ON THE MULTILINEAR GENERALIZATIONS OF THE ... - Project Euclid

ROCKY MOUNTAIN JOURNAL OF MATHEMATICS Volume 37, Number 4, 2007

ON THE MULTILINEAR GENERALIZATIONS OF THE CONCEPT OF ABSOLUTELY SUMMING OPERATORS ERHAN C ¸ ALIS ¸ KAN AND DANIEL M. PELLEGRINO ABSTRACT. In this paper we investigate the several multilinear generalizations of the concept of absolutely summing operators and their connections. We also introduce the concept of p semi-integral mappings and establish the position of p semi-integral mappings with respect to the other classes.

1. Introduction and notation. The core of the theory of absolutely summing operators lies in the ideas of A. Grothendieck in the 1950s. Further work of Pietsch [23] and Lindenstrauss and Pelczy´ nski [9] clarified Grothendieck’s insights, and nowadays the ideal of absolutely summing operators is a central topic of investigation. For details on absolutely summing operators we refer to the book by Diestel-Jarchow-Tonge [7]. A natural question is how to extend the concept of absolutely summing operators to multilinear mappings and polynomials. A first light in this direction is the work by Alencar-Matos [1], where several classes of multilinear mappings between Banach spaces were investigated. Since then, just concerning the idea of lifting the ideal of absolutely summing operators to polynomials and multilinear mappings, several concepts have appeared and so far none of the definitions proposed appears as clearly better or more useful than the rest. However, there seems to be no effort in the direction of comparing all these different classes. The aim of this paper is to investigate the connections between these classes, to introduce the class of p semi-integral mappings and to establish the position of p semi-integral mappings with respect to the other concepts. 2000 AMS Mathematics Subject Classification. Primary 46G25, Secondary 46B15. The first author is supported by CNPq (Brazil) and the second author is partially supported by CNPq Grants 471054/2006-2 and 308084/2006-3 and FAPESQ/CNPq. Part of this paper is a portion of the second author’s doctoral thesis, written under supervision of Professor M.C. Matos. Received by the editors on Sept. 1, 2004, and in revised form on March 8, 2005. c Copyright 2007 Rocky Mountain Mathematics Consortium

1137

1138

E. C ¸ ALIS ¸ KAN AND D.M. PELLEGRINO

Throughout this paper the letters E, E1 , . . . , En , G1 , . . . , Gn, F, F0 represent Banach spaces, the symbol K represents the field of all scalars (complex or real), and N denotes the set of all positive integers. Given a natural number n ≥ 2, the Banach space of all continuous n-linear mappings from E1 × · · · × En into F endowed with the sup norm will be denoted by L(E1 , . . . , En ;F ). For i = 1, . . . , n, we denote (n) by Ψi : L(E1 , . . . , En ; F ) → L(Ei ; L(E1 , .[i] . ., En ; F )) the canonical isometric isomorphism (n)

[i]

Ψi (T )(xi )(x1 · · ·xn ) = T (x1 , . . . , xn ), where .[i] . . means that the ith coordinate is not involved. For p > 0, the linear space of all sequences (xj )∞ j=1 in E such that ∞ p 1/p = ( x ) < ∞ is denoted by l (xj )∞ j p (E). We represent j=1 p j=1 by lpw (E) the linear space of the sequences (xj )∞ j=1 in E such that ∈ l for every continuous linear functional ϕ : E → K (ϕ(xj ))∞ p j=1 ∞ and define .w,p in lpw (E) by (xj )∞ = sup (ϕ(x j ))j=1 p . ϕ∈BE j=1 w,p If p = ∞ we are restricted to the case of bounded sequences and in l∞ (E) we use the sup norm. One can verify that .p , respectively .w,p , is a p-norm in lp (E), respectively lpw (E), for p < 1 and a norm in lp (E), respectively lpw (E), for p ≥ 1. If K is a Hausdorff compact topological space, C(K) denotes the Banach space, under the supremum norm, of all continuous functions on K. We begin by presenting the several classes of multilinear mappings related to the concept of absolutely summing operators: • If p ≥ 1, T ∈ L(E1 , . . . , En ; F ) is said to be p-dominated (T ∈ Ld,p (E1 , . . . , En ; F )) if there exist C ≥ 0 and regular probability measures μj on the Borel σ-algebras B(BE ) of BEj endowed with the j

weak star topologies σ(Ej , Ej ), j = 1, . . . , n, such that T (x1 , . . . , xn ) ≤ C

n j=1

p

1/p

|ϕ (xj )| dμj (ϕ)

