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The Analytic Complete Continuity Property
Journal of Mathematical Analysis and Applications 252, 967᎐979 Ž2000. doi:10.1006rjmaa.2000.7193, available online at http:rrwww.idealibrary.com on
The Analytic Complete Continuity Property Mangatiana A. Robdera Eastern Mediterranean Uni¨ ersity, Gazimaguza, Via Mersin 10, Turkey E-mail: [email protected]
and Paulette Saab Department of Mathematics, Uni¨ ersity of Missouri, Columbia, Missouri 65211 E-mail: [email protected] Submitted by William F. Ames Received July 14, 2000
We introduce the notion of the analytic complete continuity property of Banach spaces. We give different characterizations of this property. We show that this property is different from known related properties such as the complete continuity property and the analytic Radon᎐Nikodym ´ property. 䊚 2000 Academic Press
1. PRELIMINARIES The concept of the analytic Radon᎐Nikodym ´ property was first introduced by Bukhvalov and Danilevich w2x. In this note, we will weaken the convergence condition of their definition and introduce the analytic complete continuity property. We will exhibit an example which shows that the analytic complete continuity property is different from the analytic Radon᎐Nikodym ´ property. Throughout this section, ⺤ will denote the circle group, ⌺ is the -algebra of the Borel subsets of ⺤, and s 2dt the normalized Haar measure on ⺤. The dual group of ⺤ is identified with the set of integers ⺪. If X is a complex Banach space, we denote by L p Ž⺤, X . Ž1 F p - ⬁. the Banach space of Žall classes of. X-valued -Bochner p-integrable functions defined on ⺤. The space L⬁Ž⺤, X . will stand for the Banach space of Žall classes of. essentially bounded measurable X-valued functions defined on ⺤. If X s ⺓ we simply write L1 Ž⺤. and L⬁Ž⺤.. 967 0022-247Xr00 $35.00 Copyright 䊚 2000 by Academic Press All rights of reproduction in any form reserved.
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ROBDERA AND SAAB
Let X be a Banach space and let f g L1 Ž⺤, X .. Then the Fourier coefficient of f is the element of X given for each integer n by fˆŽ n . s
1
2
2
H0
f Ž t . eyi nt dt.
Ž 1.1.
If is an X-valued vector measure on ⺤ that is of bounded variation, then the Fourier coefficient of is the element of X defined for each integer n by
ˆ Ž n. s
yi nt
H⺤e
dŽ t . .
If f, g g L1 Ž⺤., the convolution of f and g is the element f ) g g L1 Ž⺤. defined by f ) gŽ t. s
1
2
f Ž t y . g Ž . d .
H 2 0
Let ⺔ s z g ⺓ : < z < - 14 be the unit disk of the complex plane. By H ⺔, X . Ž1 F p - ⬁., we denote the Hardy space of all analytic functions : ⺔ ª X satisfying 5 5 Ž p. - ⬁, where pŽ
5 5 p s sup 0Fr-1
ž
1
2
2
H0
1rp
5 Ž re it . 5 Xp dt
/
.
The space H ⬁Ž⺔, X . consists of all analytic functions : ⺔ ª X such that 5 5 Ž⬁. - ⬁, where 5 5 Ž⬁. s sup 5 Ž z . 5 X . zg⺔
For every element g H ⺔, X . Ž1 F p - ⬁. and for each 0 F r - 1, one naturally associates an element r of L p Ž⺤, X . by setting pŽ
r Ž t . s Ž re it . .
Ž 1.2.
In w2x, Bukhvalov and Danilevich introduced the notion of the analytic Radon᎐Nikodym ´ property for a Banach space. Recall that a Banach space X is said to have the analytic Radon᎐Nikodym ´ property ŽARNP. if for every g H 1 Ž⺔, X ., lim r ª 1 r exists in L1 Ž⺤, X .. In w10x Musial introduced the notion of the compact range property, also known as the complete continuity property, as follows: A Banach space X is said to have the complete continuity property ŽCCP. if every X-valued measure of bounded variation on ⺤ that is absolutely continuous with respect to Haar measure has a relatively compact range; i.e., the set
969
COMPLETE CONTINUITY PROPERTY
Ž A. : A g Ý4 is a relatively compact subset of X. It is well known that a Banach space X has the CCP if and only if every bounded linear operator T : L1 Ž⺤. ª X is completely continuous; i.e., T sends weak Cauchy sequences in L1 Ž⺤. into convergent sequences in X. Details concerning the ARNP and the CCP for Banach spaces can be found in w2, 4, 8, 10x. In the following section we shall introduce and study the notion of the analytic complete continuity property, and we shall show that it is a property weaker than both the ARNP and the CCP. For this, recall that if X is a Banach space and f g L1 Ž⺤, X ., the Pettis-norm of f is defined as follows: A f A s sup
½
1
2
H 2 0
<² f Ž t . , x*:< dt : x* g B Ž X * . .
5
A sequence f n4 in L1 Ž⺤, X . is said to be Pettis᎐Cauchy if it is Cauchy for the Pettis-norm.
2. THE ANALYTIC COMPLETE CONTINUITY PROPERTY To introduce the notion of the analytic complete continuity property, we propose requiring a weaker convergence condition in the definition of the ARNP. DEFINITION 2.1. A Banach space X is said to have the analytic complete continuity property ŽACCP. if for every g H 1 Ž⺔, X ., lim
0-r-s-1, < rys <ª0
sup
½
1 2
2
H0
² r Ž t . , x*: y ² s Ž t . , x*:< dt :
5
x* g B Ž X * . s 0. Ž 2.1. Clearly condition Ž2.1. is equivalent to lim
m, nª⬁
sup
½
1
2
H 2 0
<² r Ž t . , x*: y ² r Ž t . , x*:< dt : x* g B Ž X * . s 0, m n
5
Ž 2.2. where Ž rm . is an increasing sequence in the interval x0, 1w, converging to 1. The first examples of Banach spaces with the ACCP we have at hand are obviously those Banach spaces with the ARNP. For instance this shows that L1 Ž⺤. has the ACCP. We shall see that Banach spaces with the complete continuity property also have the ACCP.
970
ROBDERA AND SAAB
For reason of simplicity of notation, we let ⺞ 0 denote the set of non-negative integers. If X is a Banach space, we say that a bounded linear operator T : L1 Ž⺤. ª X is an analytic operator if T Ž eyi nŽ⭈. . s 0 for all n - 0; more generally we say that an X-valued measure is an analytic measure if ˆ Ž n. s 0 for all n - 0. It follows from the F. and M. Riesz theorem w11x that every analytic measure on ⺤ is absolutely continuous with respect to the Haar measure on ⺤. The following notions were introduced and studied in w13x. A Banach space X is said to have the type I-⺞ 0 complete continuity property ŽI-⺞ 0-CCP. if every analytic operator T : L1 Ž⺤. ª X sends weak Cauchy sequences in L1 Ž⺤. into norm convergent sequences. A Banach space X is said to have the type-II-⺞ 0-complete continuity property ŽII-⺞ 0-CCP. if every X-valued analytic measure on ⺤ that is of bounded variation has relatively compact range. Before we recall the characterization given in w13x of the two types of the ⺞ 0-CCP, let us fix some notations. ˜ p Ž⺤, X **. denote the set of all Žclasses of. X **For 1 F p - ⬁, let L valued functions on ⺤ such that the mapping t ¬ sup<² f Ž t ., x*:< : x* g ˜ p Ž⺤, X **. is a Banach space with respect to B Ž X *.4 is in L p Ž⺤.. Then L the norm 5 f 5 p˜ s
ž
1
2
H 2 0
1rp p
Ž sup <² f Ž t . , x*:< : x* g B Ž X *. 4 . dt
/
.
