COMPACTNESS IN L 1, DUNFORD-PETTIS OPERATORS, GEOMETRY OF BANACH SPACES Maria Girardi University of Illinois at Urbana-Champaign Proc. Amer. Math. Soc

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1 COMPCTNESS IN L 1, DUNFOD-PETTIS OPETOS, GEOMETY OF NCH SPCES Maria Girardi University of Illinois at Urbana-Champaign Proc. mer. Math. Soc. 111 (1991) 767{777 bstract. type of oscillation modeled on MO is introduced to characterize norm compactness in L1. This result is used to characterize the bounded linear operators from L1 into a anach space X that map weakly convergent sequences onto norm convergent sequences (i.e. are Dunford-Pettis). This characterization is used to study the geometry of anach spaces X with the property that all bounded linear operators from L1 into X are Dunford-Pettis. 1:Introduction The main result of this paper is a MO-style oscillation characterization of L 1 - norm compactness. From this result we obtain a characterization of the bounded linear operators from L 1 into a anach space X that map weakly convergent sequences onto norm convergent sequences (i.e. are Dunford-Pettis). With such characterizations in hand, we study the geometry of anach spaces X with the property that all bounded linear operators from L 1 into X are Dunford-Pettis. One way to insure that a relatively weakly compact set K in L 1 is relatively norm compact is to be able to nd, for each >0, a nite measurable partition of [0; 1] such that for each f in K and each in, osc f ess sup f(!),!2 ess inf!2 f(!) : This condition is too strong to characterize norm compactness Mathematics Subject Classication. 4738, 4620, This paper is part of the Ph.D. work of the author, who is grateful for the guidance of her advisor Professor J. J. Uhl, Jr. Typeset by MS-TEX

2 MI GIDI 2 ourgain showed that the ourgain property, aweakening of the above condition, also guarantees relative norm compactness of relatively weakly compact subsets in L 1. subset K of L 1 has the ourgain property if for each >0and subset of [0; 1] with positive measure there is a nite collection F of subsets of, each with positive measure, such that for each f 2 K there isanin F with osc f <. However, this condition also is too strong to characterize norm compactness. We introduce an L 1 -oscillation that is a weakening of the L 1 -oscillation used in the above conditions. The occe oscillation of an L 1 -function f on a subset is given by occe-osc f jf, fd jd () () ; observing the convention that 0=0 is 0. subset K of L 1 satises the occe criterion if for each > 0 and subset of [0; 1] with positive measure there is a nite collection F of subsets of, each with positive measure, such that for each f 2 K there isanin F with occe-osc f <. The occe criterion is a weakening of the ourgain property. The main result of this paper is that a relatively weakly compact subset of L 1 is relatively L 1 -norm compact if and only if it satises the occe criterion. Throughout this paper, X denotes an arbitrary anach space, X the dual space of X, (X) the closed unit ball of X, and S(X) the unit sphere of X. The triple (; ;) refers to the Lebesgue measure space on [0; 1], + to the sets in with positive measure, L 1 to L 1 (; ;). ecall that a subset of L 1 is relatively weakly compact if and only if it is bounded and uniformly integrable. ll notation and terminology, not otherwise explained, are as in [DU].

3 COMPCTNESS IN L 1, D-P OPETOS, GEOMETY OF NCH SPCES 3 2 : Main esult Call a subset K of L 1 a set of small occe oscillation if for each >0there isa nite positive measurable partition P of such that for each f in K X () occe-osc f < : 2P occe oscillation, the occe criterion and norm compactness in L 1 are related by Theorem 2.1, the main theorem of this paper. Theorem 2.1. For a relatively weakly compact subset K of L 1, the following statements are equivalent. (1) K is relatively norm compact. (2) K is a set of small occe oscillation. (3) K satises the occe criterion. Proof. It is well-known and easy to check that a bounded subset K of L 1 is relatively norm compact if and only if for each >0 there is a nite measurable partition of such that k f, E (f) k L 1 < for each f in K. Here, E (f) denotes the conditional expectation of f relative to the sigma eld generated by. The equivalence of (1) and (2) follows directly from this observation, the computation below, k f, E (f) k L 1 = = X 2 j f, X 2 j f, fd () j d fd () and the denition of a set of small occe oscillation. j d = X 2 () occe-osc f ; Viewed as a function from into the real numbers, occe-osc f is not increas- () ing (see Example 2.3); however, () occe-osc f is increasing (see emark 2.4). () With this in mind, we now show that (2) implies (3) in Theorem 2.1.

