Some classes of p-summing type operators.

Transcription

1 Some classes of -summing tye oerators. Oscar Blasco and Teresa Signes Abstract Let X, Y be Banach saces and denote by l w (X, Y ), l s (X, Y ) and l (X, Y ) the saces of sequences of oerators (T n ) from X into Y such that su x =1, y =1 <Tn (x),y > <, su x =1 Tn (x) < and T n <, resectively. Given Banach saces X, Y, Z and W, we introduce and study the classes of bounded linear oerators Φ:L(X, Y ) L(Z, W) such that (T n ) (Φ(T n )) mas l s (X, Y ) into l (Z, W), l s (X, Y )intol s (Z, W) and l w (X, Y )intol s (Z, W). 1 Introduction Throughout this aer X, Y, Z, W will stand for Banach saces, L(X, Y ) for the sace of bounded linear oerators from X into Y and X for the dual of X. As usual B X denotes the closed unit ball of X. For 1 < we write l (X) for the Banach sace of all absolutely -summable sequences (x n ) in X, i.e., the sace of sequences such that (x n ) l(x) =( x n X < and l w (X) for the sace of all weakly -summable sequences in X, i.e., the sace of sequences (x n ) such that ( x,x n ) l for every x X, which becomes a Banach sace with resect to the norm (x n ) l w (X) = su x B X ( x,x n. Recall that the sace of -summing oerators Π (X, Y ) consists in those bounded linear oerators T L(X, Y ) such that T ((x n ) ) =(T (x n )) defines a bounded linear oerator from l w (X) intol (Y ). The reader is Partially suorted by grants DGESIC PB and DGI (BFM ) Partially suorted by grant DGI (BFM ) 1

2 referred to [2], [3], [7], [11], [12] or [14] for definitions and results about these classes and their alications in Banach sace theory. We shall simly recall here two related notions which are the main motivations for our future considerations. The first one aears when analyzing oerators acting on saces of vectorvalued continuous functions T : C(Ω,X) Y, where Ω is a comact Haussdorf sace. An oerator is called -dominated oerator (see [4], III.19.3) if there exist a constant C>0and a robability measure µ on Ω such that T (f) C f(t) dµ(t) for all f C(Ω,X). The connection between -summing and -dominated oerators was first given by A. Pietsch, who showed that for finite dimensional Banach saces X both notions are the same. For infinite dimensional Banach saces C. Swartz (see [13]) showed that oerators in Π 1 (C(Ω,X),Y) are always 1-dominated but the sace of 1-dominated oerators from C(Ω,X) into Y coincides with Π 1 (C(Ω,X),Y) if and only if X is finite dimensional. Later S. Kislyakov introduced (see [6], Def ) the following notion in the setting of injective tensor roducts: Given 1 < and three Banach saces E,X and Y, an oerator T from the injective tensor roduct X ˇ E into Y is said to be (, E)-summing if there exists a constant C>0 such that for all N N and u 1,u 2,..., u N X E we have N Ω T (u k ) Y C su F B X N u k (F ) E where u k (F )= n k j=1 F, x j,k e j,k for u k = n k j=1 x j,k e j,k for some e j,k E and x j,k X. The sace of such oerators is denoted by Π E (X ˇ E,Y ). Theorem of [6] gives an analogue to Pietsch s domination theorem for (, X)-summing oerators. This result actually enables us to get the - dominated oerators in a very similar fashion to the -summing ones. Since C(Ω)ˇ X = C(Ω,X) then the class of -dominated oerators actually coincides with Π X (C(Ω) ˇ X, Y ). 2

3 Given two Banach saces X and Y we shall be denoting by l (X, Y ) and l w (X, Y ) the saces l (L(X, Y )) and l w (L(X, Y )) resectively. Recall that if (e n ) is a sequence in a Banach sace E then Hence one easily gets that (1.1) (e n ) l w (E) = su (α n) B l (T n ) l w (X,Y ) = su su y B Y E α n e n. ( y,t n x. The fact that we have the strong oerator toology at our disosal allows to consider the following intermediate sace of sequences of oerators. We shall use the notation l s (X, Y ) for the sace of strongly -summable sequences of oerators (T n ), that is su x BX ( T nx Y )1/ <. Let Φ : L(X, Y ) L(Z, W) be a bounded oerator. The corresondence Φ :(T n ) (Φ(T n )) always induces a linear bounded oerator from l (X, Y )tol (Z, W), as well as from l w (X, Y )tol w (Z, W). Recall that Φ is -summing if Φ mas l w (X, Y )tol (Z, W). In this aer we study several questions concerning the class of oerators Φ such that this vector-valued extension Φ roduces a bounded linear oerator either from l s (X, Y )tol (Z, W), from l s (X, Y )tol s (Z, W), or from l w (X, Y )tol s (Z, W). In articular we study oerators Φ from L(X, Y )intoz such that Φ defines a bounded linear oerator from l s (X, Y )intol (Z). These oerators will be called (l s,l )-summing and denoted by Π (l s,l )(L(X, Y ),Z). Since X ˇ E is a subsace of L(X,E) we easily get that if Φ : L(X,E) Y is (l s,l )-summing then its restriction to X ˇ E is (, E)-summing. Also, the theory can be alied, among other things, to oerators acting on weakly -summable sequences l w (X) = L(l,X) or Pettis--integrable functions which are subsaces of L(L (µ),x). 3

