Some properties and applications of weakly equicompact sets

Some properties and applications of weakly equicompact sets E. Serrano, C. Pi˜ neiro, and J. M. Delgado Abstract. Let Xand Ybe Banach spaces. A set M⊂W(X, Y ) (the space of all weakly compact operators from Xinto Y)isweakly equicompact if, for every bounded sequence (xn)inX, there exists a subsequence (xk(n)) so that (Tx k(n)) is uniformly weakly convergent for T∈M. In this paper, the notion of weakly equicompact set is used to obtain characterizations of spaces X such that X← 1, of spaces Xsuch that BX∗is weak∗sequentially compact and also to obtain several results concerning to the weak operator and the strong operator topologies. As another application of weak equicompactness, we conclude a characterization of relatively compact sets in L(X, Y ) when this space is endowed with the topology of uniform convergence on the class of all weakly null sequences. Finally, we show that similar arguments can be applied to the study of uniformly completely continuous sets. Mathematics Subject Classification (2000). 47B07. Keywords. Weakly compact operators, weakly equicompact set, collectively weakly compact set, precompact set, uniform spaces, uniformly completely continuous. 1. Introduction. Let us consider (real or complex) Banach spaces Xand Y.As usual L(X,Y ), K(X,Y ), V(X,Y ), W(X,Y ) and CW(X,Y ) will denote, respectively, the vector space of all bounded, compact, completely continuous, weakly compact or conditionally weakly compact linear operators from Xinto Yendowed with the operator norm. In [10], the authors defined weakly equicompact (respectively conditionally weakly equicompact) sets as those subsets Mof W(X, Y ) (respectively of CW(X,Y )) satisfying that, for every bounded sequence (xn)inX, there exists a subsequence (xk(n)) so that (Txk(n)) is uniformly weakly convergent (respectively uniformly weakly Cauchy) for T∈M. They have proved the following characterization: Vol. 89 (2007) Some properties and applications of weakly equicompact sets 267 Proposition A ([10, Proposition 2.2]). Let Mbe a subset of W(X,Y )(respectively of CW(X,Y )). The following statements are equivalent: (a) Mis weakly equicompact (respectively conditionally weakly equicompact). (b) Msatisfies the following properties: (i) For every bounded sequence (xn)in X, there is a subsequence (xk(n)) so that (Txk(n))is weakly convergent (respectively weakly Cauchy) for every T∈M. (ii) M∗y∗is relatively compact in X∗for every y∗∈Y∗. Remark 1.1. If M⊂K(X,Y ) and Mis weakly equicompact, then for every bounded sequence (xn)inX, there is a subsequence (xk(n)) so that (Txk(n))is convergent for every T∈Mbut this convergence is not necessarily uniform for T∈M. There are many situations in which condition (b)-ii implies itself that a set Mis weakly equicompact (see, for example, [10, corollary 2.3]). In this paper, we give an example of a set Msuch that M∗y∗is relatively compact for every y∗∈Y∗but M is not weakly equicompact; in fact, if BX∗is not weak∗sequentially compact, there exist always a Banach space Yand a set of weak∗–weakly continuous operators in W(X∗,Y) as in the example (Theorem 2.6). It is also proved that a Banach space Xdoes not contain a copy of 1if and only if the pointwise relative compactness of M∗implies the weak equicompactness of M, regardless of the Banach space Y and the set M⊂W(X,Y ). Section 3 is devoted to obtain several results on compactness in the weak operator and strong operator topologies (in short WOT and SOT, respectively) about weakly equicompact sets. Among other results, we prove that, if a set M⊂CW(X,Y ) is conditionally weakly equicompact and Mor M∗is sequentially WOT-compact, then M∗is sequentially SOT-compact and Mis conditionally collectively weakly compact (recall that a set M⊂W(X,Y ) (respectively M⊂CW(X,Y )) is called collectively weakly compact (respectively conditionally collectively weakly compact) iff the set T∈MT(BX) is relatively weakly compact (respectively conditionally weakly compact) in Y). In section 4, we consider the locally convex topologies Tw0and Twc on L(X,Y ) of the uniform convergence, respectively, on the class w0of all weakly null sequences and the class wc of all weakly Cauchy sequences in X(the space Y endowed with the weak topology). To continue with the work that has been done previously in [10, Section 3], we prove that the relatively compact sets are, in both topologies, the weakly w0−equicompact and pointwise weakly compact sets (Theorem 4.6). Finally, in section 5, we apply our techniques to characterize relatively compact sets in V(X,Y ) (again, V(X,Y ) endowed with the topology of uniform convergence on the classes w0or wc and the space Ywith the topology induced by its norm) in terms of uniformly completely continuous sets. 268 E. Serrano, C. Pi˜ neiro, and J. M. Delgado Arch. Math. Our notation is standard. If Xis a Banach space, BXwill denote its closed unit ball and X∗will be the topological dual of X. For an arbitrary set I,we will write 1(I,X) (respectively ∞(I,X)) for the Banach space of all absolutely summable (respectively bounded) X-valued functions defined on I, endowed with the norm ξ=i∈Iξi(respectively ξ= sup {ξi:i∈I}) for each ξ= (ξi)i∈I∈1(I,X) (respectively ξ=(ξi)i∈I∈∞(I,X)). As usual, we will write 1(I) (respectively ∞(I)) instead of 1(I,IR ) (respectively ∞(I,IR )) and ejwill be (δj i)i∈Ifor each j∈I. 2. Weakly equicompact sets and pointwise compactness. To start, we are going to show some situations in which condition (b)-ii in Proposition A implies itself the weak equicompactness of a set M⊂W(X, Y ). For example, this statement is true when the Banach space Xdoes not contain a copy of 1, as an application of the Rosenthal 1–theorem [8]. Furthermore, in view of Proposition A, we can also obtain the same conclusion if BX∗is weak∗sequentially compact and the operators in M⊂W(X∗,Y) are weak∗–weakly continuous (in fact, these are the only possible examples if we force the implication regardless of the Banach space Y). Finally, we will prove that the above implication occurs as well if the Banach space Y∗is separable (Corollary 2.3). If M⊂L(X, Y ) is bounded, define Vy∗:x∈X−→ (Tx,y∗)T∈M∈∞(M) for every y∗∈Y∗and put  M={Vy∗:y∗∈BY∗}. Proposition 2.1. Let Mbe a bounded subset of W(X,Y ). The following statements are equivalent: (a) Mis weakly equicompact. (b) For every bounded sequence (xn)in X, there exists a subsequence (xk(n))so that (Vy∗xk(n))converges for all y∗∈Y∗. (c) Msatisfies the following properties: (i) For every bounded sequence (xn)in X, there is a subsequence (xk(n)) so that (Vy∗xk(n))is weak∗convergent for all y∗∈Y∗. (ii)  Mis a subset of K(X,∞(M)). Proof. (a)⇒(b) is a direct consequence of the weak equicompactness of Mand (b)⇒(c) is