Weakly equicompact sets of operators defined on Banach spaces

Weakly equicompact sets of operators defined on Banach spaces By E. Serrano, C. Piñeiro and J. M. Delgado Abstract. Let Xand Ybe Banach spaces. We say that a set M⊂W(X, Y ) (the space of all weakly compact operators from Xinto Y)isweakly equicompact if, for every bounded sequence (xn)in X, there exists a subsequence (xk(n))so that (T xk(n))is weakly uniformly convergent for T∈M. We study some properties of weakly equicompact sets and, among other results, we prove: 1) if M⊂W(X, Y ) is collectively weakly compact, then M∗is weakly equicompact iff M∗∗x∗∗ ={T∗∗x∗∗ :T∈M}is relatively compact in Yfor every x∗∗ ∈X∗∗; 2) weakly equicompactsetsareprecompactinL(X, Y ) for the topology of uniform convergenceontheweakly null sequences in X. 1. Introduction. Let us consider (real or complex) Banach spaces Xand Y. As usual K(X, Y ) (respectively, W(X, Y )) will denote the vector space of all compact (respectively, weakly compact) linear operators from Xinto Yendowed with the operator norm. In [11], the authors defined equicompact sets as those subsets Mof K(X, Y ) satisfying that, for every bounded sequence (xn)in X, there exists a subsequence (xk(n))so that (T xk(n))is uniformly convergent for T∈M. They have proved that equicompactness and collectively compactness are dual concepts in the following sense: Mis equicompact (respectively, collectively compact) iff M∗={T∗:T∈M}is collectively compact (respectively, equicompact). We recall that a set M⊂K(X, Y ) (respectively, W(X, Y )) is called collectively compact (respectively, collectively weakly compact) if  T∈M T(B X)is relatively compact (respectively, weakly compact) in Y. Families of operators arise in different applications: equations containing parameters, homotopies of operators, and so forth. In these applications, it may be very interesting to know that, given a set M⊂W(X, Y ) and a bounded sequence (xn)in Xif, for some subsequence (xk(n)), the sequence (T xk(n))is weakly uniformly convergent for T∈M. That is why we study the notion of weakly equicompact set: we say that a set M⊂W(X, Y ) is weakly equicompact if, for every bounded sequence (xn) 232 E. Serrano, C. Piñeiro and J. M. Delgado arch. math. in X, there is a subsequence (xk(n))so that (T xk(n))is weakly uniformly convergent for T∈M. Given a class Gof sequences on X, we also consider the notion of conditionally weakly G−equicompact set on L(X, Y ). Section 2 is devoted to make clear the relationship between collectively weakly compact sets and weakly equicompact sets. In this way, we prove that, if M⊂W(X, Y ) is collectively weakly compact, then M∗is weakly equicompact iff M∗∗x∗∗ ={T∗∗x∗∗ :T∈M}is relatively compact for all x∗∗ ∈X∗∗. On the other hand, it is easy to show that the implication Mweakly equicompact ⇒ M∗collectively weakly compact holds for all sets M⊂W(X, Y ) when Yhas the Schur property. We prove that this condition characterizes Schur spaces. In Section 3, we consider the locally convex topologies Tw0and Twc on L(X, Y ) of the uniform convergence on the class w0of all weak-null sequences and the class wc of all weakly Cauchy sequences in X, respectively. Following the same approach as Mayoral in [8], we prove that weakly w0−equicompact sets are precompact in L(X, Y ) for both topologies Tw0and Twc. We also prove that, if Mis a weakly w0−equicompact set, then Tw0,Twc and the weak operator topology coincide on M. Our notation is standard. If Xis a Banach space, BXwill denote its closed unit ball, X∗ will be the topological dual of Xand σ(X,X∗)denotes the weak topology on X.L(X, Y ) is the Banach space of all bounded linear operators from Xinto Y, endowed with the operator norm. For an arbitrary set I, we will write 1(I, X) (respectively, ∞(I, X)) for the Banach space of all strongly summable (respectively, bounded) X-valued functions defined on I, endowed with the norm ξ= i∈I ξ(i)(respectively, ξ=sup{ξ(i):i∈I}) for eachξ∈1(I, X) (respectively, ξ∈∞(I, X)). As usual, wewill write1(I) (respectively, ∞(I)) instead of 1(I, R)(respectively, ∞(I, R)). 