4 The aer is divided into five sections. In Section 2 the notions of (l s,l )- summing, (l w,l s )-summing and (l s,l s )-summing oerators are introduced and studied. It is shown that the classes of (l w,l s )-summing and (l s,l s )- summing oerators can be easily described in terms of the classes Π or Π (l s,l ) (see Theorems 2.13 and 2.16 below). In section 3 several characterizations of the class Π (l s,l )(L(X, Y ),Z) are achieved and several examles are exhibited. In Section 4 some results about the coincidence of classes are shown, in articular an analogue of Maurey s theorem is resented. Finally we ose two oen roblems in Section 5. 2 Definitions and reliminaries Definition 2.1 Let 1 and X, Y be Banach saces. A sequence of oerators (T n ) L(X, Y ) is said to be strongly -summable in L(X, Y ) if the vector-valued sequence (T n x) belongs to l (Y ) for every x X. We shall use the notation l s (X, Y ) for the sace of all strongly -summable sequences. The norm in l s (X, Y ) is given by ( (T n ) l s (X,Y ) = su T n x Y It is rather easy to see that (l s (X, Y ), l s (X,Y )) is a Banach sace for any 1. Easy examles of sequences in l s (X, Y ) are given in the next roositions, whose roofs are left to the reader. Proosition 2.2 Let X, Y be Banach saces and 1 r 1,r 2, with 1 r r 2 = 1.If(x n) l w r 1 (X ) and (y n ) l r2 (Y ), then (x n y n ) l s (X, Y ). Proosition 2.3 Let X, Y and Z be Banach saces and 1. If (T n ) l w (X, Y ) and S Π (Y,Z), then (ST n ) l s (X, Z). 4.

5 As mentioned in the introduction l (X, Y ) stands for l (L(X, Y )) and l w (X, Y ) for l w (L(X, Y )). Let us first notice the inclusions aearing between these saces and the new one. Note that l s (K,Y) = l (K,Y) = l (Y ) and l s (X, K) = l w (X, K) = l w (X ). Observe also that for every bounded sequence (T n )inl(x, Y )we have su T n = su su n n T n x Y = su su y B Y Hence for =, l (X, Y )=l s (X, Y )=l w (X, Y ). su y,t n x. n Proosition 2.4 Let X, Y be Banach saces and 1 <. Then l (X, Y ) l s (X, Y ) l w (X, Y ). Moreover, each of the inclusions is strict in general. PROOF. We shall only show the last statement, since the inclusions are immediate by definitions. Let (e n ) be the usual basis in l (or c 0 for = 1) and consider (e n e n ) as oerators from l into l. Since (e n ) l w (l ) and (e n ) l (l ), by Proosition 2.2 we get that (e n e n ) l s (l,l ). On the other hand, e n e n l,l = e n l e n l = 1 thus (e n e n ) / l (l,l ). Let us now find a weakly -summable sequence which is not strongly - summable. Simly choose <r< and take s so that 1/r +1/ =1/s. A direct comutation, using (1.1), shows that (e n e n ) l w (l r,l s ). However (e n e n ) / l s (l r,l s ) as it is shown by selecting any λ =(λ n ) l r \ l, since e n,λ e n ls = λ n / l. Observe that if (T n ) in L(X, Y ) is a strongly -summable sequence, using the closed grah theorem, then one can associate a linear and bounded oerator S : X l (Y ) defined by S(x) =(T n x). In this way we get l s (X, Y ) as a closed subsace of L(X, l (Y )). Let us see that they are actually isometrically isomorhic. 5