˜ p Ž⺤, X **.. The Clearly L p Ž⺤, X . can be identified with a subspace of L p ˜ Ž⺤, X **. is defined the same way Fourier coefficient of an element of L ˜ p Ž⺤, X **. consisting as in Ž1.1.. By H˜ p Ž⺤, X ., we denote the subspace of L of functions f of which fˆŽ n. g X for each n and fˆŽ n. s 0 for n - 0. It is known w2x that the formula Ž re i t . s
1 2
2
H0
f Ž s . Pr Ž t y s . ds
Ž 2.3.
realizes an isometric isomorphism between H˜ p Ž⺤, X . and H p Ž⺔, X .. Here Pr denotes the Poisson kernel w9x defined by Pr Ž e i t . s
1 y r2 1 y 2 r cos t q r 2
msq⬁
s
Ý
r < m< eimt.
msy⬁
For each m g ⺞ 0 , it is easy to see that Pˆr Ž m. s r m . Thus, according to w13x, we have: THEOREM 2.1.
Let X be a Banach space. Then the following are equi-
¨ alent:
Ža. X has the type II-⺞ 0-CCP,
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COMPLETE CONTINUITY PROPERTY
Žb. if Ž a m . m G 0 ; X and the sequence f nŽ⭈. s Ý m G 0 rnm a m e i mŽ⭈. is bounded in L1 Ž⺤, X . where Ž rn .1, 0 - rn - 1, then the sequence Ž f n . is Pettis᎐Cauchy. Our next theorem establishes that the type II-⺞ 0-CCP is actually equivalent to our notion of the analytic complete continuity property. This should not come as a surprise in light of a well known characterization of the analytic Radon᎐Nikodym ´ property. THEOREM 2.2. A Banach space X has the analytic complete continuity property if and only if it has the type II-⺞ 0-CCP. Proof. Assume that X has the ACCP, and fix a sequence Ž f nŽ⭈. s Ý m G 0 rnm a m e i mŽ⭈. . nG 1 of functions that is bounded in L1 Ž⺤, X .. For each x* g X *, consider the sequence Ž x*f nŽ⭈. s Ý m G 0 rnm x*a m e i mŽ⭈. . nG 1 of elements of L1 Ž⺤.. Then Ž x*f n . is bounded in L1 Ž⺤.. Therefore there exists a function f x* g L1 Ž⺤. such that for each n g ⺞, x*f n s Pr n) f x* w2x. This allows us to define the mapping f : ⺤ ª X ** by ² f Ž t ., x*: s f x* Ž t .. Clearly ˜1 Ž⺤, X **.. On the other hand, for each n, the Fourier coefficient of f fgL is determined by 1 2 ² fˆŽ m . , x*: s ² f Ž t . , x*: eyi m t dt 2 0
H
s
1
2
2
H0
f x* Ž t . eyi m t dt
s fˆx* Ž m . s x*a m . This show that fˆŽ m. s a m g X, and fˆŽ m. s 0 for m - 0. Thus we see that f g H˜ 1 Ž⺤, X .. Formula Ž2.3. now applies and defines an element of H 1 Ž⺔, X .. It follows that 1 2 ² r Ž t . , x*: s ² f Ž s . , x*: Pr Ž t y s . ds n n 2 0
H
s
1
2
H 2 0
f x* Ž s . Pr nŽ t y s . ds
s Pr n) f x* s x*f n . By our assumption, since the Banach space X has the ACCP, the analytic function satisfies 1 2 <² r Ž t . , x*: y ² r Ž t . , x*:< dt : x* g B Ž X * . s 0; lim sup p n 2 0 p, nª⬁
½
H
5
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ROBDERA AND SAAB
this allows us to see that the sequence Ž f nŽ⭈. s Ý m G 0 rnm a m e i mŽ⭈. . nG 1 is Pettis᎐Cauchy as desired. Conversely, assume that X has the type II-⺞ 0-CCP and let g H 1 Ž⺔, X .. Then there exists a function f g H˜ 1 Ž⺤, X . such that Ž2.3. holds. It follows that for each x* g X *, ² Ž re it . , x*: s s s
¦
1
2
H0
2
1
2
H 2 0 Ý
rm
mG0
s
¦Ý
mG0
;
f Ž s . Pr Ž t y s . ds, x* ms⬁
² f Ž s . , x*:
ž
Ý
r < m < e i mŽ tys. ds
msy⬁
$
ž² f Ž ⭈. , x*: / Ž m . e
/
imt
r m fˆŽ m . e i m t , x* .
;
That is, Ž re it . s Ý m G 0 r m fˆŽ m. e i m t. Now we set f nŽ t . s Ž rn e it ., where Ž rn .1, 0 - rn - 1. Since g H 1 Ž⺔, X ., we see that the sequence Ž f n . is bounded in L1 Ž⺤, X ., and therefore it is Pettis᎐Cauchy by our assumption. This exactly shows that the function satisfies Ž2.2. and thus finishes our proof. The type I-⺞ 0-CCP was also characterized in w13x in a similar fashion as Theorem 2.1, as follows: THEOREM 2.3. lent:
Let X be a Banach space. Then the following are equi¨ a-
Ža. X has the type I-⺞ 0-CCP. Žb. If Ž a m . m g ⺤ ; X and the sequence f nŽ⭈. s Ý m g ⺞ rnm a m e i nŽ⭈. is bounded in L⬁Ž⺤, X . where Ž rn . is an increasing sequence, 0 - rn - 1, then the sequence Ž f n . is Pettis᎐Cauchy. An argument similar to the one given in the proof of Theorem 2.2 gives us the following: THEOREM 2.4. g H ⬁Ž⺔, X . lim
m, nª⬁
sup
½
1
A Banach space has the I-⺞ 0-CCP if and only if for e¨ ery
2
H 2 0
<² r Ž t . , x*: y ² r Ž t . , x*:< dt : x* g B Ž X * . s 0, m n
5
where Ž rm . is an increasing sequence in the inter¨ al x0, 1w con¨ erging to 1.
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COMPLETE CONTINUITY PROPERTY
Our next result establishes the fact that the type I-⺞ 0-CCP and the type II-⺞ 0-CCP are indeed equivalent. First let us recall that the Ne¨ anlinna class of analytic functions valued in a Banach space X is the set given by
½
N Ž ⺔, X . s f : ⺔ ª X ; f holomorphic and sup
2
H 0Fr-1 0
logq 5 f Ž re it . 5 X
dt 2
5
-⬁ .
