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NEW CONVEXITY AND FIXED POINT PROPERTIESIN HARDY AND LEBESGUE- BOCHNER SPACES

M. Besbes, S. J. Dilworth, P. N. Dowling, C. J. Lennard.

ABSTRACT

We show that for the Hardy class of functions H 1 with domain the ball or polydisc inCN , a certain type of uniform convexity property (the uniform Kadec-Klee-Huff property)holds with respect to the topology of pointwise convergence on the interior; which coincideswith both the topology of uniform convergence on compacta and the weak ∗ topology onbounded subsets of H 1.

Also, we show that if a Banach space X has a uniform Kadec-Klee-Huff property,then the Lebesgue-Bochner space L p(µ,X) 1 ≤ p < ∞ must have a related uniformKadec-Klee-Huff property. Consequently, by known results, the above spaces have normalstructure properties and fixed point properties for non-expansive mappings.

Keywords and phrases. uniform Kadec-Klee-Huff property; Hardy spaces; Lebesgue-Bochner spaces; weak-star convergence; uniform convergence on compacta; normal structure; fixed point property for nonexpan-sive mappings; strictly pseudoconvex domain with C 2-boundary; polydisc.

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NEW CONVEXITY AND FIXED POINT PROPERTIESIN HARDY AND LEBESGUE- BOCHNER SPACES

M. Besbes, S. J. Dilworth, P. N. Dowling, C. J. Lennard.

M.Besbes S.J.Dilworth

Equipe d’Analyse Department of Mathematics

U.A. No 754 au C.N.R.S. The University of South Carolina

Universite Paris 6 Columbia, South Carolina 29208

Tour 46-4eme Etage

4, Place Jussieu

75252 - Paris Cedex 05.

P.N.Dowling C.J.Lennard

Department of Mathematics and Statistics Department of Mathematics and Statistics

Miami University University of Pittsburgh

Oxford, Ohio 45056 Pittsburgh, Pennsylvania 15260

Introduction

We show that the usual Hardy space H1(∆) has a type of uniform convexity property

- the so-called uniform Kadec-Klee-Huff (UKKH) property with respect to its usual weak∗-

topology, generalizing a result of Warschawski [W] and Newman[Ne]. Using a reformula-

tion of the UKKH property, we show that UKKH properties in a Banach spaceX always lift

to a certain kind of UKKH property in the Lebesgue-Bochner space Lp(µ,X) 0 < p <∞;

extending a result of Partington [P] for `p(X). As a consequence we establish a UKKH

2

property for certain several complex variable-H1 spaces; ‘making uniform’ a result of

Hoffmann [Ho]. From UKKH properties, normal structure and fixed point properties for

non-expansive maps in compact, convex sets follow; via results of van Dulst and Sims [D-S]

and Lennard [L2].

We remark that Kadec-Klee properties in various H1 spaces have also been studied by

Kellogg [Ke], Bryskin and Sedaev [B-S], Goldstein and Swaminathan [G-S] and Godefroy

[G1] and [G2]. UKKH properties on vector-valued Hp spaces have been recently studied by

Dowling and Lennard [D-L1] and [D-L2]. Moreover, vector-valued H1 spaces are examined

in Besbes [B3]. [B3] also establishes fixed point theorems for subspaces of vector-valued

L1 spaces.

A by no means exhaustive list of related papers dealing with Kadec-Klee and other

properties lifting from X to Lp(µ,X) is the following: Smith and Turett [S-T1] and [S-

T2], Smith [S], Lin and Lin [L-L], Lin, Lin and Troyanski [L-L-T] and Downing and Turett

[D-T].

A corollary of our Theorem 2.2 is that H1(∆) has the fixed point property for non-

expansive maps on weak∗-compact convex sets; which is a recent result of Besbes [B1].

This result is also implicit in the work of Maurey [M], who proved the corresponding result

for weakly-compact, convex sets in H1(∆).

We thank Catherine for typing the manuscript. We also thank the referee for pointing

out how to shorten and improve the presentation of the proofs of Proposition 1.2 and

Theorem 3.1. The fourth author acknowledges the support of a University of Pittsburgh

Internal Research Grant during part of the preparation of this paper.

1. Preliminaries

N denotes the positive integers, R the real numbers, C the complex numbers, T the

unit sphere in C, and ∆ the open unit disc in C. For a Banach space (X, ‖·‖X ), BX denotes

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the closed unit ball in X. (Ω,Σ, µ) will denote a positive, complete measure space and for

0 < p < ∞, Lp(X) = Lp(µ,X) is the Lebesgue-Bochner space of all (equivalence classes

of) strongly measurable functions of f : Ω→ X for which the quasi-norm ‖f‖Lp(X) <∞.

‖f‖Lp(X) = ‖f‖p :=(∫

Ω

‖f(ω)‖pX dµ(ω))1/p

.

Also, Lp(µ) denotes Lp(µ,X) when X is the scalar field.