obvious. Thus, only (c)⇒(a) needs to be proved. Let’s see that Mis weakly equicompact via Proposition A. If (xn) is a bounded sequence in X,it admits a subsequence (xk(n)) such that (Vy∗xk(n))isweak ∗convergent for every y∗∈Y∗. Since the weak∗convergence in ∞(M) coincides with the convergence coordinatewise, this means that (Txk(n)) is weakly Cauchy for all T∈M.Asthe operators in Mare weakly compact, (Txk(n)) is weakly convergent for all T∈M. Now, define the operator Uy∗:(ξT)T∈M∈1(M)−→ T∈MξTT∗y∗∈X∗for each y∗∈Y∗. Notice that Uy∗is the restriction of (Vy∗)∗to 1(M). Then, the set M∗y∗⊂Uy∗(B1(M)) is relatively compact.  Vol. 89 (2007) Some properties and applications of weakly equicompact sets 269 Remark 2.2. Notice that, given a bounded set M⊂L(X,Y ),  Msatisfies the condition (c)-ii in the above proposition if and only if M∗y∗is relatively compact for every y∗∈Y∗. Corollary 2.3. Let M⊂W(X,Y )be a bounded set. If Y∗is separable and M∗y∗ is relatively compact for every y∗∈Y∗, then Mis weakly equicompact. Proof. According to Remark 2.2, the operator φ:y∗∈Y∗−→ Vy∗∈K(X, ∞(M)) is well defined. If (y∗ n) is a sequence in Y∗so that BY∗⊂ {y∗ n:n∈IN }, then  M⊂{φ(y∗ n): n∈IN }is separable. A standard argument of diagonalization yields the weak equicompactness of Mfrom Proposition 2.1.  Now, we give an example of a set Mfailing to be weakly equicompact but such that M∗y∗is relatively compact for every y∗∈Y∗. Example 2.4. For every s∈IR , consider the operator Ts:∞(IR )−→ c0(IR ) defined by Tsx=xsesfor all x=(xr)r∈IR and put M={Ts:s∈IR }. Obviously, Mis a subset of K(∞(IR ),c 0(IR )). First of all, we are going to prove that Mis not weakly equicompact. By contradiction, every sequence (xn)inB∞(IR)(xn= (xn r)r∈IR for every n∈IN ) admits a subsequence (xk(n)) so that (Tsxk(n))nis convergent for all s∈IR (Remark 1.1). This means that (xk(n) s)nconverges for all s∈IR and, therefore, (xk(n))isweak ∗convergent in ∞(IR ). This is a contradiction because B∞(IR)is not weak∗sequentially compact [3, p. 226]. On the other hand, notice that Vy∗x=(xr·yr)r∈IR where y∗=(yr)r∈IR ∈ 1(IR ). Each operator Vy∗is compact so, according to Remark 2.2, M∗y∗is relatively compact. Remark 2.5. In the previous example, we have made use of the weak∗sequentially noncompactness of B∞(IR); it is also significant to mention that the operators in Mare weak∗–weakly continuous. The following result shows this is not accidental. In the following result, W∗(Y∗,Z) will denote the subspace of W(Y∗,Z)ofall weak∗–weakly continuous operators from Y∗into Z. Theorem 2.6. Let Ybe a Banach space. The following statements are equivalent: (a) For every Banach space Zand every N⊂W ∗(Y∗,Z),Nis weakly equicompact whenever N∗z∗is relatively compact for all z∗∈Z∗. (b) For every Banach space Xand every M⊂K(X, Y ),M∗is weakly equicompact whenever M∗∗x∗∗ is relatively compact for all x∗∗ ∈X∗∗. (c) BY∗is weak∗sequentially compact. Proof. Only (b)⇒(c) needs to be proved. For every β=(βy)y∈BY∈2(BY), we define the operator Tβ:(αy)y∈BY∈2(BY)−→ y∈BY(βyαy)y∈Yand put M=Tβ:β∈B2(BY). It is easy to check that M⊂K(2(BY),Y). Since M=M∗∗ and Mβ =Tβ(B2(BY)) for all β∈B2(BY), it follows that M∗is weakly 270 E. Serrano, C. Pi˜ neiro, and J. M. Delgado Arch. Math. equicompact. Now, given a bounded sequence (y∗ n)⊂Y∗, there exists (y∗ k(n)) such that (T∗ ezy∗ k(n))=(z,y∗ k(n)) converges for all z∈BY. Theorem 2.7. 