Notice that H p Ž⺔, X . ; N Ž⺔, X ., for every 0 - p F ⬁. As it was the case for Banach spaces with the ARNP, the key ingredient that we use to establish the equivalence of the two types of complete continuity properties relies on the following lemma of w2x. LEMMA 2.5. Any analytic function g N Ž⺔, X . is the quotient of an element in H ⬁Ž⺔, X . by an element h in H ⬁Ž⺔.. We are also going to need the following key lemma which is modeled after a similar result of Bourgain w1x. LEMMA 2.6. Let Ž a m . m G 0 ; X and let Ž f nŽ t . s Ý m G 0 rnm a m e i m t . nG 1 be bounded in L1 Ž⺤, X .. The the sequence Ž f n . nG 1 is Pettis᎐Cauchy if and only if lim n 5 H⺤ f nŽ t . wnŽ t . 2dt 5 s 0 whene¨ er Ž wn . nG 1 is an L⬁ -bounded weakly null sequence in L1 Ž⺤.. We now state our next result which establishes that the type I-⺞ 0-CCP and II-⺞ 0-CCP are equivalent properties. THEOREM 2.7. A Banach space X has the type I-⺞ 0-CCP if and only if it has the type II-⺞ 0-CCP. Proof. We only need to show the necessity. Let Ž a m . m G 0 be a bounded sequence in X, and suppose that the sequence of functions Ž f nŽ t . s Ý m G 0 rnm a m e i m t . nG 1 is bouned in L1 Ž⺤, X .. We want to show that the sequence Ž f n . nG 1 is Pettis᎐Cauchy. By Lemma 2.6, this is exactly equivalent to showing that lim n 5 H⺤ f nŽ t . wnŽ t . 2dt 5 s 0, for every L⬁ -bounded weakly null sequence Ž wn . nG 1 in L1 Ž⺤.. To this end, fix an L⬁ -bounded weakly null sequence Ž wn . nG 1 in L1 Ž⺤.. As in the proof of Theorem 2.2, there exists an element of H 1 Ž⺔, X . such that r n s f n , for each n g ⺞. By the factorization Lemma 2.5, there exist g H 1 Ž⺔. and g H ⬁Ž⺔, X . such that s . A similar argument as in the proof of Theorem 2.2 shows that the analytic function satisfies Ž re it . s Ý m G 0 r m ˆ g Ž m. e i m t , for some g g H˜⬁Ž⺤, X ., and that the sequence defined by Ž g nŽ t . s Ý m G 0 rnm ˆ g Ž m. e i m t ., where 0 F rn 1, for ⬁Ž n g ⺞, is bounded in L ⺤, X .. By our assumption the Banach space X
974
ROBDERA AND SAAB
has the type I-⺞ 0-CCP; therefore the sequence Ž g n . nG 1 is Pettis᎐Cauchy. Moreover, we have dt
dt
H⺤ f Ž t . w Ž t . 2 s H⺤g Ž t . Ž t . w Ž t . 2 . n
n
n
rn
n
Thus we are done if we prove that lim n 5 H⺤ g nŽ t .r nŽ t . wnŽ t . 2dt 5 s 0. To see this, we first notice that M s sup n 5 g n wn 5 ⬁ - ⬁. On the other hand, since the function is in the Hardy space H 1 Ž⺔., the sequence Žr . nG 1 is Cauchy in L1 Ž⺤.. Now for each pair of positive integers Ž n, p ., n we write
H⺤g
nq p
Ž t . r nq pŽ t . wnqp Ž t .
s
H⺤g
nq p
q
H⺤g
dt 2
Ž t . wnqp Ž t . r nŽ t .
nq p
dt 2
Ž t . wnqp Ž t . Ž r nq pŽ t . y r nŽ t . .
dt 2
.
Therefore, we have
H⺤g
nq p
F
Ž t . r nq pŽ t . wnqp Ž t .
H⺤g q
F
nq p
H⺤g
H⺤g
2
Ž t . wnqp Ž t . r nŽ t .
nq p
nqp
dt dt 2
Ž t . wnqp Ž t . Ž r nq pŽ t . y r nŽ t . .
Ž t . wnqp Ž t . r nŽ t .
dt 2
dt 2
q M 5r nq p y r n 5 1 .
Ž 2.4.
Fix ⑀ ) 0. Choose an integer n large enough so that for every p g ⺞, M 5r nq p y r n 5 1 F ⑀r2.
Ž 2.5.
For such an integer n, since the function r n is continuous on the circle group ⺤ and since the sequence Ž wp . p G 1 is L⬁ -bounded, we have sup 5 wnqpr n 5 ⬁ F sup 5 wp 5 ⬁ 5r n 5 ⬁ - ⬁. pG1
ž
pG1
/
This shows that the sequence Ž wnq pr n . p G 1 is L⬁ -bounded. Since the sequence Ž wnq p . p G 1 is weakly null in L1 Ž⺤., it is easily checked that the
975
COMPLETE CONTINUITY PROPERTY
sequence Ž wnq pr n . p G 1 is too. Therefore, according to Lemma 2.6, since the sequence Ž g nq p . p G 1 is Pettis᎐Cauchy, one can find p large enough so that
H⺤g
nq p
Ž t . wnqp Ž t . r nŽ t .
dt 2
- r2.
Ž 2.6.
Combining Ž2.4., Ž2.5., and Ž2.6., we have for large enough integers n and p,
H⺤g
nq p
Ž t . r nq pŽ t . wnqp Ž t .
dt 2
- .
This finishes our proof. The following theorem summarizes the characterization of Banach spaces with the ACCP as follows: THEOREM 2.8. Each of the following conditions is necessary and sufficient for a Banach space X to ha¨ e the analytic complete continuity property: 1. E¨ ery X-¨ alued analytic measure of bounded ¨ ariation has relati¨ ely compact range. 2. E¨ ery analytic operator T : L1 Ž⺤. ª X sends weak Cauchy sequences 1Ž . in L ⺤ into norm con¨ ergent sequences. 3. E¨ ery g H 1 Ž⺔, X . satisfies lim
m, nª⬁
sup
½
1
2
H 2 0
<² r Ž t . , x*: y ² r Ž t . , x*:< dt : x* g B Ž X * . s 0; m n
5
whene¨ er Ž rn . is an increasing sequence, 0 - rn - 1. 4. E¨ ery g H ⬁Ž⺔, X . satisfies lim
m, nª⬁
sup
½
1
2
H 2 0
<² r Ž t . , x*: y ² r Ž t . , x*:< dt : x* g B Ž X * . s 0; m n
5
whene¨ er Ž rn . is an increasing sequence, 0 - rn - 1. The following remarks are now immediate in light of the above theorem and some results of w13x: Remark 2.9. It is easy to see that the ACCP is a hereditary property and that a Banach space X has the ACCP if and only if each of its separable subspace does too. Remark 2.10. A Banach space with the CCP has the ACCP.