Throughout this paper, (X, ‖ · ‖X) will be a Banach space, and τ will denote a topo-

logical vector space topology on X that is weaker than the norm topology. We will often

write ‖ · ‖ instead of ‖ · ‖X .

Following Lennard [L2], based on the work of Huff [Hu], we make the following defi-

nition.

1.1 Definition. (X, ‖ · ‖X) has the uniform Kadec-Klee-Huff property with respect to τ ,

(UKKH(τ)), if for all ε > 0, there exists δ = δ(ε) ∈ (0, 1) such that whenever (xn)∞n=1 is

a sequence in BX with xn −→n

x ∈ X with respect to τ and ‖x‖ > 1 − δ, it follows that

infn6=m ‖xn − xm‖ ≤ ε.

We remark that from Lennard [L1] Corollary 1.1.4, we have that whenever (X, ‖ · ‖X )

has the UKKH(τ), it follows that ‖ · ‖ is a sequentially lower semicontinuous function

with respect to τ .

The following reformulation of the UKKH(τ) property is a key ingredient in the

proof of the main result of section 3 (Theorem 3.1).

1.2 Proposition. Let (X, ‖·‖X ) be a Banach space and τ be a t.v.s. topology on X that

is weaker than the norm topology. Suppose X has the uniform Kadec-Klee-Huff property

with respect to τ . Then X has the following property.

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For all ε > 0 there exists δ ∈ (0, 1) such that for all sequences (xn)∞n=1 in BX with

xn −→n

x ∈ X with respect to τ , whenever ‖x‖ > 1− δ, it follows that for some N ∈ N,

supn,m≥N

‖xn − xm‖ ≤ ε.

The proof of the corollary below is straightforward. The details are therefore omitted.

1.3 Corollary.

(1) Let (X, ‖ · ‖X) be a Banach space with the UKKH(τ) property.

(*) Then for all ε > 0 there exists δ1(ε) > 0 such that whenever (xn)∞n=1 is a sequence

in X with xn −→n

x ∈ X with respect to τ , if

supn∈N‖xn‖ ≤ (1 + δ1)‖x‖,

then for some N ∈ N

supn,m≥N

‖xn − xm‖ ≤ ε‖x‖.

(2) Conversely, if (X, ‖ · ‖X) satisfies (*) and ‖ · ‖ is sequentially lsc with respect to

τ , then X has the UKKH(τ) property.

Proof of Proposition 1.2: Fix ε > 0. Choose δ = δ(ε/5) as in the definition of

UKKH(τ) above. Fix a sequence (xn)∞n=1 in BX with xn −→n

x with respect to τ and

‖x‖ > 1− δ. Note that for every y ∈ X and η > 0, whenever a closed ball B(y; η) contains

xn for infinitely many n, the sequential lower semi-continuity of the norm with respect to

τ gives us that ‖x− y‖ ≤ η.

Let (xm)m∈M be a maximal family such that ‖xl−xm‖ > ε/4 whenever l 6= m in M .

Clearly, M is finite by the UKKH(τ) property. Now every xn must lie in B(xm; ε/4) for

some m ∈M . Let

Q := n ∈ N : xn ∈ B(x; ε/2) .

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Then N \Q is finite. For, if not, there exists k ∈M such that

(1) xn ∈ B(xk; ε/4) for infinitely many n ∈N ; and

(2) B(xk; ε/4) is not a subset of B(x; ε/2) .

From condition (1) and the note above, we get that ‖x − xk‖ ≤ ε/4. But condition (2)

implies that ‖x− xk‖ > ε/2 − ε/4; which is a contradiction.

Since N \Q is finite, it follows that for some N ∈ N, ‖xn − x‖ ≤ ε/2 for all n ≥ N .

Our desired conclusion follows.

For 0 < p <∞ we define the Hardy space

Hp(∆) := holomorphic f : ∆→ C| ‖f‖Hp(∆) <∞ ,

where ‖f‖Hp(∆) := sup0<r<1(1/2π∫ 2π

0|f(reiθ)|p dθ)1/p. Also we define for 1 ≤ p <∞,

Hp(T) := f ∈ Lp(T) : f(n) = 0 for all n < 0.

For 1 ≤ p <∞,Hp(∆) and Hp(T) are naturally isometrically isomorphic via the Poisson

kernel and boundary values; and we will henceforth identify the two spaces. For each

integer n, and f ∈ L1(T), f (n) is the nth Fourier coefficient of f, given by

f(n) =1

2π

∫ 2π

0

f(t)e−int dt.

2. A new convexity property of H1(∆)

We remark that the following proposition due to de Leeuw and Rudin [L-R] shows

that many natural topologies on the closed unit ball of H1(∆) are, in fact, the same. This

provides the link between the result of Warschawski [W]/Newman [Ne] and an alternative

proof given by Kellogg [Ke]. We shall use condition of (iv) below in our proof of Theorem

2.2. Weak∗-convergence on H1(∆) refers to the isometric predual C(T)/A0(∆) of H1(T).

Here C(T) is the space of all continuous functions on T, with the supremum norm, and

A0(∆) is the set of (boundary values of ) disc algebra functions with zero constant term.