4 MI GIDI 4 Let the subset K of L 1 be a set of small occe oscillation. Fix >0and in +. Since K is a set of small occe oscillation, there is a positive measurable partition = f 1 ; 2 ;::: ; n g of such that for each f in K nx i=1 ( i ) (occe-osc f i ) < () : Set F = f i \ : i 2 and ( i \ ) > 0g: Fix f in K. Since the function () occe-osc f () is increasing, if occe-osc f i \ for each set i \ in F then we would have that nx nx () = ( i \ ) ( i \ ) occe-osc f i \ i=1 i=1 nx ( i ) occe-osc f i < () : i=1 This cannot be, so there is a set i \ in F such that the occe-osc f i \ <. Thus the set K satises the occe criterion. We need the following lemma which will will prove after the proof of Theorem 2.1. This lemma is the key step in the proof that (3) implies (1). Lemma 2.2. Let the relatively weakly compact subset K of L 1 satisfy the occe criterion. If > 0 and ff n g is a sequence in K, then there are disjoint sets 1 ;::: ; p in + and a subsequence fg n g of ff n g satisfying j g n, px g j n d j j d 2 : () ( j ) We now proceed with showing that (3) implies (1) in Theorem 2.1. Let the relatively weakly compact subset K of L 1 satisfy the occe criterion. Choose a sequence ff n g in K and a sequence f k g of positive real numbers decreasing to zero. It suces to nd a sequence f k g of nite measurable partitions of and a nested sequence fffng k n g k of subsequences of ff n g such that for all positiveintegers n and k k f k n, E k (f k n) k L 1 2 k ; ()

5 COMPCTNESS IN L 1, D-P OPETOS, GEOMETY OF NCH SPCES 5 where E k (f) denotes the conditional expectation of f relative to the -eld generated by k. For then the set ff n n g is relatively L 1 -norm compact since it can be uniformly approximated in the L 1 -norm within 2 k by the relatively compact set fe k (f n n )g nk [ff n ng n<k ; hence, there is a L 1 -norm convergent subsequence of ff n g. Towards (), repeated applications of Lemma 2.2 yield that for each positive integer k: (1) disjoint subsets k 1 ;::: ;k n k of each with positive measure, and (2) a subsequence ff k n g n of ff k,1 n g n (where we write ff 0 n g for ff ng ) satisfying for each positive integer n Set j f k n, Xn k k j f k n d ( k j ) k j j d k : [ n k k =n 0 and let k be the partition generated by k 0 ;k 1 ;::: ;k n k. Still observing the convention that 0=0 is 0,two appeals to the above inequality yield for each positive k j integer n and k k f k n, E k(f k n ) k L1 = k + = k + k + Xn k j fn k, j f k n, 2 k : j=0 Xn k k 0 k 0 k j f k n d (j k) k j j d j k f n k d (j k) k j j d + j f k n j d j f k n, j f k n, Xn k Xn k k j f k n d (j k) k j j d j k f n k d (j k) k j j d j 0 k f n k d ( k) k 0 j d 0

6 MI GIDI 6 Thus the proof of Theorem 2.1 is nished as soon as we verify Lemma 2.2. Proof of Lemma 2.2. Let the relatively weakly compact subset K of L 1 satisfy the occe criterion. Fix >0 and a sequence ff n g in K. The proof is an exhaustiontype argument. Let E 1 denote the collection of subsets of such that there are disjoint subsets 1 ;::: ; p of each with positive measure and a subsequence ff nk g of ff n g satisfying j f nk, px f nk d j j j d () : ( j ) Since K satises the occe criterion, there is an in + and a subsequence ff nk g of ff n g satisfying j f nk, f n k d () j d () : Thus the collection E 1 is not empty. If there is a set in E 1 with corresponding subsets 1 ;::: ; p and a subsequence fg n g of ff n g satisfying j g n, px then we are nished since (1) implies (). g j n d j j d ; (1) ( j ) Otherwise, let j 1 be the smallest positive integer for which there is a C 1 in E 1 with j1 1 (C 1). ccordingly, there is a nite sequence fcj 1 g of disjoint subsets of C 1 each with positive measure and a subsequence ffng 1 of ff n g satisfying X j fn 1 C, 1 f n 1 d j C1 (C 1 j j ) C 1 j j d (C 1 ) : (2) Note that since condition (1) was not satised, (C 1 ) <()=1. Let E 2 denote the collection of subsets of n C 1 such that there are disjoint subsets 1 ;::: ; p of each with positive measure and a subsequence ff nk g of