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 M∗y∗is relatively compact for all y∗∈Y∗. (b) Xdoes not contain a copy of 1. Proof. Only (a)⇒(b) needs to be proved. For every x∗∈X∗, we define the operator Tx∗=x∗⊗ex∗and put M={Tx∗:x∗∈BX∗}⊂K(X,c0(BX∗)). In a similar way as in example 2.4, it can be proved that M∗y∗is relatively compact for all y∗∈c0(BX∗)∗. Hence, Mis weakly equicompact and, therefore, every bounded sequence (xn)inXhas a subsequence (xk(n)) such that (Tx∗xk(n)) is convergent for all x∗∈BX∗. Since Tx∗xk(n)−Tx∗xk(m)=|xk(n)−xk(m),x ∗|, it follows that (xk(n)) is weakly Cauchy.  3. Some results on compactness of weakly equicompact sets. Proposition 3.1. Let Mbe a conditionally weakly equicompact subset of CW(X,Y ). If Mor M∗is relatively WOT sequentially compact, then M∗is relatively SOT sequentially compact. Proof. Let (Tn) be a sequence in M.IfMis relatively WOT sequentially compact, there exist a subsequence (Tk(n)) and T∈L(X,Y ) such that (Tk(n))WOT →T, that is to say, (T∗ k(n)y∗)isweak ∗convergent to T∗y∗for every y∗∈Y∗.So(T∗ k(n)y∗−T∗y∗) is a weak∗null sequence in the relatively compact set M∗y∗−T∗y∗for every y∗∈Y∗(Proposition A). From this, it can be deduced that (T∗ k(n)y∗−T∗y∗)isa norm null sequence and, thus, (T∗ k(n)) is SOT convergent. The same argument is valid if M∗is relatively WOT sequentially compact.  Definition 3.2. Let (Tn) be a sequence in L(X,Y ). We say that (Tn) has the weak cross limit property (in short, wcl property) if every bounded sequence (xn)inX admits a subsequence (xk(n)) such that weak-limn,m→∞ Tnxk(m)−Tmxk(n)=0. Remark 3.3. Notice that if (Tn) is a sequence in L(X,Y ) with the wcl property then (Tn) is, in particular, a WOT Cauchy sequence and every subsequence of (Tn) has the wcl property. Proposition 3.4. Let Mbe a conditionally weakly equicompact subset of CW(X,Y ) and (Tn)a sequence in M. The following statements are equivalent: (a) (Tn)has the wcl property. (b) (T∗ n)is SOT convergent. Vol. 89 (2007) Some properties and applications of weakly equicompact sets 271 Proof. If (Tn) has the wcl property and (T∗ n) is not SOT Cauchy, then there exist y∗∈Y∗,ε0>0 and increasing maps h, k:IN −→ IN so that   (T∗ k(n)−T∗ h(n))y∗  > ε0for all n∈IN . Given n∈IN , choose xnin BXsuch that xn,(T∗ k(n)−T∗ h(n))y∗>ε 0 (1) Without loss of generality we can assume that (Tmxn)nis uniformly weakly Cauchy for m∈IN . Now, take a subsequence (xl(n))of(xn) satisfying weak-lim n,m→∞ Tnxl(m)−Tmxl(n)=0. If we put p(n)=k(l(n)) and q(n)=h(l(n)) for all n∈IN ,wehave: xl(n),(T∗ p(n)−T∗ q(n))y∗≤Tp(n)xl(n)−Tnxl(p(n)),y∗ +Tnxl(p(n)) −Tnxl(q(n)),y∗ +Tnxl(q(n)) −Tq(n)xl(n),y∗. Having in mind that (Tn) is a conditionally weakly equicompact set with the wcl property, we deduce that lim n→∞ xl(n),(T∗ k(l(n)) −T∗ h(l(n)))y∗=0, a contradiction with (1). Conversely, given a sequence (xn)inBX, there exists a subsequence (xk(n)) such that (Tmxk(n))nis uniformly weakly Cauchy for m∈IN . Given ε>0 and y∗∈Y∗, there is a natural number n0so that  (T∗ p−T∗ q)y∗ <ε/2 and Tmxk(p)−Tmxk(q),y∗<ε/2 for all p, q ≥n0and all m∈IN . Then, taking p, q ≥n0,wehave: Tpxk(q)−Tqxk(p),y∗≤Tpxk(q)−Tpxk(q),y∗+xk(p),(T∗ p−T∗ q)y∗<ε  Theorem 3.5. Let M⊂CW(X,Y )be conditionally weakly equicompact. If Mor M∗is relatively WOT sequentially compact then Mis conditionally collectively weakly compact. Proof. According to Proposition 3.1, M∗is relatively SOT sequentially compact. Given a sequence (Tnxn) with xn∈BXand Tn∈Mfor all n∈IN , we can assume that (T∗ n) is SOT convergent and (Txn) is uniformly weakly Cauchy for T∈M. By Proposition 3.4, the sequence (Tn) has the wcl property, so there is a subsequence (xk(n)) such that weak-lim n,m→∞ Tnxk(m)−Tmxk(n)=0.