976
ROBDERA AND SAAB
Remark 2.11. A Banach lattice X has the ACCP if and only if it contains no subspace isomorphic to c 0 . 3. EXAMPLES The next two theorems provide us with more examples of Banach spaces with the ACCP. Let G denote the Ghoussoub᎐Rosenthal class, i.e., the smallest class of separable Banach spaces closed under G␦-embeddings and containing L1 Ž⺤. w5x. Here recall that if X and Y are Banach spaces a bounded linear operator T : X ª Y is said to be a G␦-embedding if T is one-to-one and T Ž F . is a G␦ set in Y, for any closed bounded subset F of X. Then we have the following: THEOREM 3.1. E¨ ery Banach space in the Ghoussoub᎐Rosenthal class G has the analytic complete continuity property. Proof. Let X be in the class G . Let T : L1 Ž⺤. ª X be an analytic operator. Then T factors through L1 Ž⺤.rH01 Ž⺤.. By w5, Corollary V.6x, every operator from L1 Ž⺤.rH01 Ž⺤. into X is completely continuous. Consequently, the operator T is also completely continuous. This proves the theorem. In their paper, Ghoussoub and Rosenthal also introduced the class of Banach spaces X such that every separable subspace of X belongs to G . Thus, in light of Remark 2.9, we can extend the result of the previous theorem to the non-separable case and therefore broaden even more the class of Banach spaces with the ACCP. THEOREM 3.2. E¨ ery Banach space with the property that each one of its separable subspaces is in the Ghoussoub᎐Rosenthal class has the analytic complete continuity property. We now mention some results concerning the quotient space L1 Ž⺤.rH01 Ž⺤.. It is a well known fact that the Banach space L1 Ž⺤.rH01 Ž⺤. fails the ARNP. Our next result is an improvement of such a fact: PROPOSITION 3.3. The Banach space L1 Ž⺤.rH01 Ž⺤. fails the ACCP. Proof. Consider the natural quotient map q: L1 Ž⺤. ª L1 Ž⺤.rH01 Ž⺤.. The operator q is clearly an analytic operator. However, it is also easily seen that the image by q of the weakly null sequence Ž e i nt . n) 0 is a non-null sequence in L1 Ž⺤.rH01 Ž⺤.. In light of Remark 2.11 we obtain the following Žsee w11x.: COROLLARY 3.4. The Banach space L1 Ž⺤.rH01 Ž⺤. does not embed in L ⺤. Ž or any Banach lattice containing no isomorphic copies of c 0 .. 1Ž
COMPLETE CONTINUITY PROPERTY
977
We finish this paper by showing an example of a Banach space that has the ACCP but fails both the ARNP and the CCP. First we recall the following example which is provided by w7, Proposition V.7x. We shall outline the main idea of the construction given in w7x. Let T : L1 Ž⺤. ª C Ž⺤. be the operator defined by T Ž e i nt . s Ž e i nt . rin,
n g ⺪ _ 04 ,
T Ž 1 . s 0. Let A 0 Ž⺤. denote the closed linear span of the set e i nt ; n ) 04 in the space C Ž⺤.. Then one observes that T Ž H01 Ž⺤.. ; A 0 Ž⺤.. It follows that the operator T induces a bounded linear operator T 噛 : L1 Ž ⺤ . rH01 Ž ⺤ . ª C Ž ⺤ . rA 0 Ž ⺤ . . In w7x, Ghoussoub et al. proved that the Davis᎐Fiegel᎐Johnson᎐Pelczynski interpolation space w3x associated to the operator T 噛 has the point of continuity property but fails the ARNP. On the other hand, Ghoussoub and Maurey established in w6x that every Banach space with the point of continuity property has the CCP. Combining all of these with Remark 2.10, one easily sees that the Davis᎐Fiegel᎐Johnson᎐Pelczynski interpolation space associated to the operator T 噛 provides an example of a Banach space with the ACCP but fails the ARNP. Finally to exhibit an example of a space that has the ACCP but fails both the ARNP and the CCP we show the following result which follows the ideas of w4x. THEOREM 3.5. Let X be a Banach space; then X has the ACCP if and only if L1 Ž⺤, X . has the ACCP. Proof. We only need to prove the necessity. Suppose X has the ACCP, and let Ž a m . m G 1 ; L1 Ž⺤, X . be such that the sequence f nŽ⭈. s Ý m G 0 rnm a m e i mŽ⭈. is bounded in L1 Ž⺤, L1 Ž⺤, X ... We need to show that the sequence Ž f n . nG 1 is Pettis᎐Cauchy. Recall that if Pr is the Poisson kernel then Pˆr Ž m. s r m and for each positive integer m Pr n r r nq 1 ) f nq1 s f n , and 5 Pr
n r r nq 1
5 L1 Ž ⺤ . s 1.
Thus, 5 f n 5 L1 Ž ⺤ , L1 Ž ⺤ , X .. F 5 f nq1 5 L1 Ž ⺤ , L1 Ž ⺤ , X .. .
978
ROBDERA AND SAAB
Therefore, lim 5 f n 5 L1 Ž ⺤ , L1 Ž ⺤ , X .. s sup 5 f n 5 L1 Ž ⺤ , L1 Ž ⺤ , X .. . n
n
By the same reasoning, if we consider f nŽ⭈. s buy fn Ž t . Ž s . s
rnm a m Ž s . e i m t ,
Ý mG0
then by Fubini’s theorem, one has dt ds
H⺤ sup H⺤5 f Ž t . Ž s . 5 2 2 n n
s
dt ds
H⺤ lim H⺤5 f Ž t . Ž s . 5 2 2 n n
dt ds
s lim
5 f Ž t . Ž s. 5 HH 2 2 ⺤ ⺤
s lim
5 f Ž t . Ž s. 5 HH 2 2 ⺤ ⺤
n
n
n
ds dt
n
s sup 5 f n 5 L1 Ž ⺤ , L1 Ž ⺤ , X .. - ⬁. n
Therefore, sup n 5 f nŽ⭈.Ž s .5 L1 Ž ⺤, X . - ⬁ for a.e. s g ⺤. Since X has the ACCP Lemma 2.6 implies that for an arbitrary L⬁-bounded weakly null sequence Ž n . nG 1 in L1 Ž⺤., lim n
dt
H⺤ f Ž t . Ž s . Ž t . 2 n
n
s0
for a.e. s g ⺤. On the other hand, dt
F 5 n 5 ⬁ 5 f n Ž ⭈ . Ž s . 5 L1 Ž ⺤ , X .
H⺤ f Ž t . Ž s . Ž t . 2 n
n
for a.e. s g ⺤. The dominated convergence theorem gives us lim n
dt
H⺤ f Ž t . Ž t . 2 n
n
L1 Ž ⺤ , X .
s 0.
Another application of Lemma 2.6 finishes the proof.
COMPLETE CONTINUITY PROPERTY
979
EXAMPLE 3.6. The function space L1 Ž⺤, Z ., where Z is the Davis᎐ Fiegel᎐Johnson᎐Pelczynski interpolation space associated to the operator T 噛 : L1 Ž ⺤ . rH01 Ž ⺤ . ª C Ž ⺤ . rA 0 Ž ⺤ . , provides us with an example of a Banach space with the ACCP but fails both the ARNP and the CCP. Remark 3.7. Recently, Randrainatoanina w12x has shown that the result of Theorem 3.5 extends to any compact metric group G other than the circle group ⺤.
REFERENCES 1. J. Bourgain, Dunford᎐Pettis operators on L1 and the Radon᎐Nikodym property, Israel J. Math. 37 Ž1980.. 2. A. V. Bukhvalov and A. A. Danilevich, Boundary properties of analytic and harmonic functions with values in Banach space, Mat. Zametki 31, 2: 203᎐214, Ž1982.. 3. J. Diestel and J. J. Uhl, Jr., ‘‘Vector Measures,’’ Math. Surveys, Vol. 15, Am. Math. Soc., Providence, 1977. 4. P. Dowling, Radon᎐Nikodym properties associated with subsets of countable discrete abelian groups, Trans. Amer. Math. Soc. 327 Ž1991., 879᎐890. 5. N. Ghossoub and H. P. Rosenthal, Martingales, G␦-embeddings and quotients of L1 , Math. Ann. 264 Ž1983., 321᎐332. 6. N. Ghossoub and B. Maurey, G␦-embeddings in Hilbert space, J. Funct. Anal. 61 Ž1985., 72᎐97. 7. N. Ghossoub, B. Maurey, and W. Schachermayer, Pluriharmonically dentable complex Banach spaces, J. Reine Angew. Math. 402 Ž1989., 76᎐127. 8. M. K. Girardi, Ph.D. thesis, University of Illinois, Urbana, IL, 1990. 9. Y. Katznelson, ‘‘An Introduction to Harmonic Analysis,’’ Dover, New York, 1976. 10. K. Musial, Martingales of Pettis integrable functions, Lecture Notes Math. 794 Ž1980., 324᎐339. 11. A. Pełczynski, Banach spaces of analytic functions, presented at the C.B.M.S. Regional ´ Conference, Kent, Ohio, 1977. 12. N. Randrianantoanina, Banach spaces with complete continuity properties, preprint. 13. M. A. Robdera and P. Saab, Complete continuity property types of Banach spaces associated with subsets of a discrete abelian group, Glasgow Math. J., to appear.