7 COMPCTNESS IN L 1, D-P OPETOS, GEOMETY OF NCH SPCES 7 ff 1 n g satisfying j f nk, px f j nk d j j d () : ( j ) Since K satises the occe criterion and (nc 1 ) > 0, we see that E 2 is not empty. If there is a set in E 2 with corresponding nite sequence fcj 2 g of subsets of and a subsequence fg n g of ffn 1g satisfying X C j g n, 2 g n d j nc1 (C 2 j j ) C 2 j j d ( n C 1 ) ; (3) then we are nished. For in this case, () holds since inequalities (2) and (3) insure that j g n, X k=1;2 j C k j g n d (C k j ) C k j j d (C 1 )+( n C 1 ) = : Otherwise, let j 2 be the smallest positive integer for which there is a C 2 in E 2 with j2 1 (C 2). ccordingly, there is a nite sequence fcj 2 g of disjoint subsets of C 2 each with positive measure and a subsequence ffng 2 of ffng 1 satisfying X j fn 2 C, 2 f n 2 d j C2 (C 2 j j ) C 2 j j d (C 2 ) : Note that since condition (3) was not satised, (C 2 ) <( n C 1 ). Continue in this way. If the process stops in a nite number of steps then we are nished. If the process does not stop, then diagonalize the resultant sequence f ff k n g1 n=1 g1 k=1 of sequences to obtain the sequence ff n n g1 n=1 and set C 1 = S 1 k=1 C k. Note that ( n C 1 )=0. For if ( n C 1 ) > 0 then, since K satises the occe criterion, there is a subset of n C 1 with positive measure and a subsequence fh n g of ff n n g satisfying j h n, h n d () j d < ():

8 Since for each positive integer m mx k=1 MI GIDI 8 1 m[ ( j k k=1 C k ) () and j m > 1, we can choose an integer m>1 such that ut this implies that is in E m since 1 j m, 1 () : n C 1 n and fh n g 1 n=m,1 is a subsequence of ff n n g 1 n=m,1, which in turn is a subsequence of m,1 [ ff m,1 n g 1 n=1. This contradicts the choice of j m.thus ( n C 1 )=0. Since K is relatively weakly compact, it is uniformly integrable. Thus, there k=1 C k exists >0 such that if () <then j f j d <for each f 2 K. Pick an integer m so that ( n m[ k=1 C k ) < : Since ffn ng1 n=m is a subsequence of ff n mg1, for each n=1 f in ff n ng1 n=m wehave X Cj jf, kfd (Cj k) Cj kjd 1km j = mx k=1 mx k=1 C k j f, X j C k j fd (C k ) + 2: (C k j ) C k j jd + n S m k=1 C k jfj d So the disjoint subsets fc k j : j 1 and k =1;::: ;mg along with the subsequence ff n n g1 n=m of ff ng satisfy the conditions of the lemma. This completes the proof of Theorem 2.1. We close this section with a few observations about occe oscillation.

9 COMPCTNESS IN L 1, D-P OPETOS, GEOMETY OF NCH SPCES 9 Example 2.3. Let =[0;1=4] and =[0;1]. Dene the function f from [0,1] into the real numbers by f(t) = C (t) where C =[1=8;1]. It is straightforward to verify that occe-osc f =1=2 but occe-osc f =7=32. emark 2.4. In the proof of Theorem 2.1, we used the fact that the function () occe-osc f () () j f, () fd () j d is an increasing function from into the reals numbers. To see this, let be a subset of with and in +. Let m = fd and m = () fd () : Note that j f, m j d = j f, m j d, j f, m j d + n j f, m j d + j m, m j d j m, m j d ; and j m, m j d = () j m, m j = j () () =j () () =j =j n fd, fd, fdj fd, fd, (),() () (f,m )d j : n n fdj fd j Thus, as needed, j f, m j d j f, m j d :