2. Weakly equicompact subsets of W(X,Y). We need the following characterization of relatively compact subsets of X∗. We include a sketch of the proof because we have not found it in the literature. Lemma 2.1. A bounded subset Aof X∗is relatively compact iff every bounded sequence (xn)in Xhas a subsequence (xk(n))so that (xk(n),a)is uniformly convergent for a∈A. Proof. Given a bounded subset Aof X∗, consider the operator S:X−→ ∞(A) defined by Sx =(x,a)a∈Afor all x∈X. Notice that Ahas the desired property iff Sis a compact operator. But, in this case, the restriction of S∗to 1(A),R:ξ∈1(A) −→  a∈A ξ(a)a ∈X∗, is compact too. Finally, note that A=R({ξb∈1(A) :b∈A})⊂R(B1(A))=aco(A) whereξbisthecharacteristicfunctionoftheset{b}andaco(A) istheclosureoftheabsolutely convex hull of A. Vol. 86, 2006 Weakly equicompact sets of operators defined on Banach spaces 233 The next proposition provides a characterization of weakly equicompact sets. Proposition 2.2. LetMbeasubsetofW(X, Y ). Thefollowingstatementsareequivalent: (a) Mis weakly equicompact. (b) Msatisfies the following properties: (i) M∗y∗is relatively compact in X∗for every y∗∈Y∗. (ii) For every bounded sequence (xn)in X, there is a subsequence (xk(n))so that (T xk(n))is weakly convergent for every T∈M. Proof. (b) ⇒(a) is a straightforward argument. For the other implication, we only have to show that (i) holds for every weakly equicompact set M⊂W(X, Y ). To see this, let (xn)be a bounded sequence in X. By hypothesis, there exists a subsequence (xk(n))such that (T xk(n))is uniformly weakly convergent for T∈M. So, for every y∗∈Y∗,(Tx k(n),y∗)=(xk(n),T∗y∗)converges uniformly for T∈M. According to Lemma 2.1, M∗y∗must be relatively compact.  Recall that a set M⊂L(X, Y ) is uniformly completely continuous (or uniformly sequentially weak-norm continuous) if, for each weakly-null sequence (xn)in X, the sequences (T xn)converge in norm to 0 uniformly for T∈M. A standard argument shows that relatively compact sets in X∗are uniformly completely continuous, but the converse is true only when Xdoes not contain a copy of 1[5, Theorem 2]. Corollary 2.3. If Xdoes not contain acopy of1, the followingstatementsareequivalent for a set M⊂W(X, Y ): (a) Mis weakly equicompact. (b) M∗y∗is uniformly completely continuous in X∗for every y∗∈Y∗. (c) (T xn)w −→ 0uniformly for T∈Mwhenever (xn)w −→ 0in X. Proof. Only (c) ⇒(a) needs to be proved. Let (xn)be a bounded sequence in X. Extracting a subsequence by the Rosenthal–Dor Theorem if necessary (see [10] and [4]), we can suppose that (xn)is weakly Cauchy. If (Txn)is not weakly Cauchy uniformly for T∈M, then there exist ε>0, y∗∈Y∗, a sequence (Tn)in Mand strictly increasing maps h, k :N−→ Nsuch that |Tnxh(n) −Tnxk(n),y∗| >ε for all n∈N.(1) Nevertheless, (xh(n) −xk(n))w −→ 0, which contradicts (1).  Remark 2.4. In the above result, notice that (c) does not imply (a) in general. Indeed, it is widely known that a Banach space Xhas the Schur property if and only if bounded subsets of X∗are uniformly completely continuous. From this, it is clear that the set M= B∞⊗B1⊂W(1, 1)satisfies condition (c). Nevertheless, taking γ=(1,1,1,...), the set M∗γ=B∞is not relatively compact. The next proposition shows that the hypothesis about Xcannot be omitted in Corollary 2.3. 