This duality is established via the F. and M. Riesz theorem.

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2.1 Proposition. Suppose (fn)∞n=1 is a norm bounded sequence in H1(∆) and f ∈

H1(∆). The following conditions are equivalent.

(i) fn −→n

f weak ∗.

(ii) fn(z) −→n

f(z) for all z ∈ ∆.

(iii) fn(z) −→n

f(z) uniformly on compact subsets of ∆.

(iv) fn(k) −→n

f(k) for all k ∈ 0, 1, 2, 3, . . . .

2.2 Theorem. H1(∆) has the uniform Kadec-Klee-Huff property for its usual weak∗-

topology.

Proof: Fix ε ∈ (0, 1).Z = H2 × H2 is a Hilbert space, and so has the UKKH property

with respect to its weak topology. Choose δ to correspond to ε/2 as in Definition 1.1

above, with X := Z and τ = the weak topology on Z.

Let (fn)∞n=1 be a sequence in BH1 with fn −→n

f ∈ H1 weak∗, and suppose

‖f‖H1 > 1− δ.

‖ · ‖H1 is lower semicontinuous with respect to the weak∗ topology. So ‖f‖H1 ≤ 1.

Fix n ≥ 1. Then,

fn = gn · hn,

where gn, hn ∈ H2 and ‖gn‖2H2 = ‖hn‖2H2 = ‖fn‖H1 ≤ 1. So (gn)∞n=1 and (hn)∞n=1 are

sequences in the weakly compact set (in H2), BH2 . Thus there exist subsequences (gnk)∞k=1

and (hnk)∞k=1 of (gn)n and (hn)n respectively such that for some g, h ∈ H2

gnk −→k

g ∈ H2, weakly in H2; and

hnk −→k

h ∈ H2, weakly in H2.

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Consequently ‖g‖H2 ≤ 1 and ‖h‖H2 ≤ 1. Now, for all p ∈N ∪ 0

gnk(p) = 〈gnk , eip(·)〉 −→k〈g, eip(·)〉 = g(p)

and similarly

hnk(p) −→kh(p).

But for all k ≥ 1, for all p ∈ N ∪ 0,

fnk(p) = (gnk · hnk)∧(p) =p∑j=0

gnk(p− j)hnk(j)

−→k

p∑j=0

g(p− j)h(j) = (g · h)∧(p)

By Proposition 2.1,

fnk −→k

f weak∗ in H1 ⇐⇒ fnk(p) −→kf(p) for all p ∈ N ∪ 0.

So,f(p) = (g · h)∧(p) for all p ∈ N ∪ 0, and consequently f = g · h. Thus,

1− δ < ‖f‖H1 =∫

T

|f | dθ2π

=∫

T

|g||h| dθ2π≤ ‖g‖H2 · ‖h‖H2 .

Now, since ‖g‖H2 ≤ 1 and ‖h‖H2 ≤ 1, it follows that

(]) 1− δ < ‖g‖H2 and 1− δ < ‖h‖H2 .

Note that

infn6=m‖fn − fm‖H1 ≤ inf

k 6=l‖fnk − fnl‖H1 .

We may clearly assume, without loss of generality, that

gn −→n

g weakly in H2 and hn −→n

h weakly in H2.

Consider Z = H2 ×H2, where

‖(p, q)‖Z :=

√‖p‖2H2 + ‖q‖2H2

2for all (p, q) ∈ Z ; and

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〈(p, q), (p1, q1)〉Z :=12[〈p, p1〉+ 〈q, q1〉

]for all (p, q), (p1, q1) ∈ Z .

(Z, 〈·, ·〉Z ) is a Hilbert space. (gn, hn) −→n

(g, h) weakly in Z. Further,

‖(gn, hn)‖Z ≤ 1 for all n ,

and by (])

‖(g, h)‖Z =

√‖g‖2 + ‖h‖2

2>

√(1− δ)2 + (1− δ)2

2= 1− δ .

So, by our choice of δ, infn6=m ‖(gn, hn)− (gm, hm)‖Z ≤ ε/2 i.e.

infn6=m

√‖gn − gm‖2H2 + ‖hn − hm‖2H2

2≤ ε/2 .

Fix n 6= m, n,m ∈ N. Then

‖fn − fm‖H1 = ‖gnhn − gmhm‖H1

= ‖gn(hn − hm) + (gn − gm)hm‖H1

≤ ‖gn‖H2 · ‖hn − hm‖H2 + ‖gn − gm‖H2‖hm‖H2

≤ ‖hn − hm‖H2 + ‖gn − gm‖H2

≤√

2√‖hn − hm‖2H2 + ‖gn − gm‖2H2 .