BE

j

for every xj ∈ Ej and j = 1, . . . , n. It is well known that T ∈ L(E1 , . . . , En ; F ) is p-dominated if and only if there exist Banach spaces G1 , . . . , Gn , absolutely p-summing linear operators uj ∈

1139

ABSOLUTELY SUMMING OPERATORS

L(Ej ; Gj ) and a continuous n-linear mapping R ∈ L(G1 , . . . , Gn ; F ) so that T = R ◦ (u1 , . . . , un ). • If p ≥ 1, T ∈ L(E1 , . . . , En ; F ) is of absolutely p-summing type (n) (T ∈ [Πas(p) ](E1 , . . . , En ; F )) if Ψj (T ) is absolutely p-summing for every j ∈ {1, . . . , n}. • If p ≥ 1, T ∈ L(E1 , . . . , En ; F ) is p semi-integral (T ∈ Lsi,p (E1 , . . . , En ; F )) if there exist C ≥ 0 and a regular probability measure μ on the Borel σ−algebra B(BE1 × · · · × BEn ) of BE1 × · · · × BEn endowed with the product of the weak star topologies σ(El , El ), l = 1, . . . , n, such that T (x1 , . . . , xn ) ≤C

1/p |ϕ1 (x1 ) · · · ϕn (xn )| dμ(ϕ1 , . . . , ϕn ) p

BE ×···×BE 1

n

for every xj ∈ Ej and j = 1, . . . , n. The infimum of the C defines a norm .si,p for the space of p semi-integral mappings. • If p ≥ 1, T : E1 × · · · × En → F is fully, or multiple, p-summing if there exists C > 0 such that

∞ j1 ,... ,jn =1

(1) T (xj1 , . . .

(n) , xjn )p

1/p ≤C

n

(k)

(xj )∞ j=1 w,p

k=1

(k)

w for every (xj )∞ j=1 ∈ lp (Ek ), k = 1, . . . , n. The space of all fully psumming n-linear mappings from E1 × · · · × En into F will be denoted by Lf as,p (E1 , . . . , En ; F ), and the infimum of the C for which the inequality always holds defines a norm .f as,p for Lf as,p (E1 , . . . , En ; F ).

• If p ≥ 1, T ∈ L(E1 , . . . , En ; F ) is strongly p-summing (T ∈ Lsas,p (E1 , . . . , En ; F )) if there exists C ≥ 0 and a regular probability measure μ on the Borel σ-algebra B(BL(E1 ,... ,En ;K) ) of BL(E1 ,... ,En ;K) with the weak star topology such that T (x1 , . . . , xn ) ≤ C

1/p |φ(x1 , . . . , xn )| dμ(φ) p

BL(E1 ,... ,En ;K)

for every xj ∈ Ej and j = 1, . . . , n.

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• If p, q1 , . . ., qn > 0, T ∈ L(E1 , . . ., En ; F ) is absolutely (p; q1 , . . ., qn )summing, or (p; q1 , . . . , qn )-summing, at the point (a1 , . . . , an ) in E1 × · · · × En when (1)

(n)

(T (a1 + xj , . . . , an + xj ) − T (a1 , . . . , an ))∞ j=1 ∈ lp (F ) (s)

w for every (xj )∞ In the case that T j=1 ∈ lqs (E), s = 1, . . . , n. is (p; q1 , . . . , qn )-summing at every (a1 , . . . , an ) ∈ E1 × · · · × En we say that T is (p; q1 , . . . , qn )-summing everywhere. Notation: ev Lev as(p,q1 ,... ,qn ) (E1 , . . . , En ; F ) or Las,p (E1 , . . . , En ; F ) if p = q1 = · · · = qn .