The Analytic Complete Continuity Property Mangatiana A. Robdera Eastern Mediterranean Uni¨ ersity, Gazimaguza, Via Mersin 10, Turkey E-mail: [email protected]
and Paulette Saab Department of Mathematics, Uni¨ ersity of Missouri, Columbia, Missouri 65211 E-mail: [email protected] Submitted by William F. Ames Received July 14, 2000
We introduce the notion of the analytic complete continuity property of Banach spaces. We give different characterizations of this property. We show that this property is different from known related properties such as the complete continuity property and the analytic Radon᎐Nikodym ´ property. 䊚 2000 Academic Press
1. PRELIMINARIES The concept of the analytic Radon᎐Nikodym ´ property was first introduced by Bukhvalov and Danilevich w2x. In this note, we will weaken the convergence condition of their definition and introduce the analytic complete continuity property. We will exhibit an example which shows that the analytic complete continuity property is different from the analytic Radon᎐Nikodym ´ property. Throughout this section, ⺤ will denote the circle group, ⌺ is the -algebra of the Borel subsets of ⺤, and s 2dt the normalized Haar measure on ⺤. The dual group of ⺤ is identified with the set of integers ⺪. If X is a complex Banach space, we denote by L p Ž⺤, X . Ž1 F p - ⬁. the Banach space of Žall classes of. X-valued -Bochner p-integrable functions defined on ⺤. The space L⬁Ž⺤, X . will stand for the Banach space of Žall classes of. essentially bounded measurable X-valued functions defined on ⺤. If X s ⺓ we simply write L1 Ž⺤. and L⬁Ž⺤.. 967 0022-247Xr00 $35.00 Copyright 䊚 2000 by Academic Press All rights of reproduction in any form reserved.
968
ROBDERA AND SAAB
Let X be a Banach space and let f g L1 Ž⺤, X .. Then the Fourier coefficient of f is the element of X given for each integer n by fˆŽ n . s
1
2
2
H0
f Ž t . eyi nt dt.
Ž 1.1.
If is an X-valued vector measure on ⺤ that is of bounded variation, then the Fourier coefficient of is the element of X defined for each integer n by
ˆ Ž n. s
yi nt
H⺤e
dŽ t . .
If f, g g L1 Ž⺤., the convolution of f and g is the element f ) g g L1 Ž⺤. defined by f ) gŽ t. s
1
2
f Ž t y . g Ž . d .
H 2 0
Let ⺔ s z g ⺓ : < z < - 14 be the unit disk of the complex plane. By H ⺔, X . Ž1 F p - ⬁., we denote the Hardy space of all analytic functions : ⺔ ª X satisfying 5 5 Ž p. - ⬁, where pŽ
5 5 p s sup 0Fr-1
ž
1
2
2
H0
1rp
5 Ž re it . 5 Xp dt
/
.
The space H ⬁Ž⺔, X . consists of all analytic functions : ⺔ ª X such that 5 5 Ž⬁. - ⬁, where 5 5 Ž⬁. s sup 5 Ž z . 5 X . zg⺔
For every element g H ⺔, X . Ž1 F p - ⬁. and for each 0 F r - 1, one naturally associates an element r of L p Ž⺤, X . by setting pŽ
r Ž t . s Ž re it . .
Ž 1.2.
In w2x, Bukhvalov and Danilevich introduced the notion of the analytic Radon᎐Nikodym ´ property for a Banach space. Recall that a Banach space X is said to have the analytic Radon᎐Nikodym ´ property ŽARNP. if for every g H 1 Ž⺔, X ., lim r ª 1 r exists in L1 Ž⺤, X .. In w10x Musial introduced the notion of the compact range property, also known as the complete continuity property, as follows: A Banach space X is said to have the complete continuity property ŽCCP. if every X-valued measure of bounded variation on ⺤ that is absolutely continuous with respect to Haar measure has a relatively compact range; i.e., the set
969
COMPLETE CONTINUITY PROPERTY
Ž A. : A g Ý4 is a relatively compact subset of X. It is well known that a Banach space X has the CCP if and only if every bounded linear operator T : L1 Ž⺤. ª X is completely continuous; i.e., T sends weak Cauchy sequences in L1 Ž⺤. into convergent sequences in X. Details concerning the ARNP and the CCP for Banach spaces can be found in w2, 4, 8, 10x. In the following section we shall introduce and study the notion of the analytic complete continuity property, and we shall show that it is a property weaker than both the ARNP and the CCP. For this, recall that if X is a Banach space and f g L1 Ž⺤, X ., the Pettis-norm of f is defined as follows: A f A s sup
½
1
2
H 2 0
<² f Ž t . , x*:< dt : x* g B Ž X * . .
5
A sequence f n4 in L1 Ž⺤, X . is said to be Pettis᎐Cauchy if it is Cauchy for the Pettis-norm.
2. THE ANALYTIC COMPLETE CONTINUITY PROPERTY To introduce the notion of the analytic complete continuity property, we propose requiring a weaker convergence condition in the definition of the ARNP. DEFINITION 2.1. A Banach space X is said to have the analytic complete continuity property ŽACCP. if for every g H 1 Ž⺔, X ., lim
0-r-s-1, < rys <ª0
sup
½
1 2
2
H0
² r Ž t . , x*: y ² s Ž t . , x*:< dt :
5
x* g B Ž X * . s 0. Ž 2.1. Clearly condition Ž2.1. is equivalent to lim
m, nª⬁
sup
½
1
2
H 2 0
<² r Ž t . , x*: y ² r Ž t . , x*:< dt : x* g B Ž X * . s 0, m n
5
Ž 2.2. where Ž rm . is an increasing sequence in the interval x0, 1w, converging to 1. The first examples of Banach spaces with the ACCP we have at hand are obviously those Banach spaces with the ARNP. For instance this shows that L1 Ž⺤. has the ACCP. We shall see that Banach spaces with the complete continuity property also have the ACCP.
970
ROBDERA AND SAAB
For reason of simplicity of notation, we let ⺞ 0 denote the set of non-negative integers. If X is a Banach space, we say that a bounded linear operator T : L1 Ž⺤. ª X is an analytic operator if T Ž eyi nŽ⭈. . s 0 for all n - 0; more generally we say that an X-valued measure is an analytic measure if ˆ Ž n. s 0 for all n - 0. It follows from the F. and M. Riesz theorem w11x that every analytic measure on ⺤ is absolutely continuous with respect to the Haar measure on ⺤. The following notions were introduced and studied in w13x. A Banach space X is said to have the type I-⺞ 0 complete continuity property ŽI-⺞ 0-CCP. if every analytic operator T : L1 Ž⺤. ª X sends weak Cauchy sequences in L1 Ž⺤. into norm convergent sequences. A Banach space X is said to have the type-II-⺞ 0-complete continuity property ŽII-⺞ 0-CCP. if every X-valued analytic measure on ⺤ that is of bounded variation has relatively compact range. Before we recall the characterization given in w13x of the two types of the ⺞ 0-CCP, let us fix some notations. ˜ p Ž⺤, X **. denote the set of all Žclasses of. X **For 1 F p - ⬁, let L valued functions on ⺤ such that the mapping t ¬ sup<² f Ž t ., x*:< : x* g ˜ p Ž⺤, X **. is a Banach space with respect to B Ž X *.4 is in L p Ž⺤.. Then L the norm 5 f 5 p˜ s
ž
1
2
H 2 0
1rp p
Ž sup <² f Ž t . , x*:< : x* g B Ž X *. 4 . dt
/
.