So infn6=m ‖fn − fm‖H1 ≤ 2(ε/2) = ε.

We remark that the previous proof generalizes a proof of Kellogg [Ke] of the weak∗-

Kadec-Klee property ( called there ‘pseudo uniform convexity’ ) of H1(∆).

We also remark that the product space construction in the previous proof can be

eliminated by observing that H2(∆) is more than UKKH(weak) - it is uniformly convex.

So for all ε > 0 there exists δ ∈ (0, 1) such that whenever (fn)∞n=1 is a sequence in

BH2 with fn −→n

f ∈ H2 weakly and ‖f‖H2 > 1− δ; not only does it follow that

infn6=m‖fn − fm‖H2 ≤ ε ;

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but, more usefully, we have that for some N ∈ N,

supn,m≥N

‖fn − fm‖H2 ≤ ε.

Of course, by Proposition 1.2 above, this stronger conclusion holds in all UKKH-

spaces.

We thank Song Ying Li and Frank Beatrous for remarking that the proof of Newman

[Ne] that H1(∆) has the Kadec-Klee property for the topology of uniform convergence on

compacta extends to Hp(∆) for all 0 < p < 1. The corresponding UKKH property for

Hp(∆) can also be proven this way. This result has been generalized to certain vector-

valued Hp spaces by Dowling and Lennard [D-L1].

An alternative proof that H1(∆) has UKKH(weak∗), and a direct proof that H1(∆)

has weak∗ normal structure (see Section 5 for the definition) follow from the theorem

immediately below.

2.3 Theorem. Let (xn)∞n=1 be a sequence in H1(∆) and suppose that xn −→n

x ∈ H1

w.r.t. the weak∗ topology. Then for every y ∈ H1, we have

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lim‖xn − x‖2H1 + ‖x+ y‖2H1 ≤ lim‖xn + y‖2H1 .

Proof: We can suppose that x = 0. Also we may assume that limn→∞ ‖xn‖H1 =

lim‖xn‖H1 . Let y ∈ H1 and put yn := xn + y. Each yn is a function in H1, and so

there exist un and vn in H2 such that yn = un · vn and ‖un‖2H2 = ‖vn‖2H2 = ‖yn‖H1 .

Moreover, (yn)n is norm bounded in H1. Just as in the proof of Theorem 2.2, there ex-

ists a strictly increasing sequence (nk)∞k=1 in N such that unk −→k

u weakly in H2 and

vnk −→k

v weakly in H2; and consequently we have that unk · vnk −→k

u · v = y weak∗ in

H1.

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The idea for the end of the proof comes from Haagerup and Pisier [H-P]. We have

that, for each k,

xnk = ynk − y = unk (vnk − v) + v (unk − u) ;

from which it follows, similarly to the proof of Theorem 1.2, that

‖xnk‖2H1 + 2‖y‖2H1 ≤ 2(‖unk‖2H2‖vnk − v‖2H2 + ‖v‖2H2‖unk − u‖2H2 + ‖u‖2H2‖v‖2H2

).

Since unk −→k

u weakly in the Hilbert space (H2, 〈·, ·〉), we have that for large k, unk − u

and u are almost orthogonal. Indeed,

‖unk − u‖2H2 + ‖u‖2H2 = ‖unk‖2H2 + 2(‖u‖2H2 −Re〈unk , u〉

)≤ ‖unk‖2H2 + γk ,

where γk := 2| ‖u‖2H2 −Re〈unk , u〉 | −→k

0. Similarly,

‖vnk − v‖2H2 + ‖v‖2H2 ≤ ‖vnk‖2H2 + δk ,

where 0 < δk −→k

0. Thus,

12‖xnk‖2H1 + ‖y‖2H1 ≤ ‖unk‖2H2‖vnk − v‖2H2 + ‖v‖2H2

(‖unk‖2H2 + γk

)≤ ‖unk‖2H2

(‖vnk‖2H2 + δk

)+ ‖v‖2H2γk

= ‖ynk‖2H1 + ‖unk‖2H2δk + ‖v‖2H2γk .

Now (unk)k is uniformly bounded in H2; and so we see that

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lim‖xnk‖2H1 + ‖y‖2H1 ≤ lim‖yn‖2H1 .

Since we are assuming that lim ‖xn‖H1 = lim‖xn‖H1 , the proof is complete.

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2.4 Remark. The idea in the above theorem is very general, and can be extended to

prove that H1(∆, `1) has UKKH(weak∗). Note that H1(∆, `1) is isometrically isomorphic

to `1(H1(∆)). That `1(H1(∆)) has UKKH(weak∗) follows from Theorem 2.2 and Theorem

3.1 below, or essentially from Partington [P]. It is easy to check that these two UKKH

results are identical.