• If 1/p ≤ 1/q1 + · · · + 1/qn and T is (p; q1 , . . . , qn )-summing at (0, . . . , 0) ∈ E1 × · · · × En , we say that T is (p; q1 , . . . , qn )summing, and we write T ∈ Las(p,q1 ,... ,qn ) (E1 , . . . , En ; F ). When p = q1 = · · · = qn we write Las,p (E1 , . . . , En ; F ). It is well known that Ld,p (E1 , . . . , En ; F ) = Las((p/n),p,... ,p) (E1 , . . . , En ; F ). Except perhaps for the concept of p semi-integral mappings, all of the above concepts are well known and individually investigated. The p semi-integral mappings were introduced in [18] motivated by the work of Alencar-Matos [1]. Dominated mappings were first explored by Schneider [24] and Matos [11] and more recently in [3, 5, 14, 16]. Multilinear mappings of absolutely summing type are motivated by abstract methods of creating ideals and are explored in [18]. The ideal of fully summing multilinear mappings was introduced by Matos [12] and investigated by Souza [25] in her doctoral thesis under his supervision. It was also independently introduced by Bombal et al. (with a different name “multiple summing”) and developed in [2, 22]. The ideal of strongly summing multilinear mappings was introduced by Dimant [8] and the concept of absolutely summing multilinear mappings appears firstly in Alencar-Matos [1], Matos [11] and have been vastly studied (we mention [13, 16 19, 22], for example). In the next section we investigate the class of p semi-integral mappings and in the last section we study the connections between the classes previously introduced. 2. p semi-integral mappings. We begin with a characterization of p semi-integral mappings that will be useful in the subsequent section.

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Theorem 1. T ∈ Lsi,p (E1 , . . . , En ; F ) if and only if there exists C ≥ 0 such that (2.1)

m

1/p T (x1,j , . . . , xn,j )

p

j=1

≤C

sup

m

ϕl ∈BE ,l=1,... ,n j=1

1/p |ϕ1 (x1,j ) . . . ϕn (xn,j )|

p

l

for every m ∈ N, xl,j ∈ El with l = 1, . . . , n and j = 1, . . . , m. Moreover, the infimum of the C in (2.1) is T si,p . Proof. If T is p semi-integral, it is not hard to obtain (2.1). Conversely, suppose that (2.1) holds. The proof follows the idea of the case p = 1 in [1]. Define • Γ1 = {f ∈ C(BE1 × · · · × BEn ); f < C −p }. • Γ2 = co {f ∈ C(BE1 × · · · × BEn ); there are xl ∈ El , l = 1, . . . , n, so that T (x1 , . . . , xn ) = 1 and f (ϕ1 , . . . , ϕn ) = |ϕ1 (x1 ) · · · ϕ(xn )|p }. where co {.} denotes the convex hull. Let us show that Γ1 ∩ Γ2 = φ. m If h ∈ Γ2 , then h = m j=1 αj fj , αj > 0, j=1 αj = 1 and fj (ϕ1 , . . . , ϕn ) = |ϕ1 (x1,j ) · · · ϕn (xn,j )|p for every ϕl ∈ BEl . By hypothesis we have h =

m

sup

ϕl ∈BE ,l=1,... ,n j=1

1/p |ϕ1 (αj x1,j ) · · · ϕn (xn,j )|p

l

≥ C −p

m

1/p

(αj )p T (x1,j , . . . , xn,j )p = C −p .

j=1

Hence h ∈ / Γ1 . By the Hahn-Banach separation theorem, there exist λ > 0 and ψ ∈ C(BE1 × · · · × BEn ) so that ψ = 1 and

ψ(f ) < λ ≤ ψ(g)

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for every f ∈ Γ1 , g ∈ Γ2 . Since each f < 0 belongs to Γ1 , we have ψ(mf ) < λ for every m ∈ N. Thus ψ(f ) ≤ 0 and ψ is a positive functional and thus, by the Riesz representation theorem, there exists a regular probability measure μ, defined on the Borel sets of BE1 × · · · × BEn so that f dμ.