(2) 272 E. Serrano, C. Pi˜ neiro, and J. M. Delgado Arch. Math. For a fixed y∗in Y∗,wehave Tk(n)xk(n)−Tk(m)xk(m),y∗≤Tk(n)xk(n)−Tnxk(n),y∗ (3) +Tnxk(n)−Tnxk(m),y∗ (4) +Tnxk(m)−Tmxk(n),y∗ (5) +Tmxk(n)−Tk(m)xk(n),y∗ (6) +Tk(m)xk(n)−Tk(m)xk(m),y∗ (7) The SOT convergence of (T∗ n) in case of (3) and (6), the conditionally weak equicompactness applied to (4) and (7) and, finally, (2) allows to state that the sequence (Tk(n)xk(n)) is weakly Cauchy.  Example 3.6. Set Tβ:(αn)∈c0−→ (βnαn)∈c0(β=(βn)∈c0). If we define M={Tβ:β∈Bc0}, it is easy to deduce that M⊂K(c0,c 0) is weakly equicompact and Mand M∗are relatively WOT sequentially compact. Nevertheless, Mis not collectively weakly compact since β∈Bc0Tβ(Bc0)=Bc0. This shows that the conclusion in Theorem 3.5 cannot be improved, in general. 4. Relatively compacts sets in Lwc(X,Yw).In [10, Section 2], the authors generalize the concept of weakly equicompact set. Given a class Gof bounded sequences in X, we say that a subset Mof L(X, Y ) (the set of linear maps from Xinto Y)isconditionally weakly G-equicompact (respectively weakly G-equicompact)if every sequence (xn)∈Ghas a subsequence (xk(n))∈Gso that (Txk(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. We will occasionally invoke the following result: Lemma 4.1 ([10, Lemma 2.6]). If M⊂L(X,Y ), the following statements are equivalent: (a) If (xn)∈w0, then (Txn)w →0uniformly for T∈M. (b) If (xn)∈wc, then (Txn)is weakly Cauchy uniformly for T∈M. (c) Mis weakly w0-equicompact. (d) Mis conditionally weakly wc-equicompact. Now, we recall the definition of the topology of uniform convergence on a class Gof bounded sets. Let Xand Ybe Hausdorff locally convex spaces, F(X, Y ) the set of all maps from Xinto Yand GacoverofXformed by bounded subsets of X. For each A∈G, we define U(A, W )={f∈F(X,Y ):f(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 subbasis of a topology, denoted by TG, compatible with the commutative group structure of F(X, Y ) and the induced topology on L(X,Y ) is locally convex. We will write LG(X,Y ) for this locally convex space or LG(X, Yτ) if we need to emphasize the topology τof Y(see [2, TVS III.13] and [5, Chapter 8]). Vol. 89 (2007) Some properties and applications of weakly equicompact sets 273 A Hausdorff uniform space Eis said to be precompact if its completion, denoted by  E, is compact. Within the framework of topological vector spaces this means that, for a subset Aof a topological vector space E, the following are equivalent: (a) Ais precompact, (b)Ais relatively compact in  Eand (c) for each 0-neighborhood Vin Ethere exists a finite subset Fso that A⊂F+V[4, Theorem 3.5.1]. Actually, the topology of uniform convergence is treated in the framework of uniform spaces (see [1, Chapter X]). We will use the following version of Ascoli’s classical theorem Ascoli’s Theorem ([1, Theorem X.17.2]). Consider two uniform spaces Xand Y, a cover Gof Xformed by precompact subsets and a set H⊂F(X,Y )such that the restriction of each h∈Hto each A∈G,h|A, is uniformly continuous. Then His precompact in the topology 