Uniformly summing sets of operators on spaces of vector-valued continuous functions

Uniformly summing sets of operators on spaces of vector–valued continuous functions By J.M. Delgado and C. Pi˜neiro Abstract. Let X,Ybe Banach spaces. We say that a set M⊂p(X, Y ) is uniformly p−summing if the series  nTx npis uniformly convergent for T∈Mwhenever (xn)belongs to p w(X). We consider uniformly summing sets of operators defined on a C(, X)-space and prove, in case Xdoes not contain a copy of c0, that Mis uniformly summing iff M#={T#:C() −→ 1(X, Y ) :T∈M}is, where T(ϕx)=(T #ϕ)x for all ϕ∈C() and x∈X. We also characterize the sets Mwith the property that Mis uniformly summing viewed in 1(C(), L(X, Y )). 1. Introduction. Families of operators between Banach spaces arise in different applications: equations containing a parameter, homotopies of operators, etc. In these applications, it may be very interesting to know that, given a set Mof p-summing operators from a Banach space Xinto another Banach space Yand a weakly p-summable sequence (xn) in X, the series  nTx npis uniformly convergent in T∈M; in this case, we will say that Mis an uniformly p-summing set. A basic argument shows that uniformly p-summing sets are bounded for the p-summing norm. In fact, if Xdoes not contain any copy of c0, bounded sets and uniformly 1-summing sets are the same. In [1], the authors have obtained a characterization of uniformly summing sets of operators defined on C()-spaces. In this paper, we consider sets of operators on vector-valued continuous functions spaces. Throughout this paper Xand Ywill be Banach spaces. If Xis a Banach space, BX= {x∈X:x1}will denote its closed unit ball and X∗will be the topological dual of X. Given a real number p∈[1,∞), a (linear) operator T:X−→ Yis said to be p-summing if there exists a positive constant Csuch that n  i=1Tx ip1/p C·sup   n  i=1|x∗,x i|p1/p :x∗∈BX∗   142 J.M. Delgado and C. Pi ˜neiro arch. math. for every finite set {x1,...,x n}⊂X. The least Cfor which the above inequality always holds is denoted by πp(T ) (the p-summing norm of T). The linear space of all p-summing operators from Xinto Yis denoted by p(X, Y ) which is a Banach space endowed with the p-summing norm. As usual, p w(X) will be the Banach space of weakly p-summable sequences in X, i.e., the sequences (xn)⊂Xsatisfying  n|x∗,x n|p<∞for all x∗∈X∗; the norm in p w(X) is p(xn)=sup    n|x∗,x n|p1/p :x∗∈BX∗  =sup (αn)∈Bq n αnxn. If p=1, recall that a sequence (xn)belongs to 1 w(X) iff the series  n xnis weakly unconditionally Cauchy (in short, w.u.c.), that is to say, iff sup n∈ xn<∞, where the supremum runs over all finite subsets of N. The set of all strongly p-summable sequences in Xis denoted by p a(X); the norm in this space is (xn)p=( nxnp)1/p. If T∈p(X, Y ), the correspondence  T:(xn)−→ (T xn)always induces a bounded operator from p w(X) into p a(Y ) with  T=πp(T ) [2, Proposition 2.1]. Some properties of uniformly summing sets are the following: •Let (Tk)be a sequence in p(X, Y ). Then,  Tk k →0 pointwise if and only if Tk k →0 pointwise and (Tk)is uniformly p-summing. •Let M⊂p(X, Y ) be a uniformly p-summing set. If Mis endowed with the strong operator topology, then the map T∈M−→  nTx np∈Ris continuous for every (xn)∈p w(X). C(, X) will denote the vector space of the continuous functions defined on the Hausdorff compact space and valued in a Banach space X(C(, R)=C()), endowed with the supremum norm. The σ-algebra of all Borel subsets of will be denoted by .Aswe mentioned earlier, uniformly summing sets of operators defined on C() are characterized in terms of their representing measures in [1, Theorem 2.1]. Concretely, if mTis the representing measure of the operator