ψ(f ) = BE ×···×BE

n

1

Defining fm by fm = C −p − (1/m), we have fm ∈ Γ1 for every m ∈ N. Thus 1 ≤ λ for every m, fm dμ = C −p − m BE ×···×BE 1

n

and hence λ ≥ C −p . Therefore, if T (x1 , . . . , xn ) = 1, defining f (ϕ1 , . . . , ϕn ) := |ϕ1 (x1 ) · · · ϕ(xn )|p , we have f ∈ Γ2 and

f dμ = ψ(f ) ≥ C −p = C −p T (x1 , . . . , xn )

(2.2) BE ×···×BE 1

n

i.e., Cp BE ×···×BE 1

|ϕ1 (x1 ) · · · ϕ(xn )|p dμ ≥ T (x1 , . . . , xn ),

n

and, since T (x1 , . . . , xn ) = 1, we obtain T (x1 , . . . , xn ) ≤ C

1/p |ϕ1 (x1 ) · · · ϕ(xn )| dμ . p

BE ×···×BE

n

1

If T (x1 , . . . , xn ) = 0, it suffices to replace x1 by x1 T (x1 , . . . , xn )−1 in (2.2), and we deduce T (x1 , . . . , xn ) ≤ C

1/p |ϕ1 (x1 ) · · · ϕ(xn )| dμ . p

BE ×···×BE 1

n

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The preceding theorem has various straightforward consequences whose proof we omit: Proposition 1. (i) (Ideal property). If T ∈ Lsi,p (E1 , . . . , En ; F ), Ak ∈ L(Dk ; Ek ), k = 1, . . . , n, and S ∈ L(F ; G), then S ◦ T ◦ (A1 , . . . , An ) is p semi-integral and S ◦ T ◦ (A1 , . . . , An )si,p ≤ ST si,p

n

Ak .

k=1

(ii) If T ∈ Lsi,p (E1 , . . . , En ; F ) and i : F → F0 is an isometric embedding, then i ◦ T si,p = T si,p . (iii) If L(E1 , . . . , En ; F ) = Lsi,p (E1 , . . . , En ; F ), then L(Ej1 , . . . Ejn ; F ) = Lsi,p (Ej1 , . . . Ejn ; F ), for every j1 , . . . , jk in {1, . . . , n} with jr = js for r = s. For r ≥ 1 we have the following characterization of r semi-integral mappings defined in C(K1 ) × · · · × C(Kn ) whose proof is based on an argument used in [1]. Theorem 2. Let K1 , . . . , Kn be Hausdorff compact topological spaces. If T ∈ L(C(K1 ), . . . , C(Kn ); F ) and r ≥ 1, then the following conditions are equivalent: (i) T is r semi-integral. (ii) There exist C ≥ 0 and a regular probability measure μ on the Borel σ−algebra B(K1 × · · · × Kn ) of K1 × · · · × Kn such that T (f1 , . . . , fn ) ≤C

K1 ×···×Kn

1/r |f1 (x1 ) · · · fn (xn )|r dμ(x1 , . . . , xn )

for every fl ∈ C(Kl ) with l = 1, . . . , n.

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(iii) There exist D ≥ 0 such that m