Results analogous to Theorem 2.3 hold in other settings, e.g. the trace class C1. See

Besbes [B2] for further details. Also, Besbes [B4] shows that if a separable dual Banach

space satisfies the weak∗ Opial condition then it has weak∗ normal structure. Moreover,

[B4] has shown that H1(∆) fails the weak∗ Opial condition. So this alternative avenue for

establishing that H1(∆) has weak∗ normal structure is not open.

2.5 Remark. The coefficient of 1/2 in Theorem 2.3 is the best possible. Indeed, fix α > 0.

Define, for each n ∈ N, xn(z) := zn, x(z) := 0 and y(z) := α, where z ∈ ∆. Suppose that

these functions satisfy the inequality of Theorem 2.3, with the constant 1/2 replaced by

C > 0, for all α > 0. It is straightforward to check that when we let α increase arbitrarily,

it follows that C ≤ 1/2.

3. Uniform Kadec-Klee properties inherited by Lp(X) from X .

3.1 Theorem. Let (X, ‖ · ‖X) be a Banach space and τ be a topological vector space

topology on X that is weaker than the norm topology. Suppose that (X, ‖ · ‖X) has the

uniform Kadec-Klee-Huff property with respect to τ.

Let (Ω,Σ, µ) be a measure space, and 1 ≤ p < ∞. Then Lp(µ,X) has the following

property.

For each ε > 0, there exists δ ∈ (0, 1) such that whenever (fn)∞n=1 is a sequence in

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BLp(X) and f ∈ Lp(X) with

(a) fn(ω) −→n

f(ω)with respect to τ for almost all ω ∈ Ω, and

(b) ‖f‖Lp(X) > 1− δ;

then for some N ∈ N,

supn,m≥N

‖fn − fm‖Lp(µ,X) ≤ ε.

Proof: Fix ε with 0 < ε < 1. Let δ1 correspond to ε/3 as in Corollary 1.3.1 (*). Choose

η satisfying

0 < η <ε

30and

(1 + η

1− η

)1/p

≤ 1 + δ1 ; and let δ :=ηp+1

p.

Now fix (fn)∞n=1 in BLp(X) and f ∈ Lp(X) satisfying conditions (a) and (b) of The-

orem 3.1. Let Ω1 := ω ∈ Ω : fn(ω) −→n

f(ω) with respect to τ. Define ϕn, ψ ∈ Lp(µ),

for all n ∈ N and ω ∈ Ω, by ϕn(ω) := ‖fn(ω)‖ and ψ(ω) := ‖f(ω)‖. Each ‖fn‖Lp(µ,X) =(∫Ωϕpn dµ

)1/p and ‖f‖Lp(µ,X) =(∫

Ωψp dµ

)1/p. So,

(1)∫

Ω

ϕpn dµ ≤ 1 for all n ∈N; and

(2)∫

Ω

ψp dµ > (1− δ)p > 1− ηp+1 .

Let gk := infn≥k ϕn, for all k ∈ N, and g := lim infn ϕn = limk gk. By hypothesis (a)

and the τ -lower semi-continuity of the norm, we know that ψ ≤ g almost everywhere, and

therefore∫

Ωgp dµ > 1− ηp+1. Thus, by the monotone convergence theorem, there exists

N1 ∈ N such that for all k ≥ N1,∫

Ωgpk dµ > 1− ηp+1. For every n ∈N define

Cn := ω ∈ Ω : ϕpn(ω)− gpn(ω) > η gp(ω) .

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Then it is easy to see that for all n ≥ N1,∫Cngp dµ ≤ 1

η

∫Ω

(ϕpn − gpn) dµ < ηp. Also, for

all n ≥ N1,

1 ≥∫

Ω

ϕpn dµ ≥∫Cn

ϕpn dµ+∫Ccn

gpn dµ ≥∫Cn

ϕpn dµ+∫

Ω

gpn dµ−∫Cn

gp dµ ;

and thus,∫Cnϕpn dµ < ηp + ηp+1 < 2pηp.

Define, for each n ∈ N,

f ′n := χCnf + χCcnfn .

Then for each n ≥ N1, ‖fn − f ′n‖Lp(X) ≤ ‖χCnψ‖p + ‖χCnϕn‖p < 3η. Moreover, for the

modified sequence (f ′n)n we have that f ′n(ω) −→n

f(ω) with respect to τ , for all ω ∈ Ω1;

while for each n, ‖f ′n(·)‖ ≤ (1 + η)1/pg on Ω1. Recall that Ω1 is a set of full measure.

Define V := ω ∈ Ω1 : ψ(ω) > (1− η)1/pg(ω). Fix ω ∈ V . Then

supn∈N‖f ′n(ω)‖ ≤

(1 + η

1− η

)1/p

‖f(ω)‖ ≤ (1 + δ1)‖f(ω)‖ .