6 Proosition 2.5 Let X, Y be Banach saces and 1. For each n N, let Q n : l (Y ) Y be the oerator Q n ((y i ) ) =y n. The corresondence T (Q n T ) is an isometric isomorhism from L(X, l (Y )) onto l s (X, Y ). PROOF. Given T L(X, l (Y )), the sequence T n = Q n T, n =1, 2,..., belongs to l s (X, Y ) and ( (T n ) l s (X,Y ) = su Q n Tx Y = su Tx l(y ) = T X,l(Y ). Therefore, the corresondence T (Q n T ) induces an isometry. To see the isomorhism let us take (S n ) l s (X, Y ) and note that S : X l (Y ) defined by S(x) =(S n x) gives a bounded oerator and Q n S = S n. As a consequence, we get that l w (X ) is isometrically isomorhic to L(X, l ) for any 1 and any Banach sace X (see [5], Pro ). Let us also recall that, for every Banach sace Z, l w (Z) is also isometrically isomorhic to L(l,Z)if1< < and L(c o,z) for = 1. In these last cases the isomorhims are given by associating to each oerator T the sequence x n = T (e n ). Of course the connection between both results goes through the adjoint oerator. Combining both identifications we get that the strongly -summable sequences in L(l q,x) are recisely the weakly q-summable sequences in l (X). Corollary 2.6 Let X be a Banach sace, 1 and 1 <q<. Then the ma Ψ:l w q (l (X)) l s (l q,x), defined by Ψ((x k ) k )=(T n ) n where T n are defined by T n (e k )=Q n (x k ) for all n, k N, is an isometric isomorhism. Similar result is true for q =1relacing l by c o. The concet of strongly -summing sequence can now be used to define new classes of -summing tye oerators. Definition 2.7 Let 1 <. LetX, Y, Z and W be Banach saces. 6

7 An oerator Φ:L(X, Y ) Z is said to be (l s,l )-summing if there exists a constant C>0 such that (2.1) Φ(T i ) Z C su T i x Y x B X for any finite family of oerators T 1,...,T n in L(X, Y ). An oerator Φ:X L(Y,Z) is said to be (l w,l s )-summing if there exist a constant C>0 such that (2.2) su Φ(x i )(y) Z y B Y C su x B X for every choice of elements x 1,...,x n in X. An oerator Φ:L(X, Y ) L(Z, W) is said to be there exist a constant C>0 such that (2.3) su Φ(T i )(z) W z B Z x,x i C su T i x Y (l s,l s )-summing if for every choice of oerators T 1,...,T n in L(X, Y ). The least constants in (2.1), (2.2) and (2.3) are denoted by π (l s,l )(Φ), π (l w,l s ) (Φ) and π (l s,l s ) (Φ) resectively. We shall denote by Π (l s,l )(L(X, Y ),Z) the sace of all (l s,l )-summing oerators from L(X, Y ) to Z, byπ (l w,l s ) (X, L(Y,Z)) the sace of all (l w,l s )- summing oerators from X to L(Y,Z) and by Π (l s,l s ) (L(X, Y ), L(Z, W)) the sace of all strongly (l s,l )-summing oerators from L(X, Y ) to L(Z, W). Π (l s,l )(L(X, Y ),Z), Π (l w,l s ) (X, L(Y,Z)) and Π (l s,l s ) (L(X, Y ), L(Z, W)) become Banach saces with the norms π (l s,l )( ), π (l w,l s ) ( ) π (l s,l s ) ( ), resectively. The corresonding definitions for = would simly lead to the sace of bounded oerators in all cases. Alternative definition for (l s,l )-summing oerators is the following one:. Remark 2.8 equivalent: Let 1 < and Φ L(L(X, Y ),Z). The following are 7

8 i) Φ is (l s,l )-summing. ii) Φ mas sequences (T n ) l s (X, Y ) into sequences (Φ(T n )) l (Z). iii) The linear oerator Φ :l s (X, Y ) l (Z) defined by Φ((T n ) ) = (Φ(T n )) is continuous. Similar equivalences are true for (l w,l s )-summing oerators and (l s,l s )- summing oerators. Remark 2.9 It is rather easy to see that, when some of the saces is finite dimensional, the classes Π (l s,l )(L(X, Y ),Z), Π (l w,l s )(X, L(Y,Z)) reduce either to bounded oerators or to -summing oerators and that the class Π (l s,l s )(L(X, Y ), L(Z, U)) reduces to one of the revious cases. Remark 2.10 Recall that dimx = imlies Π (X, X) L(X, X) and observe that and Π (l s,l )(L(K,X ),Y)=L(X,Y) Π (l s,l )(L(X, K),Y)=Π (X,Y). Therefore, for any infinite dimensional X, we have Π (l s,l )(L(K,X ),Y) Π (l s,l )(L(X, K),Y), for some Banach sace Y, Π (l s,l )(L(X, K), L(X, K)) Π (l w,l s ) (L(X, K), L(X, K)), Π (l w,l s ) (L(K,X), L(K,X)) Π (l s,l )(L(K,X), L(K,X)). Remark 2.11 If A and B are saces of oerator then Π (A, B) Π (l w,l s ) (A, B) Π (l s,l )(A, B), Π (l w,l s )(A, B) Π (l s,l )(A, B) Π (l s,l s )(A, B). It is not difficult to show that the inclusions are strict in general. 8