1/r T (f1,j , . . . , fn,j )r

j=1

≤D

m

sup

xl ∈Kl ,l=1,... ,n j=1

1/r |f1,j (x1 ) · · · fn,j (xn )|r

for every m ∈ N, fl,j ∈ C(Kl ) with l = 1, . . . , n and j = 1, . . . , m. Proof. A modification on the proof of Theorem 1 shows the equivalence between (ii) and (iii). Hence we only need to prove the equivalence between (i) and (iii), which follows from Theorem 1 and from the identity m

sup

ϕl ∈BC(K

l

) ,l=1,... ,n

|ϕ1 (f1,j ) · · · ϕn (fn,j )|r

j=1

=

m

sup

xl ∈Kl ,l=1,... ,n j=1

|f1,j (x1 ) · · · fn,j (xn )|r

for every fl,j ∈ C(Kl ) with l = 1, . . . , n, j = 1, . . . , m, and m ∈ N. Note that, for xl ∈ Kl , l = 1, . . . , n, we have m

|f1,j (x1 ) · · · fn,j (xn )| = r

j=1

m

|δx1 (f1,j ) · · · δxn (fn,j )|r

j=1

where δxl ∈ BC(Kl ) is given by δxl (f ) = f (xl ), for every f ∈ C(Kl ), l = 1, . . . , n. Hence

sup

m

xl ∈Kl ,l=1,... ,n j=1

|f1,j (x1 ) · · · fn,j (xn )|r ≤

sup

ϕl ∈BC(K

l

) ,l=1,... ,n

m j=1

|ϕ1 (f1,j ) · · · ϕn (fn,j )|r .

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Now we show the converse. We have m

|ϕ1 (f1,j ) · · · ϕn (fn,j )|r

j=1

m

r−1

= sup

|ϕ1 (f1,j ) · · · ϕn (fn,j )| λj ϕ1 (f1,j ) · · · ϕn (fn,j )

|λj |=1 j=1

m

= sup

ϕ1 ⊗ · · · ⊗ ϕn λj |ϕ1 (f1,j ) · · · ϕn (fn,j )|r−1 f1,j ⊗ · · · ⊗ fn,j

|λj |=1

j=1

m ≤ ϕ1 ⊗ · · · ⊗ ϕn sup λj |ϕ1 (f1,j ) · · · ϕn (fn,j )|r−1 f1,j ⊗ · · · ⊗ fn,j |λj |=1

≤ ≤

sup

|λj |=1,xl ∈Kl ,l=1,... ,n j=1 m

sup

xl ∈Kl ,l=1,... ,n j=1

⎡

≤

j=1

m

λj |ϕ1 (f1,j ) · · · ϕn (fn,j )|r−1 f1,j (x1 ) · · · fn,j (xn )

sup

xl ∈Kl ,l=1,... ,n

⎣

|ϕ1 (f1,j ) · · · ϕn (fn,j )|r−1 |f1,j (x1 ) · · · fn,j (xn )|

m

r/r−1 |ϕ1 (f1,j ) · · · ϕn (fn,j )|r−1

j=1

(r−1)/r

⎤ 1/r m ⎦. × |f1,j (x1 ) · · · fn,j (xn )|r j=1

Hence, m

1−(r−1/r) |ϕ1 (f1,j ) · · · ϕn (fn,j )|

r

j=1

≤

sup

xl ∈Kl ,l=1,... ,n

m

1/r |f1,j (x1 ) · · · fn,j (xn )|r

,

j=1

and the proof is done. 3. Connections between the different classes. Recall that an n-linear mapping T is said to be completely continuous if it takes

1146


sequences weakly converging to zero into sequences norm converging to zero. It is well known that every absolutely p-summing operator is weakly compact and completely continuous. So, a natural question is to ask whether their multilinear generalizations still preserve these properties. Concerning completely continuous mappings, it is not hard to prove that every p semi-integral mapping is completely continuous. On the other hand, contrary to the linear case, the absolutely summing (and strongly summing) multilinear mappings are not completely continuous, in general. ∞For example, T : l2 × l2 → K given by ∞ , (y ) ) = T ((xj )∞ j j=1 j=1 j=1 xj yj is absolutely 1-summing and strongly 2-summing but fails to be completely continuous. In this section we will obtain certain connections between the different classes investigated in this paper and apply our results to give an alternative direct answer for a question (concerning weak compactness for strongly summing mappings) posed by Dimant [8] and recently answered in Carando-Dimant [6]. Theorem 3. Let E and F be Banach spaces. Then (i) [Πas(p) ](n E; F ) = Las(p;p,∞,... ,∞) (n E; F ) ∩ · · · ∩ Las(p;∞,... ,∞,p) × ( E; F ). n