˜ p Ž⺤, X **.. The Clearly L p Ž⺤, X . can be identified with a subspace of L p ˜ Ž⺤, X **. is defined the same way Fourier coefficient of an element of L ˜ p Ž⺤, X **. consisting as in Ž1.1.. By H˜ p Ž⺤, X ., we denote the subspace of L of functions f of which fˆŽ n. g X for each n and fˆŽ n. s 0 for n - 0. It is known w2x that the formula Ž re i t . s
1 2
2
H0
f Ž s . Pr Ž t y s . ds
Ž 2.3.
realizes an isometric isomorphism between H˜ p Ž⺤, X . and H p Ž⺔, X .. Here Pr denotes the Poisson kernel w9x defined by Pr Ž e i t . s
1 y r2 1 y 2 r cos t q r 2
msq⬁
s
Ý
r < m< eimt.
msy⬁
For each m g ⺞ 0 , it is easy to see that Pˆr Ž m. s r m . Thus, according to w13x, we have: THEOREM 2.1.
Let X be a Banach space. Then the following are equi-
¨ alent:
Ža. X has the type II-⺞ 0-CCP,
971
COMPLETE CONTINUITY PROPERTY
Žb. if Ž a m . m G 0 ; X and the sequence f nŽ⭈. s Ý m G 0 rnm a m e i mŽ⭈. is bounded in L1 Ž⺤, X . where Ž rn .1, 0 - rn - 1, then the sequence Ž f n . is Pettis᎐Cauchy. Our next theorem establishes that the type II-⺞ 0-CCP is actually equivalent to our notion of the analytic complete continuity property. This should not come as a surprise in light of a well known characterization of the analytic Radon᎐Nikodym ´ property. THEOREM 2.2. A Banach space X has the analytic complete continuity property if and only if it has the type II-⺞ 0-CCP. Proof. Assume that X has the ACCP, and fix a sequence Ž f nŽ⭈. s Ý m G 0 rnm a m e i mŽ⭈. . nG 1 of functions that is bounded in L1 Ž⺤, X .. For each x* g X *, consider the sequence Ž x*f nŽ⭈. s Ý m G 0 rnm x*a m e i mŽ⭈. . nG 1 of elements of L1 Ž⺤.. Then Ž x*f n . is bounded in L1 Ž⺤.. Therefore there exists a function f x* g L1 Ž⺤. such that for each n g ⺞, x*f n s Pr n) f x* w2x. This allows us to define the mapping f : ⺤ ª X ** by ² f Ž t ., x*: s f x* Ž t .. Clearly ˜1 Ž⺤, X **.. On the other hand, for each n, the Fourier coefficient of f fgL is determined by 1 2 ² fˆŽ m . , x*: s ² f Ž t . , x*: eyi m t dt 2 0
H
s
1
2
2
H0
f x* Ž t . eyi m t dt
s fˆx* Ž m . s x*a m . This show that fˆŽ m. s a m g X, and fˆŽ m. s 0 for m - 0. Thus we see that f g H˜ 1 Ž⺤, X .. Formula Ž2.3. now applies and defines an element of H 1 Ž⺔, X .. It follows that 1 2 ² r Ž t . , x*: s ² f Ž s . , x*: Pr Ž t y s . ds n n 2 0
H
s
1
2
H 2 0
f x* Ž s . Pr nŽ t y s . ds
s Pr n) f x* s x*f n . By our assumption, since the Banach space X has the ACCP, the analytic function satisfies 1 2 <² r Ž t . , x*: y ² r Ž t . , x*:< dt : x* g B Ž X * . s 0; lim sup p n 2 0 p, nª⬁
½
H
5
972
ROBDERA AND SAAB
this allows us to see that the sequence Ž f nŽ⭈. s Ý m G 0 rnm a m e i mŽ⭈. . nG 1 is Pettis᎐Cauchy as desired. Conversely, assume that X has the type II-⺞ 0-CCP and let g H 1 Ž⺔, X .. Then there exists a function f g H˜ 1 Ž⺤, X . such that Ž2.3. holds. It follows that for each x* g X *, ² Ž re it . , x*: s s s
¦
1
2
H0
2
1
2
H 2 0 Ý
rm
mG0
s
¦Ý
mG0
;
f Ž s . Pr Ž t y s . ds, x* ms⬁
² f Ž s . , x*:
ž
Ý
r < m < e i mŽ tys. ds
msy⬁
$
ž² f Ž ⭈. , x*: / Ž m . e
/
imt
r m fˆŽ m . e i m t , x* .
;
That is, Ž re it . s Ý m G 0 r m fˆŽ m. e i m t. Now we set f nŽ t . s Ž rn e it ., where Ž rn .1, 0 - rn - 1. Since g H 1 Ž⺔, X ., we see that the sequence Ž f n . is bounded in L1 Ž⺤, X ., and therefore it is Pettis᎐Cauchy by our assumption. This exactly shows that the function satisfies Ž2.2. and thus finishes our proof. The type I-⺞ 0-CCP was also characterized in w13x in a similar fashion as Theorem 2.1, as follows: THEOREM 2.3. lent:
Let X be a Banach space. Then the following are equi¨ a-
Ža. X has the type I-⺞ 0-CCP. Žb. If Ž a m . m g ⺤ ; X and the sequence f nŽ⭈. s Ý m g ⺞ rnm a m e i nŽ⭈. is bounded in L⬁Ž⺤, X . where Ž rn . is an increasing sequence, 0 - rn - 1, then the sequence Ž f n . is Pettis᎐Cauchy. An argument similar to the one given in the proof of Theorem 2.2 gives us the following: THEOREM 2.4. g H ⬁Ž⺔, X . lim
m, nª⬁
sup
½
1
A Banach space has the I-⺞ 0-CCP if and only if for e¨ ery
2
H 2 0
<² r Ž t . , x*: y ² r Ž t . , x*:< dt : x* g B Ž X * . s 0, m n
5
where Ž rm . is an increasing sequence in the inter¨ al x0, 1w con¨ erging to 1.
973
COMPLETE CONTINUITY PROPERTY
Our next result establishes the fact that the type I-⺞ 0-CCP and the type II-⺞ 0-CCP are indeed equivalent. First let us recall that the Ne¨ anlinna class of analytic functions valued in a Banach space X is the set given by
½
N Ž ⺔, X . s f : ⺔ ª X ; f holomorphic and sup
2
H 0Fr-1 0
logq 5 f Ž re it . 5 X
dt 2
5
-⬁ .