9 Remark 2.12 (i) For each x X, the evaluation ma e x : L(X, Y ) Y given by e x (T )=Tx is (l s 1,l 1 )-summing. (ii) For each y Y, the oerator Φ y : X L(X, Y ) given by Φ y (x )= x y is (l w 1,l s 1)-summing. (iii) For each S L(Y,Z) the oerator Φ S : L(X, Y ) L(X, Z) given by Φ S (T )=ST is (l s 1,l s 1)-summing. Moreover, if S Π (Y,Z), 1 <, then Φ S is (l w,l s )-summing. We now show that actually Π (l w,l s ) (X, L(Y,Z)) and L(Y,Π (X, Z)) can be identified. Theorem 2.13 Let Φ:X L(Y,Z) be a bounded oerator and let us define Φ # : Y L(X, Z) by Φ # (y)(x) :=Φ(x)(y), x X, y Y. The corresondence Φ Φ # is an isometric isomorhism between Π (l w,l s )(X, L(Y,Z)) and L(Y,Π (X, Z)). PROOF. Take Φ Π (l w,l s ) (X, L(Y,Z)) and y Y, then Φ # (y)(x i ) Z y Y Π (l w,l s )(Φ) su x B X x,x i for every choice of elements {x 1,...,x n } in X. Thus, Φ # (y) Π (X, Z) and π (Φ # (y)) y Y π (l w,l s )(Φ). Hence Φ # L(Y,Π (X, Z)) and Φ # π (l w,l s )(Φ). On the other hand, if Ψ L(Y,Π (X, Z)) then su Ψ # (x i )(y) Z y B Y = su Ψ(y)(x i ) Z y B Y su {π (Ψ(y))} y B Y Ψ su x B X su x B X x,x i x,x i for every finite sequence {x 1,...,x n } in X. Hence Ψ # Π (l w,l s )(X, L(Y,Z)) and π (l w,l s ) (Ψ # ) Ψ. Since (Ψ # ) # = Ψ the roof is finished. 9

10 Therefore, the class Π (l w,l s ) inherits some roerties from those in Π.For examle, Grothendieck theorem (see [3], Theorem 1.13) imlies the following: Corollary 2.14 Let X be a Banach sace. Every oerator from l 1 into L(X, l 2 ) is (l w 1,l s 1)-summing, i.e. Π (l w 1,l s 1 ) (l 1, L(X, l 2 )) = L(l 1, L(X, l 2 )). Corollary 2.15 Let X be a Banach sace, K be a comact Hausdorff sace and (Ω,µ) be a σ-finite measure sace. Every oerator from C(K) into L(X, L 1 (Ω,µ)) is (l w 2,l s 2)-summing, i.e. Π (l w 2,l s 2 ) (C(K), L(X, L 1 (µ))) = L(C(K), L(X, L 1 (µ))). Furthermore, arguing as in Theorem 2.13 we get also the following result whose roof is left to the interested reader. Theorem 2.16 The corresondence Φ Φ # is an isometric isomorhism between Π (l s,l s ) (L(X, Y ), L(Z, W)) and L(Z, Π (l s,l )(L(X, Y ),W)). We would like to oint out also certain behaviour of these classes as oerator ideals and some comosition results. The roof of the following roosition is straightforward. Proosition 2.17 Let 1 < and let X, Y, Z, W, U, X 0 and Z 0 be Banach saces. 1) If Φ Π (l s,l )(L(X, Y ),Z) and Ψ L(Z, Z 0 ), then the comosition ΨΦ belongs to Π (l s,l )(L(X, Y ),Z 0 ) with π (l s,l )(ΨΦ) Ψ Z,Z0 π (l s,l )(Φ). That is, L Π (l s,l ) Π (l s,l ). 2) If Φ L(X 0,X) and Ψ Π (l w,l s ) (X, L(Y,Z)), then the comosition ΨΦ belongs to Π (l w,l s ) (X 0, L(Y,Z)) with π (l w,l s ) (ΨΦ) π (l w,l s ) (Ψ) Φ X0,X. That is, Π (l w,l s ) L Π (l w,l s ). 3) If Φ Π (l s,l s ) (L(X, Y ), L(Z, W)) and Ψ Π (l s,l )(L(Z, W),U), then ΨΦ Π (l s,l )(L(X, Y ),U) with π (l s,l )(ΨΦ) π (l s,l )(Ψ) π (l s,l s ) (Φ). That is, Π (l s,l ) Π (l s,l s ) Π (l s,l ). 10