234 E. Serrano, C. Piñeiro and J. M. Delgado arch. math. Proposition 2.5. Let Xbe a Banach space. The following statements are equivalent: (a) Xcontains no copy of 1. (b) For some Banach space Y={0}and every M⊂K(X, Y ),if(T xn)w −→ 0uniformly for T∈Mwhenever (xn)w −→ 0in X, then Mis weakly equicompact. Proof. Inorder to see that Xdoes not have a copy of 1, it suffices to show that every uniformly completely continuous set A⊂X∗is relatively compact [5, Theorem 2]. Choose y0∈Yand y∗ 0∈Y∗such that y0,y∗ 0=1. It is easy to show that the set M=A⊗y0is uniformly completely continuous and, by hypothesis, weakly equicompact. In particular, M∗y∗ 0=y0,y∗ 0A=Ais relatively compact.  The study of sets with property 2.3−(c) is interesting even in the case that X← 1 and, actually, it suggests a way to generalize the notion of weak equicompactness. Given a class Gof bounded sequences in X, we say that M⊂L(X, Y ) is conditionally weakly G−equicompact (respectively weakly G−equicompact) if every sequence (xn)∈Ghas a subsequence (xk(n))∈Gso that (T xk(n))is weakly Cauchy (respectively weakly convergent) uniformly for T∈M. We will denote by w0the class of all weakly null sequences in Xand by wc the class of all weakly Cauchy sequences. Using standard arguments, it is easy to prove the next lemma: Lemma 2.6. If M⊂L(X, Y ), the following statements are equivalent: (a) If (xn)∈w0, then (T xn)w −→ 0uniformly for T∈M. (b) If (xn)∈wc, then (T xn)is weakly Cauchy uniformly for T∈M. (c) Mis weakly w0−equicompact. (d) Mis conditionally weakly wc−equicompact. For a fixed bounded subset Mof W(X, Y ), put H= T∈M T(B X)and consider now the operator U:1(M, X) −→ Ydefined by Uξ = T∈M T(ξ(T))for every ξ∈1(M, X).It is easy to show that HU(B1(M,X))co(H ) [11, Proposition 2.2] from which we can conclude the following result: Proposition 2.7. A set M⊂W(X, Y ) is collectively weakly compact iff Uis weakly compact. Now, we are ready to present one of our main results: Theorem 2.8. Let M⊂W(X, Y ) be collectively weakly compact. The set M∗is weakly equicompact iff M∗∗x∗∗ is relatively compact for all x∗∗ ∈X∗∗. Vol. 86, 2006 Weakly equicompact sets of operators defined on Banach spaces 235 Proof. First of all, notice that the adjoint of the operator Uis U∗:Y∗−→ ∞(M, X∗), definedby U∗y∗(T ) =T∗y∗foreachy∗∈Y∗and T∈M. Let (y∗ n)bea bounded sequence in Y∗. Since U∗is weakly compact, there exist a subsequence (y∗ k(n))and g∈∞(M, X∗) such that (U∗y∗ k(n))w −→ g. For every x∗∗ ∈X∗∗ and T∈M, the map FT,x∗∗ :f∈ ∞(M, X∗)−→  f(T),x∗∗∈R(or C) is a linear functional in ∞(M, X∗)∗. Then, g(T ), x∗∗=g, FT,x∗∗ =lim nU∗y∗ k(n)(T ), x∗∗=lim nT∗y∗ k(n),x∗∗ so (T ∗y∗ k(n))w −→ g(T ) for all T∈M. Now, a call to Proposition 2.2 concludes the proof.  Proposition 2.9. Let Ybe a Banach space. The following statements are equivalent: (a) For every Banach space Xand every M⊂W(X, Y ),M∗is collectively weakly compact whenever Mis weakly equicompact. (b) For some nonreflexive Banach space Xand every M⊂W(X, Y ),M∗is collectively weakly compact whenever Mis weakly equicompact. (c) Y has the Schur property. Proof. Toshow(b)⇒(c), suppose Xis a nonreflexive Banach space so that for every M⊂W(X, Y ),M∗is collectively weakly compact whenever Mis weakly equicompact. For the sake of a contradiction, assume that there exists a sequence (yn)w −→ 0 with yn=1 for every n∈N. Since Xis nonreflexive, we can choose a sequence (x∗ n) with x∗ n=1 for every n∈Nand such that the set {x∗ n:n∈N}is not relatively weakly compact. PutM={x∗ n⊗yn:n∈N}. To see that Misweakly equicompact, takeabounded