T:C() −→ Yand |mT|is the variation measure, we have: Theorem ([1, Theorem 2.1]).The following statements are equivalent for a bounded subset Mof 1(C(), Y ): (a) Mis uniformly summing. (b) The family of positive measures {|mT|:T∈M}is uniformly countably additive. (c) If (En)is a sequence of pairwise disjoint Borel subsets of , the series  nmT(En) is uniformly convergent in T∈M. Vol. 87, 2006 Uniformly summing sets of operators functions 143 This result and the fact that, given an operator T:C(, X) −→ Y, there exists an unique operator T#:C() −→ L(X, Y ) such that T(ϕx)=(T #ϕ)x for all ϕ∈C() and x∈X [4, III.19.2] are the keys to show the relationship between the uniform sumability of a set Min 1(C(, X), Y ) and the set M#={T#:T∈M}. In Section 2, we prove that M#is uniformly summing (in 1(C(), 1(X, Y ))) whenever Mis. It is also proved that, when Xdoes not contain any copy of c0, any set M⊂1(C(, X), Y ) is uniformly summing iff M#⊂1(C(), 1(X, Y )) is. In fact, this property characterizes Banach spaces with no copy of c0. In Section 3, we give an interesting example of a set Msuch that M#is uniformly summing as a subset of 1(C(), L(X, Y )) but it is not uniformly summing in 1(C(), 1(X, Y )). Finally, we characterize those sets Mwith the property that M#is uniformly summing viewed in 1(C(), L(X, Y )). 2. Uniformly summing subsets of 1(C(, X), Y ) 1(C(, X), Y ) 1(C(, X), Y ).Given an operator T:C(, X) −→ Y, for each E∈we consider the operator mT(E) ∈L(X, Y ∗∗)defined by mT(E)x =T∗∗(χEx),(1) for all x∈X,χExbeing the continuous linear form on C(, X)∗defined by χEx,ν= ν(E),x. The map mT:−→ L(X, Y ∗∗)is a finitely additive vector measure of bounded semivariation mT() =Tfor which the equality Tf =  f(ω)dm T(ω)(2) holds for every f∈C(, X). The measure mTis called the representing measure of T. The next result, due to Swartz [7], characterizes the summing operators on C(, X). Theorem (C. Swartz, 1973).Let T:C(, X) −→ Ybe an operator with representing measure mT. The following statements are equivalent: (a) Tis 1-summing. (b) For each ϕ∈C(),T#ϕbelongs to 1(X, Y ) and the operator T#:C() −→ 1(X, Y ) is 1-summing. (c) For each E∈,mT(E) belongs to 1(X, Y ) and mThas bounded π1-variation. Furthermore, the equality π1(T ) =π1(T #)=π1−|mT|() holds, π1−|mT|being the variation of the vector measure E∈→ mT(E) ∈1(X, Y ). Our first result proves that the implication (a)⇒(b) in Swartz’s theorem is also true for uniformly summing subsets of 1(C(, X), Y ). Proposition 2.1 If Mis an uniformly summing subset of 1(C(, X), Y ) then M#is an uniformly summing subset of 1(C(), 1(X, Y )). 144 J.M. Delgado and C. Pi ˜neiro arch. math. Proof. By contradiction, suppose there is a sequence (ϕn)in 1 w(C()) such that  n π1(T #ϕn)is not uniformly convergent in T∈M. Then, there exists ε>0, a sequence (Tk)in Mand a strictly increasing sequence of natural numbers (nk)so that: nk+1  n=nk+1 π1(T # kϕn)>2ε(3) for all k∈N. Fixed k∈N, for every n∈{nk+1,...,n k+1}we can choose a finite set {xn 1,...,xn pn}⊂BXsatisfying pn  i=1(T # kϕn)xn i>π 1(T # kϕn)−ε 2n (4) and 1((xn i)pn i=1)1. Now we consider the sequence (ϕ1x1 1,...,ϕ 1x1 p1,ϕ 2x2 1,..., ϕ2x2 p2,......)and denote it by (fn). It is easy to prove that (fn)belongs to 1 w(C(, X)). Nevertheless, given any j∈Nthere exists k=k(j) so that nkjand it follows from (3) and (4) that:  n>j Tkfn nk+1  n=nk+1pn  i=1Tk(ϕnxn i) = nk+1  n=nk+1pn  i=1(T # kϕn)xn i>ε. This is a contradiction because Mis an uniformly summing set.  