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 H|A={h|A:h∈H}is uniformly equicontinuous. For simplicity, we will denote by Ywthe space (Y,σ(Y,Y ∗)) and by Y· the space (Y,·); we will also denote by w0(respectively wc) the class formed by the ranges of all weakly null sequences (respectively weakly Cauchy sequences). We refer to Tw w0and Tw wc as the topologies of uniform convergence when we consider, respectively, the classes w0and wc in Xand the weak topology in Y;Lw0(X, Yw) and Lwc(X,Yw) denote L(X, Y ) endowed with the topologies Tw w0and Tw wc respectively (recall that L(Xw,Y w)=L(X·,Y ·)). Obviously WOT Tw w0Tw wc. Moreover: the topologies Tw w0and Tw wc only coincide iff Xis weakly sequentially complete. However, the authors obtained the following result in [10]: Theorem B ([10, Theorem 3.2]). Let M⊂L(X, Y )be a bounded set. The following statements are equivalent: (a) Mis precompact in Lwc(X,Yw). (b) Mis precompact in Lw0(X, Yw). (c) Mis weakly w0-equicompact. The aim of this section is to deduce a characterization of compact sets in the spaces Lwc(X,Yw) and Lw0(X, Yw) via Theorem B. We design by Ls(X,Y ) the space L(X,Y ) endowed with the topology of simple convergence, that is to say, the topology TS, where Sis the family of the finite sets in X(notice that the topologies WOT and SOT are particular cases). We will need the following properties, too: 1) Ls(X,Y ) is closed in Fs(X,Y ) [2, Proposition III.16.4]. 2)  Ls(X,  Y)=L s(X,  Y) [5, 8−§39.6(7)]. 274 E. Serrano, C. Pi˜ neiro, and J. M. Delgado Arch. Math. Notice that the natural inclusion i:Lw0(X,Yw)−→ Ls(X,  Yw) is continuous and, thus, there exists a unique continuous extension i:  Lw0(X,Yw)−→ Ls(X,  Yw). In particular, this yields that if M⊂L(X,Y ), then M  Lw0(X,Yw)⊆MLs(X,  Yw). A similar argument shows that M  Lwc(X,Yw)⊆MLs(X,  Yw). Lemma 4.2. If M⊂L(X,Y )is bounded and Mx is relatively weakly compact for all x∈X, then MLs(X,  Yw)⊂L(X,Y ). Proof. Since  Yw ∗=Y∗[4, 3.4.4], the weak topology in  Ywis precisely σ( Yw,Y∗) and {W(0; y∗,ε):y∗∈Y∗,ε>0}is a 0-neighborhood subbasis for this topology. Now, take a map Qin MLs(X,  Yw). Given x∈X,y∗∈Y∗and ε>0, there exists an operator Tε∈Msuch that |Qx −Tεx, y∗| <ε; in other words, Qx ∈Mxσ(  Yw,Y ∗). By hypothesis, Mxσ(Y,Y ∗)is σ( Yw,Y∗)-closed. So we have Mxσ(  Yw,Y ∗)⊆Mxσ(Y,Y ∗)σ(  Yw,Y ∗) =Mxσ(Y,Y ∗) from which Qx ∈Mxσ(Y,Y ∗). This yields that Q∈L(X,Y ). Now, a straightforward argument shows that the operator Qbelongs to L(X,Y ) and Q≤supT∈MT.  Corollary 4.3. If M⊂L(X,Y )is bounded and Mx is relatively weakly compact for all x∈X, then M  Lwc(X,Yw),M  Lw0(X,Yw)⊂L(X, Y ). The following lemma can be proved using a standard argument: Lemma 4.4. Let Gan arbitrary class of bounded sequences in X.IfM⊂L(X,Y ) is conditionally weakly G-equicompact, then MLs(X,  Yw)is conditionally weakly G-equicompact. Corollary 4.5. Let Gan arbitrary class of bounded sequences in X.IfM⊂L(X, Y ) is conditionally weakly G-equicompact, then M  Lwc(X,Yw)and M  Lw0(X,Yw)are conditionally weakly G-equicompact. Now, we are able to state one of our main results: Theorem 4.6. Let M⊂L(X,Y )be a bounded set. The following statements are equivalent: (a) Mis relatively compact in Lwc(X,Yw). (b) Mis relatively compact in Lw0(X,Yw). (c) Msatisfies the following properties: (i) Mis weakly w0-equicompact. (ii) Mx is relatively weakly compact for all x∈X.