11 4) If Φ Π (l w,l s ) (X, L(Y,Z)) and Ψ Π (l s,l s ) (L(Y,Z), L(W, U)), then ΨΦ Π (l w,l s ) (X, L(W, U)) with π (l w,l s ) (ΨΦ) π (l w,l s ) (Ψ) π (l w,l s ) (Φ). That is, Π (l s,l s ) Π (l w,l s ) Π (l w,l s ). 5) If Φ Π (l w,l s )(X, L(Y,Z)) and Ψ Π (l s,l )(L(Y,Z),W), then ΨΦ Π (X, W) with π (ΨΦ) π (l s,l )(Ψ) π (l w,l s )(Φ). That is, Π (l s,l ) Π (l w,l s ) Π. The classical theorem of Pietsch stated that if Φ is q-summing and Ψ is -summing then ΨΦ is r-summing, with 1 = min{1, } (see [3], Theorem r q 2.22 or [5], Theorem ). Next we establish that when Ψ is (l w,l s )- summing then ΨΦ is (l w r,l s r)-summing. The roof follows along the lines of Theorem 2.22 in [3]. Proosition 2.18 Let Φ Π q (X, Y ) and Ψ Π (l w,l s )(Y,L(Z, W)) with 1, q <. Define 1 r< by 1 r = min{1, q }. Then ΨΦ is (lw r,l s r)- summing with π s r(ψφ) π (l w,l s ) (Ψ)π q (Φ). That is, Π (l w,l s ) Π q Π (l w r,l s r ). PROOF. We assume first that then 1 = Let (x q r q n) l w r (X) be given. Alying Lemma 2.23 in [3] we get sequences (σ n ) l q and (y n ) l w (Y ) such that (σ n ) lq (x n ) r/q l w(x), (y n) r l w (Y ) π q (Φ) (x n ) r/ l w r (X) and Φ(x n )=σ n y n for all n. By Hölder s inequality, using that Ψ is (l w,l s )- summing and that [ΨΦ(x n )](z) =σ n [Ψ(y n )(z)] for every z Z, weget ( k su [ΨΦ(x n )](z) r W z B Z r ( k σ n q q ( k su Ψ(y n )(z) W z B Z (x n ) r/q l w r (X)π (l w,ls ) (Ψ) su y B Y ( k π (l w,l s )(Ψ) π q (Φ) (x n ) l w r (X). y,y n Then ΨΦ belongs to Π (l w r,l s r ) (X, L(Z, W)). If q > 1 and we assume that 1 < q, then Φ Π (X, Y ) with π (Φ) π q (Φ) and alying the first art with and we get ΨΦ Π (l w 1,l s 1 ) (X, L(Z, W)), which comletes the roof. 11

12 3 (l s,l )-summing oerators Let us now resent several examles of such oerators. The first one connects the notion of -summing and (l s,l )-summing oerators. Let 1 < and let T : X Y be a bounded oerator. Denote T : L(l,X) l (Y ) the ma given by T (S) =(TSe n ). Recall that T is -summing if and only if T is bounded and π (T )= T. We first study when T is (l s,l )-summing oerator. The answer is given in the following theorem. Theorem 3.1 Let X, Y be Banach saces, 1 << and T L(X, Y ). Then T Π (l s,l )(L(l,X),l (Y )) if and only if T Π (l (X),l (Y )) where T : l (X) l (Y ) is given by T ((x j ) j=1) =(T (x j )) j=1. Moreover π ( T )= π (l s,l )( T ). Same result holds for =1with the relacement of l by c 0. PROOF. Assume that T Π (l s,l )(L(l,X),l (Y )). Given (x k ) k l w (l (X)) we define the oerators T n : l X such that T n (e k )=Q n (x k ) for all n, k N. Since Corollary 2.6 gives that (T n ) l s (l,x) then ( T (T n )) n = ((TT n (e k )) k ) n l (l (Y )). Now we have T (x k ) l (Y ) = T (Q n (x k )) Y = T (T n ) l (Y ) (π (l s,l )( T )) (T n ) l s (l,x) =(π (l s,l) ( T )) (x k ) l w (l (X)). For the converse assume that T is -summing and, as above, we write T (T n ) l (Y ) = T (x k ) l (Y ) (π ( T )) (x k ) l w (l(x)) =(π ( T )) (T n ) l s (l,x). Examle 3.2 Let 1 < <, X, Y be Banach saces. T Π (X,Y), then the oerator Φ T : L(X, l ) l (Y ) u (Tu (e i )) Assume that 12