(n)

(ii) If T ∈ Lsi,p (E1 , . . . , En ; F ), then Ψi (T ) ∈ Las,p (Ei; L(E1 , .[i] . ., (n) En ; F )) and Ψi (T )(x) is p semi-integral for every x in Ei . (iii) Ld,p (n E; F ) ⊂ Lsi,p (n E; F ) ⊂ [Πas(p) ](n E; F ). (iv) Lsi,p (n E; F ) ⊂ Ld,np (n E; F ). n n (v) Lsi,p (n E; F ) ⊂ Lf as,p (n E; F ) ⊂ Lev as,p ( E; F ) ⊂ Las,p ( E; F ).

(vi) Lsi,p (n E; F ) ⊂ Lsas,p (n E; F ). Proof. (i) The case n = 3 is illustrative. If T ∈ [Πas(p) ](3 E; F ) and w ∞ ∞ (xj )∞ j=1 ∈ lp (E), (yj )j=1 ∈ l∞ (E), (zj )j=1 ∈ l∞ (E) are nonidentically

1147


null, we have 1/p ∞ p T (xj , yj , zj ) j=1

∞ p 1/p yj zj ∞ ∞ = (yj )j=1 ∞ (zj )j=1 ∞ T xj , (y )∞ , (z )∞ j j=1 ∞ j j=1 ∞ j=1 ∞ p 1/p (3) ∞ Ψ (z ≤ (yj )∞ ) (T )(x ) j j=1 ∞ j j=1 ∞ 1 (3) ≤ Ψ1 (T )

j=1

as,p

∞ ∞ (yj )∞ j=1 ∞ (zj )j=1 ∞ (xj )j=1 w,p ,

and thus T ∈ Las(p;p,∞,∞) (3 E; F ). The other cases are similar. The converse is not difficult. (ii) Let T ∈ Lsi,p (E1 , . . . , En ; F ). Fix a natural number i ∈ {1, . . . , n} and let xj ∈ Ei , j = 1, . . . , m. For ε > 0, there exist xk,j ∈ BEk , k = 1, .[i] . ., n such that Ψni (T )(xj )p ≤ Ψni (T )(xj )(x1,j , .[i] . ., xn,j )p +

ε . m

Hence m

Ψni (T )(xj )p

j=1

≤ε+

m

Ψni (T )(xj )(x1,j , .[i] . ., xn,j )p

j=1

≤ ε + T psi,p

sup

m

ϕl ∈BE ,l=1,... ,n j=1

[i]

|ϕi (xj )ϕ1 (x1,j )· · ·ϕn (xn,j )|p

l

≤ ε + T psi,p sup

m

ϕi ∈BE j=1 i

|ϕi (xj )|p .

Since ε is arbitrary, we have m j=1

p Ψni (T )(xj )p ≤ T psi,p (xj )m j=1 w,p .

1148


(n)

Hence Ψi (T ) ∈ Las,p (Ei; L(E1 , .[i] . ., En ; F )) for every i = 1, . . . , n. (n)

Now we show that Ψi (T )(x) is p semi-integral for every x ∈ Ei , i = 1, . . . , n. Since m

(n) Ψi (T )(x)(x1,j , .[i] . ., xn,j )p

1/p

j=1

=

m

1/p T (x1,j , . . . , x, . . . , xn,j )

p

j=1

≤ T si,p

m

sup

ϕl ∈BE ,l=1,... ,n j=1

≤ T si,p x

1/p |ϕ1 (x1,j ) · · · ϕi (x) · · · ϕn (xn,j )|p

l

m

sup

[i] ϕl ∈BE ,l=1,...,n j=1

1/p

[i]

|ϕ1 (x1,j )· · ·ϕn (xn,j )|

p

,

l

(n)

we get Ψi (T )(x)si,p ≤ T si,p x, for every x ∈ Ei , i = 1, . . . , n. (iii) The proof that Lsi,p (n E; F ) ⊂ [Πas(p) ](n E; F ) is a consequence of (ii). If T ∈ Ld,p (n E; F ), then