Hence, by Corollary 1.3.1(*) and our choice of δ1, there exists N(ω) ∈ N such that

supn,m≥N(ω) ‖f ′n(ω)− f ′m(ω)‖ ≤ ε

3‖f(ω)‖. Next, for each N ∈ N, set

UN :=ω ∈ Ω : for all n,m ≥ N , ‖f ′n(ω)− f ′m(ω)‖ ≤ ε

3g(ω)

.

We have that V ⊆ ∪∞N=1UN and U1 ⊆ U2 ⊆ U3 ⊆ . . . . So we can choose N2 ∈ N

with∫V \UN2

gp dµ ≤ (ε/12)p. Note that for almost all ω ∈ V c it is true that ηgp(ω) ≤

gp(ω)− ψp(ω). Consequently, for all n,m ≥ N2,

‖f ′n − f ′m‖pLp(X) ≤

∫UN2

(ε3g(ω)

)pdµ(ω) +

∫(V \UN2 )∪V c

2p(1 + η)gp(ω) dµ(ω)

≤(ε

3

)p ∫Ω

gp dµ+ 4p( ε

12

)p+ 4p

∫V c

1η

(gp − ψp) dµ

<(ε

3

)p+ 4p

( ε12

)p+ 4pηp ; and so

‖f ′n − f ′m‖Lp(X) <ε

3+ε

3+

4ε30

. Finally, we see that for every n,m ≥ N3 = maxN1,N2,

‖fn − fm‖Lp(X) <2ε3

+10ε30

= ε.

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Note that in the proof above, δ depends only on ε, p ∈ [1,∞) and the space X, and

not on the particular measure space (Ω,Σ, µ) under consideration.

Also note that for 1 ≤ p <∞, whenever (fn)∞n=1 is a sequence in B`p(X) with

fn −→n

f weakly, i.e. σ(`p(X), `p′ (X∗))

it follows that

fn(k) −→n

f(k) weakly in X for all k ∈ N.

Consequently, Theorem 3.1 has the following corollary, due to Partington [P].

3.2 Corollary. Let (X, ‖ · ‖) be a Banach space with the UKKH property with respect

to the weak topology on X. Let 1 ≤ p < ∞. Then `p(X) has the UKKH property with

respect to its weak topology.

We remark that the analogous theorem to Theorem 3.1 concerning the lifting of Kadec-

Klee properties also holds. Its somewhat easier proof, based on the paper of Hoffman [Ho]

is omitted.

3.3 Remark. Theorem 3.1 is also true for p in the range 0 < p < 1. The small modifica-

tions needed to make the argument above go through in this case involve only the choice

of δ = δ(ε).

4. A Uniform Kadec-Klee property for H1 of the polydisc and ball in CN

This section extends (‘uniformizes’) results of Hoffman [Ho] and Godefroy [G1]. Fix

N ∈ N. As usual ∆ = z ∈ C : |z| < 1, ∆N is called the unit polydisc in CN , while

dmN is the usual normalized Lebesgue product measure on TN.

If ω = (eiθ1 , . . . , eiθn) ∈ ∆N and λ ∈ C, we define λω := (λeiθ1 , . . . , λeiθn). Also, let

H1(∆N ) := holomorphic f : ∆N → C∣∣ ‖f‖H1(∆N ) <∞ , where

15

‖f‖H1(∆N ) := sup0<r<1

∫TN|f(rω)| dmN (ω) , for all f ∈ H1(∆N ) .

Let σ be the topology of uniform convergence on the compact subsets of ∆N . σ is a

tvs topology on H1(∆N ) that is, moreover, locally convex.

For f : ∆N → C define the slice function fω : ∆→ C for each ω ∈ TN by

fω(λ) := f(λω), λ ∈ ∆.

It is a fact that if f ∈ H1(∆N ), then fω ∈ H1(∆) for mN -almost all ω ∈ TN ; and we

have the following ‘slice’ formula:

(*) ‖f‖H1(∆N ) =∫

TN‖fω‖H1(∆) dmN (ω).

Note that if (fn)∞n=1 is a sequence in H1(∆N ), f ∈ H1(∆N ) and

fn(z) −→n

f(z)

uniformly on the compact subsets of ∆N , then for all ω ∈ TN ,

fn,ω(λ) −→n

fω(λ)

uniformly on the compact subsets of ∆.

Since we are dealing with a sequence, it follows from Proposition 1.1 that for mN -

almost all ω ∈ Ω, fn,ω −→n

fωweak∗ in H1(∆). For each h ∈ H1(∆N ) define h ∈

L1(mN ,H1(∆)) by

h(ω) = hω, ω ∈ TN .

h 7→ h is a linear mapping that isometrically embeds H1(∆N ) inside L1(mN ,H1(∆)); by

formula (*) above. Moreover if fn −→n

f with respect to σ in H1(∆N ), then for almost all

ω ∈ TN ,

fn(ω) −→n

f(ω) weak∗ in H1(∆).

16

4.1 Theorem. (H1(∆N ), ‖ · ‖H1(∆N )) has the uniform Kadec-Klee-Huff property with

respect to the topology σ of uniform convergence on compact subsets of ∆N .