10 MI GIDI 10 emark 2.5. The occe oscillation has been implicitly studied. y denition, a function f in L 1 is of MO (bounded mean oscillation) provided sup I occe-osc f I < 1 where the sup is over all intervals contained in [0; 1]. 3 : pplications Fix a bounded linear operator T from L 1 into X and consider the uniformly bounded subset T ((X )) = f T (x ) : k x k 1g of L 1. The oscillation behavior of elements in T ((X )) provides information about T. Ghoussoub, Godefroy, Maurey, and Schachermayer [GGMS] showed that the operator T is strongly regular if and only if the subset T ((X )) of L 1 has the ourgain property. ecall that an operator is called Dunford-Pettis if it maps weakly convergent sequences onto norm convergent sequences. With Theorem 2.1, we obtain an analogous characterization of Dunford-Pettis operators. From the oscillation characterizations, it is easy to see that a strongly regular operator is Dunford-Pettis. Corollary 3.1. bounded linear operator T from L 1 into a anach space X is Dunford-Pettis if and only if the subset T ((X )) of L 1 satises the occe criterion. Corollary 3.1 follows directly from Theorem 2.1 and the fact below. Fact 3.2. The following statements are equivalent. (1) T is a Dunford-Pettis operator. (2) T maps weak compact sets to norm compact sets. (3) T ((L 1 )) is a relatively norm compact subset of X. (4) The adjoint of the restriction of T to L 1 from X into L 1 is a compact operator. (5) s a subset of L 1, T ((X )) is relatively norm compact.

11 COMPCTNESS IN L 1, D-P OPETOS, GEOMETY OF NCH SPCES 11 The equivalence of (2) and (3) follows from the fact that the subsets of L 1 that are relatively weakly compact are precisely those subsets that are bounded and uniformly integrable, which in turn, are precisely those subsets that can be uniformly approximated in L 1 -norm by uniformly bounded subsets. The other implications are easy to verify [cf. DU]. Ghoussoub, Godefroy, Maurey, and Schachermayer also obtained an internal geometric description of the class of anach spaces with the property that all bounded linear operators from L 1 into X are stongly regular. Using Corollary 3.1, we obtain internal geometric descriptions of the class of anach spaces with the property that all bounded linear operators from L 1 into X are Dunford-Pettis. To provide a avor of the techniques involved in these descriptions, we present an outline of one the arguments. Theorem 3.3. If each bounded subset of a anach space X is weak-norm-one dentable, then each bounded linear operator from L 1 into X is Dunford-Pettis. ecall that subset D is weak-norm-one dentable if for each >0 there is a nite subset F of D such that for each x in S(X ) there is x in F satisfying x=2 co (D n V ;x (x)) co fz 2 D : j x (z, x) jg: Sketch of Proof. Let all bounded subsets of X be weak-norm-one dentable. Fix a bounded linear operator T from L 1 into X. y Corollary 3.1, it is enough to show that the subset T ((X )) of L 1 satises the occe criterion. To this end, x >0 and in +. Let F denote the vector measure from into X given by F (E) =T( E ). Since the subset f F (E) (E) : E and E 2 + g of X is bounded, it is weak-norm-one dentable. ccordingly, there is a nite collection F of subsets of each in + such that for each x in the unit ball of X there is a

12 MI GIDI 12 set in F such that if F () () = 1 2 F (E 1 ) (E 1 ) F (E 2 ) (E 2 ) for some subsets E i of with E i 2 +, then 1 2 x F (E 1 ) (E 1 ), x F () () x F (E 2 ) (E 2 ), x F () () < : Fix x 2 (X ) and nd the associated in F. y denition, the set T ((X )) will satisfy the occe criterion provided that occe-osc (T x ). For a nite positive measurable partition of, denote f = X E2 F (E) (E) E : Set E + = [ E 2 : x F (E) (E) x F () () ; and E, = [ E 2 : x F (E) (E) < x F () () : We may assume that the L 1 -function T x is not constant a.e. on. Furthermore, we may nd an increasing sequence f n g of positive measurable partitions of satisfying _ ( n )=\ and (E + n )= () 2 =(E, n ) : It is easy to verify that since F () =() has the proper form, " x f n, x F () d () = () < () : 1 2 x F (E + n ) (E + n ), x F () () x F (E, n ) (E, n ), x F () () #