13 is (l s,l )-summing with π (l s,l )(Φ T )=π (T ). PROOF. For any choice of u 1,...,u N -summing we have N { m Φ T (u n ) l (Y ) = su m L(X, l ), since T : X Y is N } Tu n(e i ) Y (π (T )) su m { m su { m = (π (T )) su su m { N (π (T )) su u n x l }. N } u n(e i ),x N } e i,u n x Therefore Φ T is (l s,l )-summing and π (l s,l )(Φ T ) π (T ). Therefore Φ T is (l s,l )-summing and π (l s,l )(Φ T ) π (T ). It is straightforward to get equality of norms since π (l s,l )(Φ T ) Φ T = π (T ). Examle 3.3 Let 1 <, let X be a reflexive Banach sace, Y be a searable Banach sace and L (µ, Y ) denote the sace of Bochner -integrable functions. If µ is a finite Borel measure on the comact toological sace ( B X,σ(X, X )) then the oerator Θ : L(X, Y ) L (µ, Y ) defined by Θ (T )(x) =T (x), x B X,is(l s,l )-summing with π (l s,l )(Θ ) µ( B X. PROOF. We first note that x T (x) is weakly continuous function on B X and hence weakly measurable. Using the searability of Y we get that it is a bounded measurable function and then in L (µ, Y ). This shows that the oerator Θ is well defined and bounded. If T 1,...,T n are in L(X, Y ), then Θ (T i ) L (µ,y ) = T i (x) Y dµ(x) B X { µ( B X ) su T i (x) Y 13 }.

14 Hence we have that Θ is (l s,l )-summing and π (l s,l )(Θ ) µ(b X. Examle 3.4 Let (Ω,µ) be a σ-finite measure sace, X, Y be Banach saces, 1 < and denote by L (µ, X) the sace of Bochner -integrable functions. For each f L (µ, X), the oerator Φ f : L(X, Y ) L (µ, Y ) defined by Φ f (T )(w) =T (f(w)), w Ω, is (l s,l )-summing and π (l s,l )(Φ f ) f L(µ,X). PROOF. Observe first that the oerator is well defined. Let T 1,...,T n oerators from X into Y.IfE = {w Ω:f(w) 0} then Φ f (T i ) L (µ,y ) = = E E T i (f(w)) Y dµ(w) T i ( f(w) f(w) ) Y f(w) dµ(w) f L su (µ,x) T i x Y Examle 3.5 Let 1 < and let X, Y be Banach saces. Any sequence (T n ) l (X, Y ) induces an oerator T : L(l 1,X) l (Y ) S (T n S(e n )) which is (l s,l )-summing with π (l s,l )( T ) (T n ) l(x,y ). PROOF. Take S 1,...,S N L(l 1,X), then ). N T (S k ) = N T n S k (e n ) Y ( ( N T n ) su S k (a) X a B l1 ). This gives that π (l s,l )( T ) (T n ) l(x,y ). To finish the section we give several equivalent formulations for the notion of (l s,l )-summing oerator. 14

15 Note that we can write the fact (T n ) n N l s (X, Y ) using duality: su T i x Y = su (y i ) B l n (Y ) Ti yi X. Proosition 3.6 Let 1 < and Φ L(L(X, Y ),Z). The following statements are equivalent: i) Φ Π (l s,l )(L(X, Y ),Z). ii) There exists a constant C>0 such that (3.1) Φ(T i ) Z C su (y i ) B l n (Y ) Ti yi X for every T 1,...,T n in L(X, Y ). Moreover π (l s,l )(Φ) = inf{c : C verifying (3.1)}. Proosition 3.7 Let 1 < and Φ L(L(X, Y ),Z). The following statements are equivalent: i) Φ Π (l s,l )(L(X, Y ),Z). ii) For each u L(X, l (Y )) the oerator is (l s,l )-summing. Φ u : L(l (Y ),Y) Z T Φ(Tu) iii) There exist a constant c such that π (l s,l )(Φ u ) c u for each n N and each u L(X, l n (Y )). Moreover π (l s,l )(Φ) = su{π (l s,l )(Φ u ):u L(X, l (Y ), u =1} = inf{c : c verifies (iii)}. 15