Proof: Fix ε > 0. Let (X, ‖ · ‖) be (H1(∆N ), ‖ · ‖H1(∆N )) and (Ω,Σ, µ) be (TN ,Bo(TN ),

mN ), where Bo(TN ) = the Borel subsets of TN . Let p = 1. Choose δ ∈ (0, 1) as in

Theorem 3.1, applied to this special case.

Suppose (fn)∞n=1 is a sequence in BH1(∆N ), fn −→n

f ∈ H1(∆N ) with respect to σ

and ‖f‖H1(∆N ) > 1− δ. By the remarks preceding the theorem,

fn(ω) −→n

f(ω)weak∗ in H1(∆),

for almost all ω ∈ TN ; and ‖f‖L1(mN ,H1(∆)) > 1− δ.

Now, by Theorem 2.2, (H1(∆), ‖ · ‖H1(∆)) has the UKKH(weak∗) property.

Thus, by Theorem 3.1, there exists N ∈ N such that

supn,m≥N

‖fn − fm‖L1(mN ,H1(∆)) ≤ ε.

Put differently,

supn,m≥N

‖fn − fm‖H1(∆N ) ≤ ε.

4.2 Note. As remarked in Hoffman [Ho], an analogue of formula (*) holds for any f ∈

H1(BCN ). (See Rudin [R] for more details on this setting.) Thus the analogue of Theorem

4.1, where ∆N is replaced everywhere by BCN , is also true. Here BCN is the closed unit

ball for the Euclidean norm on CN .

Godefroy [G1] considered the generalization of the domain U = BCN to that of a

strictly pseudoconvex domain U with C2-boundary, based on the work of Øvrelid [Ø] and

Henkin [He].In this case we also require that U is a closed unit ball for some norm on CN ;

17

and we again let σ be the topology of uniform convergence on the compact sets of Uo. [G1]

Theorem 22 shows that H1(U) has the Kadec-Klee property w.r.t. σ. Moreover, by [G1]

Lemma 18, a slice formula for the H1(U) norm, analogous to (*), holds in this setting.

Consequently, one has the following theorem. The proof is similar to that of Theorem 4.1.

4.3 Theorem. (H1(U), ‖ · ‖H1(U)) has the UKKH property w.r.t the topology σ, if U is

a strictly pseudoconvex domain with C2-boundary in CN , such that U is the closed unit

ball for some norm on CN .

4.4 Remark. [G1] Lemma 19 shows that BH1(U) is σ-compact, for U as in Theorem 4.3.

Similarly, one can see that BH1(U) is σ-compact for U = ∆N . Now, in both cases, σ is a

compact topology on BH1(U) that is stronger than the topology ρ of pointwise convergence

on Uo. Hence these topologies coincide on BH1(U); and so we have that H1(U) has the

UKKH(ρ ) property.

Let U be as in Remark 4.4. By a result of Ng [Ng] (also see Kaijser [Ka]), since σ is

a locally convex topology on H1(U) that is compact when restricted to BH1(U), H1(U) is

isometrically isomorphic to the dual of the Banach space V , where

V :=ϕ ∈ H1(U)∗| ϕ :

(BH1(U), σ

)→ (the scalars,usual topology) is continuous

.

Moreover, the weak∗ topology σ(H1(U), V ) coincides with the topology σ on BH1(U). Thus

we have the next result.

4.5 Theorem. Let U be a strictly pseudoconvex domain with C2-boundary in CN , such

that U is the closed unit ball for some norm on CN ; or let U = ∆N . Then (H1(U), ‖·‖H1(U))

has the weak∗ UKKH property with respect to the predual V defined above.

5. UKKH leads to normal structure and fixed point theorems

18

By results of van Dulst and Sims [D-S] for the weak and weak∗ topologies on a

Banach space , and Lennard [L2] for other topologies (including convergence in measure

on L1[0, 1]), the UKKH(τ) property implies normal structure w.r.t. τ , NS(τ). (Also see

Istratescu and Partington [I-P]). Moreover, NS(τ) implies the fixed point property w.r.t

τ , FPP (τ), by Kirk [Ki1], [Ki2].

Let (X, ‖ · ‖) be a Banach space and τ be a topological vector space topology on

X that is weaker than the norm topology. X has normal structure w.r.t. τ (NS(τ)), if for

all norm bounded, τ -compact, convex subsets C of X with two or more points, we have

that

rad(C) < diam(C) .

Here rad(C) := infy∈C supx∈C ‖x−y‖ and diam(C) := supx,y∈C ‖x−y‖. X is said to have

the fixed point property with respect to τ (FPP (τ)), if for all non-empty, τ -compact,

convex, norm bounded subsets C of X, every non-expansive mapping T : C → C has a

fixed point. T is non-expansive if

‖Tx− Ty‖ ≤ ‖x− y‖ for all x, y ∈ C.