sequence(xn)inX; using astandardargument ofdiagonalization, there exists asubsequence (xk(n))so that (xk(n),x∗ p)is convergent, for every p∈N. Finally, given y∗∈Y∗, the set M∗y∗={yn,y∗x∗ n:n∈N}isrelativelycompactsincelim nyn,y∗x∗ n=0. According to Proposition 2.2, Mis weakly equicompact; nevertheless, it is clear that M∗={yn⊗x∗ n: n∈N}is not collectively weakly compact which contradicts (b). If Yhas the Schur property and M⊂W(X, Y ) is weakly equicompact, it is not difficult to prove that Mis equicompact; that is to say, for each bounded sequence (xn)in X, there exists a subsequence (xk(n))so that (T xk(n))is uniformly norm convergent for T∈M.By [11, Lemma 2.4], it follows that M∗is collectively compact and, in particular, collectively weakly compact; this proves (c) ⇒(a). It is easy to prove that every relatively compact set M⊂W(X, Y ) is collectively weakly compact and weakly equicompact. Unfortunately, there exist weakly compact subsets of K(X, Y ) which are not weakly equicompact. Example 2.10. Let X=2and Y=1. For each β=(βn)∈2we denote by Tβ the operator defined by (αn)∈2−→ (αnβn)∈1. 236 E. Serrano, C. Piñeiro and J. M. Delgado arch. math. Then Tβ∈K(2, 1)and the set M={Tβ:β∈B2}is weakly compact in K(2, 1). Nevertheless, Mis not weakly equicompact. In fact, (Tβ)∗:∞−→ 2is defined by (Tβ)∗(γn)=(γnβn), for all γ=(γn)∈∞. So, taking γ=(1,1,1,...), the set M∗γ=B2is not relatively compact. Therefore, Mis not weakly equicompact. The following proposition shows that the implication Mrelatively weakly compact ⇒ Mweakly equicompact is true for all M⊂W(X, Y ) iff X∗has the Schur property. Proposition 2.11. Let Xbe a Banach space. The following statements are equivalent: (a) For every Banach space Yand every M⊂W(X, Y ),Mis weakly equicompact whenever it is relatively weakly compact. (b) ForsomeBanachspaceY={0}andeveryM⊂W(X, Y ),Misweakly equicompact whenever it is relatively weakly compact. (c) X∗has the Schur property. Proof. (b) ⇒(c) IfA⊂X∗isrelativelyweaklycompact, itis obvious thatM=A⊗y0 (y0∈Y\{0}) is relatively weakly compact in W(X, Y ). By hypothesis, Mmust be weakly equicompact, so a call to Lemma 2.1 tells us that Ais relatively compact. (c) ⇒(a) Recall that a dual Banach space X∗has the Schur property iff Xhas the Dunford-Pettis property and does not contain a copy of 1. So, to prove that a set M⊂W(X, Y ) is weakly equicompact, by Corollary 2.3, it suffices to prove that Mis weakly w0−equicompact. Then, suppose M⊂W(X, Y ) is relatively weakly compact and (xn)w −→ 0inX. For a contradiction, assume (T xn)w −→ 0 but the convergence is not uniform for T∈M. Thus there exist y∗∈Y∗,ε>0, strictly increasing sequences (pn) and (qn)of natural numbers, and a sequence (Tn)in Mso that |xpn−xqn,T∗ ny∗|=|Tn(xpn−xqn), y∗| >ε, for all n∈N.(2) Nevertheless, (xpn−xqn)w −→ 0 and (T ∗ ny∗)admits a weakly convergent subsequence (notice that M∗is relatively weakly compact). Since Xhas the Dunford-Pettis property, (2) is a contradiction.  3. Compactness of weakly equicompact sets. To start, we recall the definition of the topology of uniform convergence on a class Gof bounded sets. Let Xand Ybe Hausdorff locally convex spaces. Let Gbe a class of bounded subsets of Xwith the properties: 1) GisacoverofX;2)Gis closed for finite unions. For each A∈G, we define U(A,W) ={T∈L(X, Y ) :T (A) ⊂W}where Wruns over a 0-neighborhood basis Bof Y. It is well-known that the set {U(A,W) :A∈G,W∈B}is a 0-neighborhood basis of a locally convex topology TGon L(X, Y ). We will write LG(X, Y ) for this locally convex space (see [2, TVS III.13] and [7, Chapter 8]). For simplicity, we also denote the class Vol. 86, 2006 Weakly