The converse of this proposition is not true, in general, and we are going to give an example. We will use the Banach space c0(X) of the null sequences (xn)in Xendowed with the norm (xn)∞=supnxn. Example 2.2. There is a bounded set Min 1(c0(c0), R)such that M#is uniformly summing but Mis not. Let (e∗ k)be the unit vector basis of 1. For each m∈N, we consider the operator Tm:c0(c0)−→ Rdefined by Tmˆx= k m mk2+1e∗ k+m−1,x k for all ˆx=(xk)∈c0(c0). So, the operator T# m:c0−→ 1is defined by T# mek=m·(mk2+ 1)−1e∗ k+m−1for all k∈N. Notice that T# mek=m·(mk2+1)−11/k2holds for every m∈N. Then, given ε>0, there exists k0∈Nso that sup m∈N kk0 T# mek<ε. Vol. 87, 2006 Uniformly summing sets of operators functions 145 With this inequality in hand, it is easy to prove that, if M=(Tm), then M#is uniformly summing. Nevertheless, we are going to show that Mis not. Consider the sequence (ˆxn) in c0(c0)defined by ˆxn=(xn k)k=(en,e n+1,...,e 2n−1,0,0,...). That is to say: xn k=ek+n−1if kn 0 otherwise . The sequence (ˆxn)belongs to 1 w(c0(c0)) but, for every m∈N,wehave  nm|Tmˆxn|= nm n  k=1 m mk2+1e∗ k+m−1,e k+n−1 = m  k=1 m mk2+1m m+1; then, Mis not uniformly summing. Now we face the problem of finding out under which conditions the converse of Proposition 2.1 is true. We will use the following technical lemma [6]. Lemma 2.3. Let Xbe a separable Banach space and a metrizable compact Hausdorff space. Given ε>0,(fn)in 1 w(C(, X)) and a countably additive positive measure µon , there exists a countably partition (Ek)of formed by Borel sets so that the sequence (fn(ω) − k χEk(ω)fn(ωk))nbelongs to 1 w(X) and 1fn(ω) − k χEk(ω)fn(ωk)n <ε µ-almost every ω∈, each ωkbeing in Ekfor all k∈N. Remark 2.4. In the conditions of the above result, there exists N∈with µ(N) =0 such that, if we define the function gn:−→ Xby gn(ω) =(fn(ω) −  k χEk(ω)fn(ωk))χ\N(ω) for each n∈N, then the sequence (gn)belongs to 1 w(C(, X)∗∗)and 1(gn)<ε. Theorem 2.5. Regardless the Banach space Y, the following statements are equivalent: (a) For all M⊂1(C(, X), Y ),Mis uniformly summing iff M#is. (b) Xdoes not contain any copy of c0. Proof. (a)⇒(b) We will prove that every w.u.c series  n xnin Xis unconditionally convergent. Choose y0∈Yso that y0=1, ω0∈and ϕ0∈C() satisfying ϕ0(ω0)=1. We consider the set M={Tx∗:x∗∈BX∗}⊂1(C(, X), Y ) where 146 J.M. Delgado and C. Pi ˜neiro arch. math. Tx∗f=x∗,f(ω 0)y0for all f∈C(, X).Given(ϕn)in 1 w(C()) and ε>0, there exists N∈Nsuch that  nN|ϕn(ω0)|<ε. Then we have nN π1(T # x∗ϕn)= nNϕn(ω0)x∗⊗y0 nN|ϕn(ω0)|<ε for all x∗∈BX∗, and this proves that M#(and so, M) is uniformly summing. Now, given ε>0 and (xn)in 1 w(X) there exists n0∈Nsuch that  nn0 Tx∗(ϕ0xn)<ε for all x∗∈BX∗. Finally, notice that  nn0 |x∗,x n| =  nn0 Tx∗(ϕ0xn)<ε for all x∗∈BX∗. (b) ⇒(a) We have to prove that, if M⊂1(C(, X), Y ) is a set for which M#is uniformly summing and the series  n fnis w.u.c in C(, X), then  nTfnis uniformly convergent in T∈M. First of all, notice that we only have to prove the result when Xis a separable space. In fact, if X0is defined as the closed span of {fn(ω) :n∈N,ω∈}, then X0is a separable space with no copy of c0. We also can assume that is metrizable. To see this, consider the equivalence relation ∼on defined by ω∼ωiff fn(ω) =fn(ω)for all n∈N.We will denote the quotient set / ∼by 0and π:−→ 0will be the quotient map. Let dbe the map defined on 0×0defined by d([ω],[ω])= n 1 2nfn(ω) −fn(ω). (0,d)is a compact metric space since the map πis continuous. Finally, for every T∈M, we define T0:C(0,X) −→ Yby T0φ=T(φ◦π) and put M0={T0:T∈M}.Itis easy to show that (M0)#is uniformly summing. As M#is uniformly summing in 1(C(), 1(X, Y )), it follows from [1, Teorema 2.1] that there exists a countably additive positive measure µon so that lim µ(E)→0π1−|mT|(E) =0 uniformly in T∈M.Givenε>0, invoke Lemma 2.3 and Remark 2.4 and consider the sequence (gn)⊂1 w(C(, X)∗∗)such that 1(gn)< ε 2M.