It is known that the norm boundedness assumption for τ -compact, convex sets is redundant

(see, for example, Khamsi [Kh]).

Van Dulst and Sims [D-S] considered the following cases: (1) X is a Banach space

and τ = the weak topology on X; and (2) X is a dual Banach space and τ = the weak ∗

topology on X with respect to a given predual, such that BX is weak ∗ sequentially

compact. Lennard [L2] Theorem 4.2 (b) considered the more general case: (3) τ is such

that every τ -compact set in BX is τ -sequentially compact. This includes the case of τ =

convergence locally in measure and X = L1(µ) for (Ω,Σ, µ) a σ-finite measure space.

In all three cases, whenever X is UKKH(τ), it follows that X has NS(τ); and so

X has the FPP (τ). We remark that UKKH(τ) implies a property stronger than NS(τ)

19

: the Chebychev centre of each non-empty, τ -compact, norm bounded set in X must be

norm compact and non-empty (see [L2], [D-S] and [I-P]). We have the following theorem.

5.1 Theorem. Let U be a strictly pseudoconvex domain with C2-boundary in CN , such

that U is the closed unit ball for some norm on CN ; or let U be the polydisc ∆N . Then

H1(U) has NS(τ) and the FPP (τ) property, where τ is any of the topologies σ, ρ or

weak∗ discussed in Section 4 above.

Proof: Since all of the topologies listed in the statement of the theorem coincide and

are metrizable when restricted to BH1(U), the result follows by Theorem 4.5 and the

immediately preceding remarks.

Let us now discuss the Lebesgue-Bochner spaces. We use the notation of Theo-

rem 3.1. We remark that Theorem 3.1 will lead to the conclusion ‘Lp(µ,X) has NS(η)

and the FPP (η)’, for some tvs topology η on Lp(µ,X) that is weaker than the norm

topology, in the following case. Whenever (fn)∞n=1 is a sequence in BLp(µ,X) that con-

verges to f ∈ Lp(µ,X) with respect to η, it follows that for some subsequence (fnk)k of

(fn)n, fnk(ω) −→k

f(ω) with respect to the given weak topology τ on X, for almost all

ω ∈ Ω; and (Lp(µ,X), η) satisfies case (3) above.

A quite general class of such topologies η is described below. Fix(X, ‖ · ‖) a Banach

space and τ a tvs topology on X that is weaker than the norm topology. Fix (Ω,Σ, µ) a

finite measure space and 0 < p <∞.

Suppose that τ is a metric topology generated by a metric d. Define a metric D on

Lp(µ,X) by

D(f, g) :=∫

Ω

d(f(ω), g(ω))1 + d(f(ω), g(ω))

dµ(ω), for all f, g ∈ Lp(µ,X).

20

Then if (fn)∞n=1 is a sequence in Lp(µ,X) converging to f ∈ Lp(µ,X) in the topology η(τ)

generated by D; it follows that for some subsequence (fnk)k of (fn)n,

d(fnk(ω), f(ω)) −→k

0 for almost all ω ∈ Ω.

So fnk(ω) −→k

f(ω) with respect to τ for almost all ω ∈ Ω. Thus, applying case (3) above

to η(τ), we see that Theorem 3.1 gives us the following theorem.

5.2 Theorem. Let X, τ and µ be as described above. Whenever X has the UKKH(τ)

property and τ is metrizable, we have that Lp(µ,X) has the UKKH(η(τ)) property; and

hence Lp(µ,X) has NS(η(τ)) and the FPP (η(τ)).

For example, from above we see that Lp(µ,H1(U)) has NS(η(τ)), where τ is the

weak∗ topology on H1(U). Moreover, from [L2] we can conclude that Lp(µ,L1(ν)) has

NS(η(τ)), where τ is the topology of convergence locally in measure on the space L1(ν)

and ν is a σ-finite measure.

5.3 Note. In the above discussion, we may replace our finite measure µ by a σ-finite

measure µ. The only modification that needs to be made is in the definition of the metric

D, and this is quite straightforward.

We also remark that Smith and Turrett [S-T2] have recently shown that Lp(µ,X),

1 < p <∞, has NS(weak) if X is a reflexive Banach space with NS(weak).

We present one final normal structure result.

5.4 Theorem. Let (X, ‖ · ‖X) be a Banach space which is K-isomorphic to H1(∆). If

K <√

3/2 then X has NS(weak) and the FPP (weak).

Proof: By Theorem 1.3 it follows that, for all y, xnw−→n

x ∈ X,

12

lim‖xn − x‖2X + ‖x+ y‖2X ≤ K2lim‖xn + y‖2X .

21

Now K <√

3/2 if and only if 1/2 + 1 > K2, where 1/2, 1 and K2 are the respective

coefficients of each term in the inequality above. By Besbes [B2] Proposition 2.6, (X, ‖·‖X )

has NS(weak).

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