∈ T.

Combining this with the fact that µ(T \ A(x, τ, δ)) > 0, we deduce that

2274 L. Chen et al. / Nonlinear Analysis 70 (2009) 2270–2276

14

∑k=±1,±i

ρM

(x+ kτ

1+ δ

)≥

∫T\A(x,τ,δ)

M

(t,

14(1+ δ)

∑k

‖x(t)+ kτ(t)‖

)dt = ∞.

This shows that maxk ρM( x+kτ1+δ ) = ∞, where k takes values±1 and±i. So, we have max|λ|=1 ‖x+λτ‖M ≥ 1+δ, a contradiction.

Next we assume that ρM(x) = 1 and the assumptions (ii) and (iii) are satisfied. If the desired assertion does not hold,we can deduce as above that there exists ε1 > 0 such that for any η > 0 there exists τ ∈ LM(X) with ‖τ‖M ≥ ε1 satisfyingmax|λ|=1 ‖x+ λτ‖M < 1+ η. By (iii), there exists δ > 0 such that ‖τ|A0‖M < ε1

3 , where

A0 = A(x, τ, δ) =

t ∈ T :∑

k=±1,±i‖x(t)+ kτ(t)‖ ≤ 4(1+ δ)‖x(t)‖

.

Now we consider the following two cases.(a)

∫T\A0

M(t, ‖x(t)‖)dt > 0. In this case, we have

ρM

(x

1+ η

)=

∫A0

M(t,‖x(t)‖

1+ η

)dt +

∫T\A0

M(t,‖x(t)‖

1+ η

)dt

≤

∫A0

M

t,14

∑k=±1,±i

‖x(t)+ kτ(t)‖

1+ η

dt +1

1+ δ

∫T\A0

M

t,14

∑k=±1,±i

‖x(t)+ kτ(t)‖

1+ η

dt

=

∫TM

t,14

∑k=±1,±i

‖x(t)+ kτ(t)‖

1+ η

dt −δ

1+ δ

∫T\A0

M

(t,

14∑k

‖x(t)+ kτ(t)‖

1+ η

)dt

≤14

∑k=±1,±i

ρM

(x+ kτ

1+ η

)−

δ

1+ δ

∫T\A0

M(t,

1+ δ1+ η

‖x(t)‖)

dt.

Taking η > 0 small enough such that 1+δ1+η > 1, we have

ρM

(x

1+ η

)≤ 1−

δ

1+ δ

∫T\A0

M(t, ‖x(t)‖)dt.

Leting η→ 0, we get the following contradiction:

1 = ρM(x) ≤ 1−δ

1+ δ

∫T\A0

M(t, ‖x(t)‖)dt.

(b)∫T\A0

M(t, ‖x(t)‖)dt = 0. Assume in this case that µe0 > 0, where e0 = µ{t ∈ T \ A0 : ‖x(t)‖ > e(t)}. Then we get thecontradiction:

0 =∫T\A0

M(t, ‖x(t)‖)dt ≥∫e0

M(t, ‖x(t)‖)dt > 0.

So µe0 = 0. Combining this with (ii), we have ‖x(t)‖ = e(t) for µ-a.e. t ∈ T \ A0. Therefore(1+

δ

2

)‖x(t)‖ > e(t) for µ-a.e. t ∈ T \ A0.

Thus, there exists e1 ⊆ T \ A0 with µe1 > 0 such that∫e1

M(t,(

1+δ

2

)‖x(t)‖

)dt > 0.

As in case (a), we have

ρM

(x

1+ η

)=

∫A0

M(t,‖x(t)‖

1+ η

)dt +

∫T\A0

M(t,‖x(t)‖

1+ η

)dt

≤

∫A0

M

t,14

∑k=±1,±i

‖x(t)+ kτ(t)‖

1+ η

dt +1

1+ δ

∫T\A0

M

t,14

∑k=±1,±i

‖x(t)+ kτ(t)‖

1+ η

dt

=

∫TM

t,14

∑k=±1,±i

‖x(t)+ kτ(t)‖

1+ η

dt −δ

1+ δ

∫T\A0

M(t,

14‖x(t)+ kτ(t)‖

1+ η

)dt

≤14

∑k=±1,±i

ρM

(x+ kτ

1+ η

)−

δ

1+ δ

∫T\A0

M(t,

1+ δ1+ η

‖x(t)‖)

dt.

L. Chen et al. / Nonlinear Analysis 70 (2009) 2270–2276 2275

Taking η > 0 small enough such that 1+δ1+η > 1+ δ

2 , we have

ρM

(x

1+ η

)≤ 1−

δ

1+ δ

∫T\A0

M(t,(

1+δ

2

)‖x(t)‖

)dt.

≤ 1−δ

1+ δ

∫e1

M(t,(

1+δ

2

)‖x(t)‖

)dt.

Taking η→ 0, we get the contradiction:

1 = ρM(x) ≤ 1−δ

1+ δ

∫e1

M(t,(

1+δ

2

)‖x(t)‖

)dt. �

Remark 2. It can be verified analogously to our Theorem 3 that conditions (i), (ii) and (iii) from the theorem imply also thatx is a complex locally uniformly rotund point of B(LM). So, we have the following

Corollary 1. Let LM be a complex Musielak–Orlicz space, x ∈ S(LM). Then x is a complex strongly extreme point of B(LM) if andonly if it is a complex locally uniformly rotund point of B(LM).

Remark 3. If follows from Corollary 1 that condition (iv) from Theorem 1 in [16] which gives criteria for complex locallyuniformly rotund points of B(LM) can be removed and that condition (iii) from that theorem can be simplified.

A consequence of Theorem 3 is the following

Theorem 4. A complex Musielak–Orlicz space LM is complex midpoint locally uniformly rotund iff:

(i) M ∈ ∆2;(ii) e(t) = 0 for µ-a.e.t ∈ T;

(iii) for any x ∈ S(LM), ε > 0, there exists δ > 0 such that for every y ∈ LM , we have ‖y|A(x,y,δ)‖M < ε3 , where

A(x, y, δ) =

t ∈ T :∑

k=±1,±i‖x(t)+ ky(t)‖ ≤ 4(1+ δ)‖x(t)‖

.

Proof. The sufficiency is trivial (see [16]).Necessity. If condition (i) does not hold, then there exists x0 ∈ S(LM) such that ‖x(t)‖ < E(t) forµ-a.e. t ∈ T and ρM(x) < 1.

This contradicts condition (i) of Theorem 3, which shows that x0 is not a complex strongly extreme point.If condition (ii) does not hold, then T0 = {t ∈ T : e(t) > 0} is not a null set. Pick now b > 0 and T1 ⊆ T \ T0 such that∫

T1

M(t, b)dt = 1.

Choosing v ∈ S(X) and setting x = e(t)2 v|T0 + bv|T1 , we have ρM(x) = 1, whence ‖x‖M = 1. However,

µ{t ∈ T : ‖x(t)‖ < e(t)} ≥ µT0 > 0,

which contradicts condition (ii) of Theorem 3.The necessity of condition (iii) can be deduced by condition (iii) of Theorem 3. �

As an immediate consequence of Corollary 1 and Theorem 4, we obtain the following

Corollary 2. A complex Musielak–Orlicz space LM is complex midpoint locally uniformly rotund if and only if it is complex locallyuniformly rotund.

Acknowledgment

In case of Prof. Cui Yunan this paper was supported by NSF of CHINA, Grant number: 10571037.

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