16 PROOF. (i) (ii) Observe that Ψ u : L(l (Y ),Y) L(X, Y ), Ψ u (T )= Tu, is strongly (l s,l )-summing with π (l s,l s ) (Ψ u ) u and Φ u = ΦΨ u. Then by (iii) in Proosition 2.17, Φ u is (l s,l )-summing with π (l s,l )(Φ u ) π (l s,l )(Φ) u. (ii) (iii) It follows easily from the closed grah theorem. (iii) (i) Let T 1,...,T n be oerators in L(X, Y ) and let S n : X l n (Y ) be defined by S n (x) =(T i x) n. Note that S n X,l n (Y ) = (T i ) n l s (X,Y ). For i =1,...,n we denote by Q i,n the rojections Q i,n : l n (Y ) Y given by Q i,n (y j ) n j=1 = y i. Hence for each n N and i =1,...,n, T i = Q i,n S n and Φ(T i )=Φ(Q i,n S n )=Φ Sn (Q i,n ). By hyothesis Φ Sn is (l s,l )-summing with π (l s,l )(Φ Sn ) B S n X,l n (Y ), thus Φ(T i ) Z = Φ Sn (Q i,n ) Z { B S n X,l n (Y ) su Q i,n (λ) Y B (T i ) n l s (X,Y ). : λ B l n (Y ) } Consequently, Φ is (l s,l )-summing with π (l s,l )(Φ) B. 4 Relations between the classes As in the case of -summing oerators we have the following inclusions. Proosition 4.1 Let X, Y, Z and W be Banach saces and 1 <q. Then (i) Π (l w,l s ) (X, L(Y,Z)) Π (l w q,l s q ) (X, L(Y,Z)). (ii) Π (l s,l )(L(X, Y ),Z) Π (l s q,l q)(l(x, Y ),Z). (iii) Π (l s,l s ) (L(X, Y ), L(Z, W)) Π (l s q,l s q ) (L(X, Y ), L(Z, W)). PROOF. (i) follows from Theorem 2.13 and Π (X, Z) Π q (X, Z). To see (ii) we take Φ Π (l s,l )(L(X, Y ),Z) and T k L(X, Y ) for k = 1,..., n. Let us write n Φ(T k) q = n Φ(β kt k ) where β k = Φ(T k ) q 16.

17 Alying Hölder s inequality with conjugate indices q and ) Φ(T k ) q (π (l s,l )(Φ)) su T k (x) β k This gives that q q ) /q (π (l s,l )(Φ)) su T k (x) q β q q k we get ) (q ) q ) /q ) (q ) = (π (l s,l )(Φ)) su T k (x) q Φ(T k ) q q. /q /q. Φ(T k ) q π(l s,l )(Φ) su T k (x) q (iii) now follows using Theorem 2.16 and (ii). Next we are going to see that, under some assumtions on the Banach saces, these classes coincide, at least for certain values of and q. Let us recall that some classical result, due to B. Maurey (see [10] or [3], Theorem 11.13), states that if Y has cotye 2 and 2 << then Π (X, Y )=Π 2 (X, Y ). Using Theorem 2.13 we get the following corollary. Corollary 4.2 Let X, Y, Z be Banach saces and 2 <<. Assume that Z has cotye 2, then Π (l w,l s ) (X, L(Y,Z))=Π (l w 2,l s 2 ) (X, L(Y,Z)). It is natural to ask whether there are generalizations in the framework of (l s,l )-summing oerators. Next result is the extension of Theorem in [6] to our setting. Theorem 4.3 Let X, Y, Z and W be Banach saces and 2 < <. Assume that Y has tye 2 and Z has cotye 2. Then Π (l s,l )(L(X, Y ),Z)=Π (l s 2,l 2 )(L(X, Y ),Z). 17

18 PROOF. Let Φ Π (l s,l )(L(X, Y ),Z) and T 1,...,T n be a finite sequence of oerators in L(X, Y ). Using that Z has cotye 2 one has Φ(T i ) 2 Z 2 C 2 (Z) 1 0 r i (t)φ(t i ) dt Z ( 1 ( ) C 2 (Z) Φ r i (t)t i dt 0 Z. Observe that n r i(t)t i is a simle function 2 n j=1 S jχ [ j 1 2 n, j 2 n ) oerators S j and then 1 0 ( ) Φ r i (t)t i Z 2n dt = n 2 j=1 Φ(S j ) Z (π (l s,l )(Φ)) 2 n su (π (l s,l )(Φ)) su n j=1 S j (x) Y for some r i (t)t i (x) dt. Y Hence, using Kahane s inequality and the tye 2 condition on Y, it yields ( 1 Φ(T i ) 2 2 Z π (l s,l )(Φ)C 2 (Z) su r i (t)t i (x) dt x B X 0 Y 1 π (l s,l )(Φ)C 2 (Z)K su r i (t)t i (x) dt Y 0 π (l s,l )(Φ)C 2 (Z)T 2 (Y )K su T i (x) 2 Y 2 and the roof is finished. Corollary 4.4 Let 2 <<, X, Y, Z and W be Banach saces. If Y has tye 2 and W has cotye 2, then Π (l s,l s )(L(X, Y ), L(Z, W))=Π (l s 2,l s 2 ) (L(X, Y ), L(Z, W)). 18