(5) Vol. 87, 2006 Uniformly summing sets of operators functions 147 Again, by [1, Teorema 2.1], there is a natural number k0so that  kk0 π1(mT(Ek)) < ε 4 (6) for all T∈M. Since the series  n fn(ωk)is unconditionally convergent, we can choose n0∈Nsuch that  nn0 fn(ωk)<ε 8M (7) for all k<k 0. Then, for every Tin M,wehave  nn0 Tfn= nn0   fndmT  nn0  fn− k χEkfn(ωk)dmT + nn0   k χEkfn(ωk)dm T. Thanks to (5) we obtain:  nn0  fn− k χEkfn(ωk)dmT  nn0   gndmT = nn0 T∗∗gnπ1(T )1((gn)) < ε 2 for all T∈M. On the other hand, we have  nn0   k χEkfn(ωk)dm T  nn0 kmT(Ek)fn(ωk) = k  nn0 mT(Ek)fn(ωk)   k<k0 π1(mT(Ek)) ·1(fn(ωk))nn0 + kk0 π1(mT(Ek)) ·1(fn(ωk))nn0; 148 J.M. Delgado and C. Pi ˜neiro arch. math. now, (7) yields  k<k0 π1(mT(Ek)) ·1(fn(ωk))nn0<ε 4Mπ1−|mT|() ε 4, while (6) yields  kk0 π1(mT(Ek)) ·1(fn(ωk))nn0 kk0 π1(mT(Ek)) < ε 4 and this concludes the proof.  3. Uniformly summing sets in 1(C(), L(X, Y )) 1(C(), L(X, Y )) 1(C(), L(X, Y )).The aim of this section is to find out under which conditions a set Min 1(C(, X), Y ) is such that M#is uniformly summing in 1(C(), L(X, Y )). Proposition 2.1 tells us that, if Mis uniformly summing in 1(C(, X), Y ) then M#is uniformly summing in 1(C(), L(X, Y )). But Example 2.2 shows that the converse is not true, in general. Anyway, 1(X, Y ) and L(X, Y ) are the same in that example. That is the reason why we give another example to show that a set M#can be uniformly summing viewed in 1(C(), L(X, Y )) and, nevertheless, M#is not uniformly summing in 1(C(), 1(X, Y )). Example 3.1. There is a set Min 1(C([0,1], 2), 2)so that the set M#is uniformly summing in 1(C([0,1]), L(2, 2)) but M#is not uniformly summing in 1(C([0,1]), 1(2, 2)). Consider the sequence of real numbers (xm)defined by exmxm=m2+1 and let pmbe the integer part of xm. It is easy to show that (xm)is strictly increasing and limmxm= limmpm=∞. For each m∈N, we define βm=(βm k)= m2  k=1 p−1 mek∈2and the operator Am:(αk)∈ 2−→ (αk·βm k)k∈2. Notice that Am=1/pm. On the other hand, since the rank of Amis a subspace of 1, it follows that Amis 1-summing and, in particular, Hilbert–Schmidt; so, we have π1(Am)π2(Am)=  m2  n=1 1 p2 m  1/2 =m pm .(8) Now, denote by dω the Lebesgue measure on [0,1] and put dν =dω/√1−ω. Finally, consider M=(Tm), where Tm:C([0,1], 2)−→ 2is defined by Tmf=Am  1 0 ek, f (ω)dν k Vol. 87, 2006 Uniformly summing sets of operators functions 149 for all fin C([0,1], 2).Iff∈C([0,1], 2), put fk(·)=ek,f(·)∈C([0,1]); using H¨ older inequality we have  k 1 0 fk(ω) dν 2 ν2 1 k 1 0 |fk(ω)|2dν 4f2 C([0,1],2). This shows that Tmis well defined and Tm2/pm. A straightforward argument allows us to state that T# m:C([0,1])−→ 1(2, 2)is defined by T# mϕ=( 1 0 ϕdν)A mfor all ϕ∈C([0,1]). Each operator T# mis 1-summing; indeed,  n π1(T # mϕn)= n 1 0 ϕndνπ1(Am) =π1(Am) n|ν,ϕn|2π1(Am)1(ϕn). So, Swartz’s theorem tells us that Tmis 1-summing and π1(Tm)=π1(T # m)=2π1(Am)(9) for all m∈N. Now, notice that T# mϕ=p−1 m|ν,ϕ|for all ϕ∈C([0,1])and m∈N,soit is obvious that M#is uniformly summing viewed as a subset of 1C([0,1]), L(2, 2). Nevertheless, M#is not even bounded in 1C([0,1]), 1(2, 2). To see this, notice that the sequence exm(m2+1)−1mis convergent to 0, so there exists a natural number N such that xm<log(m2+1)(10) for all mN. Using (8), (9) and (10), we obtain lim mπ1(T # m)=lim m2π1(Am)lim 2m pm=lim m 2m xm lim m 2m log(m2+1)=∞. In the next proposition, we obtain a characterization of the subsets Min 1(C(, X), Y ) with the property that M#is uniformly summing viewed in 1(C(), L(X, Y )). Proposition 3.2. Let Mbe a subset of 1(C(, X), Y ). The following statements are equivalent: (a) M#is uniformly summing in 1(C(), L(X, Y )). (b) The series  nTfnis uniformly convergent in T∈Mwhenever (fn)is a sequence in C(, X) so that (fn(·))∈1 w(C()).