13 COMPCTNESS IN L1, D-P OPETOS, GEOMETY OF NCH SPCES 13 Since (x f n ) converges to (T x ) in L 1 -norm, occe-osc (T x ) (T x ), () (T x ) d () d : Thus T ((X )) satises the occe criterion, as needed. Using martingale techniques, one can show that the converse of Theorem 3.3 is true. For this argument and more results along these line, we refer the reader to [G]. eferences [1]. J. ourgain, On martingales in conjugate anach spaces, (unpublished). [2]. J. ourgain, Dunford-Pettis operators on L1 and the adon-nikodym property, Israel J. Math. 37 (1980), 34{47. [3]. J. ourgain, Sets with the adon-nikodym property in conjugate anach space, Studia Math. 66 (1980), 291{297. []. J. ourgain and H. P. osenthal, Martingales valued in certain subspaces of L1, Israel J. Math. 37 (1980), 54{75. [DU]. J. Diestel and J. J. Uhl, Jr., Vector Measures, Math. Surveys, no. 15, mer. Math. Soc., Providence,.I., [DU]. J. Diestel and J. J. Uhl, Jr., Progress in Vector Measures 1977{1983, Measure theory and its applications (Sherbrooke, Que., 1982), Lecture Notes in Mathematics, vol. 1033, Springer-Verlag, erlin and New York, 1983, pp. 144{192. [GGMS]. N. Ghoussoub, G. Godefroy,. Maurey, and W. Schachermayer, Some topological and geometrical structures in anach spaces, Mem. mer. Math. Soc., vol. 70, no. 378, mer. Math. Soc., Providence,.I., [G]. Maria Girardi, Dentability, Trees, and Dunford-Pettis operators on L1, (to appear in Pac. J. Math.). [GU]. Maria Girardi and J. J. Uhl, Jr., Slices, NP, Strong egularity, and Martingales, (to appear in ull. ustral. Math. Soc.). [J]. obert C. James, separable somewhat reexive anach space with nonseparable dual, ull. mer. Math. Soc. 80 (1974), 738{743. [K]. Ken Kunen and Haskell osenthal, Martingale proofs of some geometrical results in anach space theory, Pac. J. Math. 100 (1982), 153{175. [PU]. Minos Petrakis and J. J. Uhl, Jr., Dierentiation in anach spaces, Proceedings of the nalysis Conference (Singapore 1986), North-Holland, New York, []. Haskell osenthal, On the structure of non-dentable closed bounded convex sets, dvances in Math. (to appear). [U]. L.H. iddle and J. J. Uhl, Jr., Martingales and the ne line between splund spaces and spaces not containing a copy of l1, Martingale Theory in Harmonic nalysis and anach Spaces, Lecture Notes in Mathematics, vol. 339, Springer-Verlag, erlin and New York, 1981, pp. 145{156. [S]. Charles Stegall, The adon-nikodym property in conjugate anach spaces, Trans. mer. Math. Soc. 206 (1975), 213{223. [T]. M. Talagrand, Pettis integral and measure theory, Mem. mer. Math. Soc., vol. 51, no. 307, mer. Math. Soc., Providence,.I., 1984.

14 MI GIDI 14 [W]. lan Wessel, Some results on Dunford-Pettis operators, strong regularity and the adon-nikodym property, Seminaire d'nalyse Fonctionnelle (Paris VII{VI, 1985{ 1986), Publications Mathematiques de l'universite Paris VII, Paris. Department of Mathematics, University of Illinois, 1409 W. Green St., Urbana, IL 61801

MARIA GIRARDI Fact 1.1. For a bounded linear operator T from L 1 into X, the following statements are equivalent. (1) T is Dunford-Pettis. () T maps w

MARIA GIRARDI Fact 1.1. For a bounded linear operator T from L 1 into X, the following statements are equivalent. (1) T is Dunford-Pettis. () T maps w DENTABILITY, TREES, AND DUNFORD-PETTIS OPERATORS ON L 1 Maria Girardi University of Illinois at Urbana-Champaign Pacic J. Math. 148 (1991) 59{79 Abstract. If all bounded linear operators from L1 into a