Notice that H p Ž⺔, X . ; N Ž⺔, X ., for every 0 - p F ⬁. As it was the case for Banach spaces with the ARNP, the key ingredient that we use to establish the equivalence of the two types of complete continuity properties relies on the following lemma of w2x. LEMMA 2.5. Any analytic function g N Ž⺔, X . is the quotient of an element in H ⬁Ž⺔, X . by an element h in H ⬁Ž⺔.. We are also going to need the following key lemma which is modeled after a similar result of Bourgain w1x. LEMMA 2.6. Let Ž a m . m G 0 ; X and let Ž f nŽ t . s Ý m G 0 rnm a m e i m t . nG 1 be bounded in L1 Ž⺤, X .. The the sequence Ž f n . nG 1 is Pettis᎐Cauchy if and only if lim n 5 H⺤ f nŽ t . wnŽ t . 2dt 5 s 0 whene¨ er Ž wn . nG 1 is an L⬁ -bounded weakly null sequence in L1 Ž⺤.. We now state our next result which establishes that the type I-⺞ 0-CCP and II-⺞ 0-CCP are equivalent properties. THEOREM 2.7. A Banach space X has the type I-⺞ 0-CCP if and only if it has the type II-⺞ 0-CCP. Proof. We only need to show the necessity. Let Ž a m . m G 0 be a bounded sequence in X, and suppose that the sequence of functions Ž f nŽ t . s Ý m G 0 rnm a m e i m t . nG 1 is bouned in L1 Ž⺤, X .. We want to show that the sequence Ž f n . nG 1 is Pettis᎐Cauchy. By Lemma 2.6, this is exactly equivalent to showing that lim n 5 H⺤ f nŽ t . wnŽ t . 2dt 5 s 0, for every L⬁ -bounded weakly null sequence Ž wn . nG 1 in L1 Ž⺤.. To this end, fix an L⬁ -bounded weakly null sequence Ž wn . nG 1 in L1 Ž⺤.. As in the proof of Theorem 2.2, there exists an element of H 1 Ž⺔, X . such that r n s f n , for each n g ⺞. By the factorization Lemma 2.5, there exist g H 1 Ž⺔. and g H ⬁Ž⺔, X . such that s . A similar argument as in the proof of Theorem 2.2 shows that the analytic function satisfies Ž re it . s Ý m G 0 r m ˆ g Ž m. e i m t , for some g g H˜⬁Ž⺤, X ., and that the sequence defined by Ž g nŽ t . s Ý m G 0 rnm ˆ g Ž m. e i m t ., where 0 F rn 1, for ⬁Ž n g ⺞, is bounded in L ⺤, X .. By our assumption the Banach space X
974
ROBDERA AND SAAB
has the type I-⺞ 0-CCP; therefore the sequence Ž g n . nG 1 is Pettis᎐Cauchy. Moreover, we have dt
dt
H⺤ f Ž t . w Ž t . 2 s H⺤g Ž t . Ž t . w Ž t . 2 . n
n
n
rn
n
Thus we are done if we prove that lim n 5 H⺤ g nŽ t .r nŽ t . wnŽ t . 2dt 5 s 0. To see this, we first notice that M s sup n 5 g n wn 5 ⬁ - ⬁. On the other hand, since the function is in the Hardy space H 1 Ž⺔., the sequence Žr . nG 1 is Cauchy in L1 Ž⺤.. Now for each pair of positive integers Ž n, p ., n we write
H⺤g
nq p
Ž t . r nq pŽ t . wnqp Ž t .
s
H⺤g
nq p
q
H⺤g
dt 2
Ž t . wnqp Ž t . r nŽ t .
nq p
dt 2
Ž t . wnqp Ž t . Ž r nq pŽ t . y r nŽ t . .
dt 2
.
Therefore, we have
H⺤g
nq p
F
Ž t . r nq pŽ t . wnqp Ž t .
H⺤g q
F
nq p
H⺤g
H⺤g
2
Ž t . wnqp Ž t . r nŽ t .
nq p
nqp
dt dt 2
Ž t . wnqp Ž t . Ž r nq pŽ t . y r nŽ t . .
Ž t . wnqp Ž t . r nŽ t .
dt 2
dt 2
q M 5r nq p y r n 5 1 .
Ž 2.4.
Fix ⑀ ) 0. Choose an integer n large enough so that for every p g ⺞, M 5r nq p y r n 5 1 F ⑀r2.
Ž 2.5.
For such an integer n, since the function r n is continuous on the circle group ⺤ and since the sequence Ž wp . p G 1 is L⬁ -bounded, we have sup 5 wnqpr n 5 ⬁ F sup 5 wp 5 ⬁ 5r n 5 ⬁ - ⬁. pG1
ž
pG1
/
This shows that the sequence Ž wnq pr n . p G 1 is L⬁ -bounded. Since the sequence Ž wnq p . p G 1 is weakly null in L1 Ž⺤., it is easily checked that the
975
COMPLETE CONTINUITY PROPERTY
sequence Ž wnq pr n . p G 1 is too. Therefore, according to Lemma 2.6, since the sequence Ž g nq p . p G 1 is Pettis᎐Cauchy, one can find p large enough so that
H⺤g
nq p
Ž t . wnqp Ž t . r nŽ t .
dt 2
- r2.
Ž 2.6.
Combining Ž2.4., Ž2.5., and Ž2.6., we have for large enough integers n and p,
H⺤g
nq p
Ž t . r nq pŽ t . wnqp Ž t .
dt 2
- .
This finishes our proof. The following theorem summarizes the characterization of Banach spaces with the ACCP as follows: THEOREM 2.8. Each of the following conditions is necessary and sufficient for a Banach space X to ha¨ e the analytic complete continuity property: 1. E¨ ery X-¨ alued analytic measure of bounded ¨ ariation has relati¨ ely compact range. 2. E¨ ery analytic operator T : L1 Ž⺤. ª X sends weak Cauchy sequences 1Ž . in L ⺤ into norm con¨ ergent sequences. 3. E¨ ery g H 1 Ž⺔, X . satisfies lim
m, nª⬁
sup
½
1
2
H 2 0
<² r Ž t . , x*: y ² r Ž t . , x*:< dt : x* g B Ž X * . s 0; m n
5
whene¨ er Ž rn . is an increasing sequence, 0 - rn - 1. 4. E¨ ery g H ⬁Ž⺔, X . satisfies lim
m, nª⬁
sup
½
1
2
H 2 0
<² r Ž t . , x*: y ² r Ž t . , x*:< dt : x* g B Ž X * . s 0; m n
5
whene¨ er Ž rn . is an increasing sequence, 0 - rn - 1. The following remarks are now immediate in light of the above theorem and some results of w13x: Remark 2.9. It is easy to see that the ACCP is a hereditary property and that a Banach space X has the ACCP if and only if each of its separable subspace does too. Remark 2.10. A Banach space with the CCP has the ACCP.