19 In articular we get the following alications to oerators acting on l w r (X) =L(l r,x). Corollary 4.5 Let X and Y be Banach saces of tye 2 and cotye 2 resectively. If 2 << and 1 r, q <, then (i) Π (l s,l )(l w r (X),Y)=Π (l s 2,l 2 )(l w r (X),Y). (ii) Π (l s,l s ) (l w r (X),l w q (Y ))=Π (l s 2,l s 2 ) (l w r (X),l w q (Y )). 5 Oen roblems Recall the domination theorem in the setting of (, Y )-summing oerators roved by S. Kisliakov. Theorem 5.1 (see [6] ) Let 1 < and X, Y and Z be Banach saces. An oerator T : X ˇ Y Z is (, Y )-summing if and only if there are a robability measure µ on (B X,σ(X,X)) and a constant C>0such that for all u X Y one has T (u) Z C u(x ) Y dµ(x ). B X Moreover π Y (T ) is the least of the constants verifying the revious estimate. Question 1. Let 1 < and X, Y and Z be Banach saces. Assume T : L(X,Y) Z is (l s,l )-summing oerator. Does there exist a robability measure µ on (B X,σ(X,X)) and a constant C>0such that T (u) Z C u(x ) Y dµ(x ) B X for all u L(X,Y)? The reader should be aware that the classical roofs can be reeated, under certain assumtions of reflexivity and searability on the saces X and Y. The difficulty aears when dealing with general Banach saces. Now let us oint out that the authors, relying uon Theorem 5.1, have been able to show the following result about comosition oerators for (, Y )- summing oerators. 19

20 Theorem 5.2 (see [1]) Let X, Y, Z and W be Banach saces and 1, q, s where 1 q = s 1. If T ΠY (X ˇ Y,Z) and S Π s (Z, W) then the oerator ST Π Y q (X ˇ Y,W) and π Y q (ST) π s (S) π Y (T ). Since we do not have at our disosal such a domination theorem in general we do not know the answer to the following question. Question 2. Let X, Y, Z and W be Banach saces and 1, q, r where 1 q = r 1. Let T Π (l s,l) (L(X, Y ),Z) and S Π r (Z, W). Does ST belong to Π (l s q,l q)(l(x, Y ),W)? Acknowledgments: We would like to thank Bernardo Cascales for ointing out the lack of measurability of certain functions used in a reliminary version and the referees for valuable comments. References [1] O. Blasco and T. Signes. Remarks on (, q, Y )-summing oerators. Submitted. [2] A. Defant and K. Floret. Tensor norms and oerator ideals. Noth- Holland, Amsterdam, [3] J. Diestel, H. Jarchow y A. Tonge, Absolutely Summing Oerators. Cambridge University Press, [4] N. Dinculeanu, Vector Measures. Pergamon Press, New York, [5] H. Jarchow, Locally convex saces. Teuber, Stuttgart, [6] S. V. Kislyakov, Absolutely summing oerators on the disk algebra, St. Petersburg Math. J. 3 (4) (1992), [7] J. Lindenstrauss and A. Pelczyński, Absolutely summing oerators in L -saces and their alications, Studia Math. 29 (1968),

21 [8] J. Lindenstrauss, L. Tzafriri, Classical Banach Saces I, Lecture Notes in Mathematics, vol. 338, Sringer-Verlag (1973) [9] J. Lindenstrauss, L. Tzafriri, Classical Banach Saces II, Sringer- Verlag, New York (1979). [10] B. Maurey, Théorèmes de factorisation our les oérateurs à valeurs dans les esaces L. Soc. Math. France, Astérisque 11, Paris [11] A. Pietsch, Oerator Ideals. Noth-Holland, Amsterdam, [12] G. Pisier, Factorization of linear oerators and geometry of Banach saces. Amer. Math. Soc. Vol 60, Providence,R.I [13] C. Swartz, Absolutely summing and dominated oerators on saces of vector-valued continuos functions, Trans. Amer. Math. Soc. 179 (1973), [14] P. Wojtaszczyk, Banach Saces for Analysts. Cambridge University Press, Cambridge, Address of the authors: Oscar Blasco Teresa Signes Deartamento de Análisis Matemático Deartamento de Matemática Alicada Facultad de Matemáticas Facultad de Informática Universidad de Valencia Universidad de Murcia Burjassot Esinardo Valencia (Sain) Murcia (Sain) 21