equicompact sets of operators defined on Banach spaces 237 formed by the ranges of all weakly null sequences (respectively weakly Cauchy sequences) by w0(respectively wc). We will use the following version of Ascoli’s classical theorem. Theorem 3.1 [1, Theorem 2 on T.G. X.17].Consider two uniform spaces Xand Y,a cover Gof Xformed by precompact subsets and a set Hof functions from Xto Ysuch that the restriction of each h∈Hto each A∈Gis uniformly continuous. Then His precompact in the topology (uniformity) of uniform convergence on members of Gif and only if it satisfies the following two conditions: (I) His pointwise precompact, i.e., H(x) ={h(x) :h∈H}is precompact in Y, for each x∈X. (II) For each A∈G, the set HAof restrictions to Aof the functions h∈His uniformly equicontinuous. For the sake of completeness we consider the definition of a precompact set in the setting of uniform spaces. A Hausdorff uniform space Eis said to be precompact if its completion is compact. Equivalently, Eis totally bounded, that is, for every vicinity Uof Ethere is a covering of Eby finitely many sets which are small of order U. Within the framework of topological vector spaces this reads as follows: for a subset Aof a topological vector space Ethe following are equivalent: (a) Ais precompact, (b) Ais relatively compact in the completion of Eand (c) for each 0-neighborhood Vin Ethere exists a finite subset F so that A⊂F+V[6, Theorem 1 on 3.5]. Theorem 3.2. Let M⊂L(X, Y ) be a bounded set. The following statements are equivalent: (a) Mis precompact in Lwc(X, Y ). (b) Mis precompact in Lw0(X, Y ). (c) Mis weakly w0−equicompact. Proof. We shall apply Ascoli’s Theorem to the uniform spaces (X, σ (X, X∗)) and (Y, σ(Y, Y ∗)), and the families w0and wc. Notice that w0and wc are covers of Xformed by weakly precompact sets. (b) ⇒(c) According to Ascoli’s Theorem, the set MAof restrictions to Aof the functions T∈Mis uniformly equicontinuous (T:(A, σ(X, X∗)|A)→(Y, σ (Y, Y ∗))), for each weakly null sequence A. This implies that (T xn)w −→ 0 uniformly for T∈Mwhenever (xn)w −→ 0. In fact, given (xn)w −→ 0, ε>0 and y∗∈Y∗, there exists a weakneighborhood Vof 0 so that xn−xm∈V⇒ |  Tx n−Tx m,y∗| <ε, for all T∈M. (c) ⇒(a) Since every weakly bounded set is weakly precompact, we only have to prove that, for each weakly Cauchy sequence A, the set MAis uniformly equicontinuous. We take a 0-neighborhoods basis Bof (X, σ(X, X∗)) formed by absolutely convex sets. We 238 E. Serrano, C. Piñeiro and J. M. Delgado arch. math. have to prove that, given ε>0, a weakly Cauchy sequence (xn)and y∗∈Y∗, there exist U∈Bso that xn−xm∈U⇒ Tx n−Tx m∈W(0;y∗,ε), for all T∈M. Firstof all, Lemma2.6 tellsus thatthere exist n0∈Nsuchthat Tx n−Tx m∈W(0;y∗,ε), for all n, m n0and T∈M. Case1: (xn)w −→ x. Without loss of generality, we can assume that xn= xm, for n= m, and xn= xfor all n∈N. If we choose V1∈Bsatisfying V1∩({xi−xj:i= j i, j n0}∪{xi−x:in0})=∅, then Tx n−Tx m∈W(0;y∗,ε)for all n,m>n 0and for T∈M, but notice that xn−xm∈ V1whenever n, m n0and n= m. Now, choose n∗ 0>n 0 so that xn−x∈1 2V1for all n>n ∗ 0. It is easy to check that xn−xm∈ 1 2V1whenever n>n ∗ 0 and mn0. Finally, if we take V2∈Bsuch that V2∩{xn−xm:mn0<nn∗ 0}=∅, then U=(1 2V1)∩V2is the desired neighborhood. Case2: (xn)is not weakly convergent. Again, we suppose xn= xm, for n= mand choose V1∈Bsuch that V1∩{xi−xj:i= ji,jn0}=∅. We are going to show that for each mn0, there exist Um∈Band n∗ m>n 0such that xn−xm∈ 1 2Umfor all n>n ∗ m. In fact, for every V∈Bthere exist nv∈N(nv>n 0) so that xn−xn∈1 2V whenever n, n>n v. If for each V∈Bthere exist n>n vsuch that xn−xm∈1 2V, then xn∈xm+Vfor all