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Remark 2.11. A Banach lattice X has the ACCP if and only if it contains no subspace isomorphic to c 0 . 3. EXAMPLES The next two theorems provide us with more examples of Banach spaces with the ACCP. Let G denote the Ghoussoub᎐Rosenthal class, i.e., the smallest class of separable Banach spaces closed under G␦-embeddings and containing L1 Ž⺤. w5x. Here recall that if X and Y are Banach spaces a bounded linear operator T : X ª Y is said to be a G␦-embedding if T is one-to-one and T Ž F . is a G␦ set in Y, for any closed bounded subset F of X. Then we have the following: THEOREM 3.1. E¨ ery Banach space in the Ghoussoub᎐Rosenthal class G has the analytic complete continuity property. Proof. Let X be in the class G . Let T : L1 Ž⺤. ª X be an analytic operator. Then T factors through L1 Ž⺤.rH01 Ž⺤.. By w5, Corollary V.6x, every operator from L1 Ž⺤.rH01 Ž⺤. into X is completely continuous. Consequently, the operator T is also completely continuous. This proves the theorem. In their paper, Ghoussoub and Rosenthal also introduced the class of Banach spaces X such that every separable subspace of X belongs to G . Thus, in light of Remark 2.9, we can extend the result of the previous theorem to the non-separable case and therefore broaden even more the class of Banach spaces with the ACCP. THEOREM 3.2. E¨ ery Banach space with the property that each one of its separable subspaces is in the Ghoussoub᎐Rosenthal class has the analytic complete continuity property. We now mention some results concerning the quotient space L1 Ž⺤.rH01 Ž⺤.. It is a well known fact that the Banach space L1 Ž⺤.rH01 Ž⺤. fails the ARNP. Our next result is an improvement of such a fact: PROPOSITION 3.3. The Banach space L1 Ž⺤.rH01 Ž⺤. fails the ACCP. Proof. Consider the natural quotient map q: L1 Ž⺤. ª L1 Ž⺤.rH01 Ž⺤.. The operator q is clearly an analytic operator. However, it is also easily seen that the image by q of the weakly null sequence Ž e i nt . n) 0 is a non-null sequence in L1 Ž⺤.rH01 Ž⺤.. In light of Remark 2.11 we obtain the following Žsee w11x.: COROLLARY 3.4. The Banach space L1 Ž⺤.rH01 Ž⺤. does not embed in L ⺤. Ž or any Banach lattice containing no isomorphic copies of c 0 .. 1Ž
COMPLETE CONTINUITY PROPERTY
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We finish this paper by showing an example of a Banach space that has the ACCP but fails both the ARNP and the CCP. First we recall the following example which is provided by w7, Proposition V.7x. We shall outline the main idea of the construction given in w7x. Let T : L1 Ž⺤. ª C Ž⺤. be the operator defined by T Ž e i nt . s Ž e i nt . rin,
n g ⺪ _ 04 ,
T Ž 1 . s 0. Let A 0 Ž⺤. denote the closed linear span of the set e i nt ; n ) 04 in the space C Ž⺤.. Then one observes that T Ž H01 Ž⺤.. ; A 0 Ž⺤.. It follows that the operator T induces a bounded linear operator T 噛 : L1 Ž ⺤ . rH01 Ž ⺤ . ª C Ž ⺤ . rA 0 Ž ⺤ . . In w7x, Ghoussoub et al. proved that the Davis᎐Fiegel᎐Johnson᎐Pelczynski interpolation space w3x associated to the operator T 噛 has the point of continuity property but fails the ARNP. On the other hand, Ghoussoub and Maurey established in w6x that every Banach space with the point of continuity property has the CCP. Combining all of these with Remark 2.10, one easily sees that the Davis᎐Fiegel᎐Johnson᎐Pelczynski interpolation space associated to the operator T 噛 provides an example of a Banach space with the ACCP but fails the ARNP. Finally to exhibit an example of a space that has the ACCP but fails both the ARNP and the CCP we show the following result which follows the ideas of w4x. THEOREM 3.5. Let X be a Banach space; then X has the ACCP if and only if L1 Ž⺤, X . has the ACCP. Proof. We only need to prove the necessity. Suppose X has the ACCP, and let Ž a m . m G 1 ; L1 Ž⺤, X . be such that the sequence f nŽ⭈. s Ý m G 0 rnm a m e i mŽ⭈. is bounded in L1 Ž⺤, L1 Ž⺤, X ... We need to show that the sequence Ž f n . nG 1 is Pettis᎐Cauchy. Recall that if Pr is the Poisson kernel then Pˆr Ž m. s r m and for each positive integer m Pr n r r nq 1 ) f nq1 s f n , and 5 Pr
n r r nq 1
5 L1 Ž ⺤ . s 1.
Thus, 5 f n 5 L1 Ž ⺤ , L1 Ž ⺤ , X .. F 5 f nq1 5 L1 Ž ⺤ , L1 Ž ⺤ , X .. .
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Therefore, lim 5 f n 5 L1 Ž ⺤ , L1 Ž ⺤ , X .. s sup 5 f n 5 L1 Ž ⺤ , L1 Ž ⺤ , X .. . n
n
By the same reasoning, if we consider f nŽ⭈. s buy fn Ž t . Ž s . s
rnm a m Ž s . e i m t ,
Ý mG0
then by Fubini’s theorem, one has dt ds
H⺤ sup H⺤5 f Ž t . Ž s . 5 2 2 n n
s
dt ds
H⺤ lim H⺤5 f Ž t . Ž s . 5 2 2 n n
dt ds
s lim
5 f Ž t . Ž s. 5 HH 2 2 ⺤ ⺤
s lim
5 f Ž t . Ž s. 5 HH 2 2 ⺤ ⺤
n
n
n
ds dt
n
s sup 5 f n 5 L1 Ž ⺤ , L1 Ž ⺤ , X .. - ⬁. n
Therefore, sup n 5 f nŽ⭈.Ž s .5 L1 Ž ⺤, X . - ⬁ for a.e. s g ⺤. Since X has the ACCP Lemma 2.6 implies that for an arbitrary L⬁-bounded weakly null sequence Ž n . nG 1 in L1 Ž⺤., lim n
dt
H⺤ f Ž t . Ž s . Ž t . 2 n
n
s0
for a.e. s g ⺤. On the other hand, dt
F 5 n 5 ⬁ 5 f n Ž ⭈ . Ž s . 5 L1 Ž ⺤ , X .
H⺤ f Ž t . Ž s . Ž t . 2 n
n
for a.e. s g ⺤. The dominated convergence theorem gives us lim n
dt
H⺤ f Ž t . Ž t . 2 n
n
L1 Ž ⺤ , X .
s 0.
Another application of Lemma 2.6 finishes the proof.
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EXAMPLE 3.6. The function space L1 Ž⺤, Z ., where Z is the Davis᎐ Fiegel᎐Johnson᎐Pelczynski interpolation space associated to the operator T 噛 : L1 Ž ⺤ . rH01 Ž ⺤ . ª C Ž ⺤ . rA 0 Ž ⺤ . , provides us with an example of a Banach space with the ACCP but fails both the ARNP and the CCP. Remark 3.7. Recently, Randrainatoanina w12x has shown that the result of Theorem 3.5 extends to any compact metric group G other than the circle group ⺤.
REFERENCES 1. J. Bourgain, Dunford᎐Pettis operators on L1 and the Radon᎐Nikodym property, Israel J. Math. 37 Ž1980.. 2. A. V. Bukhvalov and A. A. Danilevich, Boundary properties of analytic and harmonic functions with values in Banach space, Mat. Zametki 31, 2: 203᎐214, Ž1982.. 3. J. Diestel and J. J. Uhl, Jr., ‘‘Vector Measures,’’ Math. Surveys, Vol. 15, Am. Math. Soc., Providence, 1977. 4. P. Dowling, Radon᎐Nikodym properties associated with subsets of countable discrete abelian groups, Trans. Amer. Math. Soc. 327 Ž1991., 879᎐890. 5. N. Ghossoub and H. P. Rosenthal, Martingales, G␦-embeddings and quotients of L1 , Math. Ann. 264 Ž1983., 321᎐332. 6. N. Ghossoub and B. Maurey, G␦-embeddings in Hilbert space, J. Funct. Anal. 61 Ž1985., 72᎐97. 7. N. Ghossoub, B. Maurey, and W. Schachermayer, Pluriharmonically dentable complex Banach spaces, J. Reine Angew. Math. 402 Ž1989., 76᎐127. 8. M. K. Girardi, Ph.D. thesis, University of Illinois, Urbana, IL, 1990. 9. Y. Katznelson, ‘‘An Introduction to Harmonic Analysis,’’ Dover, New York, 1976. 10. K. Musial, Martingales of Pettis integrable functions, Lecture Notes Math. 794 Ž1980., 324᎐339. 11. A. Pełczynski, Banach spaces of analytic functions, presented at the C.B.M.S. Regional ´ Conference, Kent, Ohio, 1977. 12. N. Randrianantoanina, Banach spaces with complete continuity properties, preprint. 13. M. A. Robdera and P. Saab, Complete continuity property types of Banach spaces associated with subsets of a discrete abelian group, Glasgow Math. J., to appear.