n>n vand this yields the contradiction (xn)w −→ xm. Finally, put n∗ 0=max{n∗ 1,...,n ∗ n0}andchooseV2∈BsuchthatV2∩{xn−xm:mn0<nn∗ 0}=∅. IfU=V1∩n0 m=11 2Um∩V2,wehaveTx n−Tx m∈W(0;y∗,ε),forallT∈Mwhenever xn−xm∈U. In general, the weak operator topology (in short, WOT) Tw0and Twc are different. However, we have obtained the following results. Proposition 3.3. If M⊂L(X, Y ) is weakly w0−equicompact, then the weak operator topology, Tw0and Twc coincide on M. Proof. Obviously, Twc is stronger than the weak operator topology on L(X, Y ). So, we only have to prove that every Twc−neighborhood of 0 in Mis a WOT–neighborhood of 0. If A={xn:n∈N}runs over the family wc in Xand W={y∈Y:|y,y∗ i|  1:i=1,...,p}over a basis of weak-neighborhoods of 0 in Y, then the sets U(A,W) ∩ M={T∈M:Tx n∈Wfor all n∈N}form a basis of Twc−neighborhoods of 0 in M.GivenU(A,W) ∩M, since (xn)is weakly Cauchy, there exists n0∈Nso that |Tx n−Tx m,y∗ i| 1/2 for all n, m n0,i∈{1,...,p}and T∈M.IfV= V(0;x1,...,x n0,y∗ 1,...,y∗ p,1 2),wehave |Tx n,y∗ i| |Tx n−Tx n0,y∗ i|+|Tx n0,y∗ i| <1 2+1 2=1 for all n>n 0,i∈{1,...,p}and T∈V∩M. So, we have obtained that V∩MU(A,W)∩M. Vol. 86, 2006 Weakly equicompact sets of operators defined on Banach spaces 239 We recall that a set K⊂Xis called limited if lim nx,x∗ n=0 uniformly for x∈K whenever (x∗ n)is a weak*-null sequence in X∗. Proposition 3.4. The convergent sequences for Twc,Tw0and the weak operator topology coincide iff Xhas the following property: “For all (xn)w −→ 0in X, the set {xn:n∈N} is limited.” Proof. (a)Necessity. Suppose that there exists a weakly null sequence (xn)in Xso that the set {xn:n∈N}is not limited. Then, we can choose a sequence (x∗ n)w∗ −→ 0inX∗ such that lim nxk,x∗ n=0 but the convergence is not uniform for k∈N. Choose y0∈Yso that y0=1; then the sequence (x∗ n⊗y0)is WOT-null but is not Tw0−convergent. (b) Sufficiency. A standard argument shows that {xn:n∈N}is limited whenever (xn)is a weakly Cauchy sequence in X. Let (Tn)WOT →0. In particular, (Tnx,y∗)is convergent to 0, for all x∈Xand y∗∈Y∗. Equivalently, the sequence (T ∗ ny∗)is weak*-null, for all y∗∈Y∗. Nevertheless, assume that (Tn)is not Twc−null. Then we can obtain a sequence (xn)weakly Cauchy so that (Tnxm)nis not weakly uniformly convergent for m∈N. So, there exist y∗∈Y∗,ε>0, a strictly increasing sequence (pn)in Nand a sequence (mn) of natural numbers satisfying |xmn,T∗ pn(y∗)|=|Tpn(xmn), y∗| >ε, for all n∈N.(3) As the set {xn:n∈N}is limited and (T ∗ pn(y∗)) w∗ −→ 0, (3) is a contradiction.  Notice that the above property implies the Dunford-Pettis property. Nevertheless, c0is an example of a Banach space with the Dunford-Pettis property that does not satisfy the hypothesis of Proposition 3.4. In fact, the unit vector basis (en)of c0is weakly null but the set {en:n∈N}is not limited. Finally, we mention that the weak topology, Tw0and Twc are different in general. For an example, take a Banach space Xwithout the Dunford-Pettis property. Then we can choose weakly null sequences (xn)in Xand (x∗ n)in X∗satisfying lim n→∞xn,x∗ n=0. Thus the sequence (x∗ n⊗y0)is weakly null and, nevertheless, it is not Tw0−convergent to 0. More precisely, we have obtained the following relationship between both classes of convergent sequences (the proof is similar to the proof of the above proposition). Proposition 3.5. The weakly convergent sequences in L(X, Y ) are Twc−convergent iff Xhas the Dunford-Pettis property. Acknowledgement.Theauthors thank the referee for his interesting suggestions. References [1] N. Bourbaki, Topologie Générale, Tome II. Paris 1974. [2] N. Bourbaki, Topological Vector Spaces. Chap. 1–5, Berlin 2003.