1/p

T (x1 , . . . , xn ) ≤ C

|ϕ1 (x1 )| dμ1 (ϕ1 ) p

BE

1

1/p

···

|ϕn (xn )| dμn (ϕn ) p

BE

n

=C BE ×···×BE 1

|ϕ1 (x1 )

n

· · · ϕn (xn )| d(μ1 ⊗ · · · ⊗ μn )(ϕ1 , . . . , ϕn ) p

and hence T ∈ Lsi,p (n E; F ).

1/p ,

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(iv) Suppose that T is p semi-integral. Then, from Theorem 1 we deduce m

1/p T (x1,j , . . . , xn,j )

p

j=1

≤C

m

sup

ϕl ∈BE ,l=1,... ,n j=1 l

≤C

⎡ ⎣

sup

1/p |ϕ1 (x1,j ) · · · ϕn (xn,j )|

p

m

ϕl ∈BE ,l=1,... ,n

1/np |ϕ1 (x1,j )|

np

j=1

l

···

m

1/np |ϕn (xn,j )|np

⎤ ⎦

j=1

m = C (x1,j )m j=1 w,np · · · (xn,j )j=1 w,np . (v) If T ∈ Lsi,p (n E; F ), then m

T (x1,j1 , . . . , xn,jn )p

j1 ,...jn =1

≤C

m

p

BE ×···×BE j ,...j =1 n n 1 1

|ϕ1 (x1,j1 ) · · · ϕn (xn,jn )|p dμ(ϕ1 , . . . , ϕn ).

So

1/p

m

T (x1,j1 , . . . , xn,jn )

p

j1 ,...jn =1

≤C

m

BE ×···×BE j ,...j =1 n n 1 1

|ϕ1 (x1,j1 ) · · · ϕn (xn,jn )|p dμ 1/p × (ϕ1 , . . . , ϕn )

≤C

sup

ϕl ∈BE ,l=1,... ,n l

m j1 ,...jn =1

1/p |ϕ1 (x1,j1 ) · · · ϕn (xn,jn )|

p

1150

=C


sup

ϕl ∈BE ,l=1,... ,n

⎡ 1/p m ⎣ |ϕ1 (x1,j1 )|p j1 =1

l

···

m

1/p |ϕn (xn,jn )|p

⎤ ⎦,

jn =1

and thus T ∈ Lf as,p (n E; F ). Now let us consider T ∈ Lf as,p (n E; F ). ∞ The case n = 2 is illustrative and indicates the proof. If (xj )∞ j=1 , (yj )j=1 w ∈ lp (E), we have ∞

1/p T (a + xj , b + yj ) − T (a, b)p

j=1

≤

∞

1/p T (a, yj )p

+

j=1

+

∞

1/p

≤

1/p

T (zk , yj )

p

+

j,k=1

+

1/p T (xj , b)p

j=1

T (xj , yj )p

j=1

∞

∞

∞

∞

T (xj , wk )

j,k=1

1/p T (xj , yk )

1/p p

< ∞,

p

j,k=1 ∞ where (zk )∞ k=1 = (a, 0, 0, . . . ) and (wk )k=1 = (b, 0, 0, . . . ).

The proof of (vi) is easy and we omit it. Remark 1. Obviously, each one of the assertions in Theorem 3 holds for spaces E1 , . . . , En instead of E, . . . , E. The inclusion Lsi,1 (E1 , . . . , En ; F ) ⊂ Lsas,1 (E1 , . . . , En ; F ) is sometimes strict. In fact, if T : l1 × l1 → K is given by ∞ T ((xi )∞ j=1 , (yj )j=1 )

=

∞ j=1

yj

j

xk ,

k=1

then T fails to be semi-integral, see [1], but T is obviously strongly 1-summing.

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The inclusion Lsi,1 (E1 , . . . , En ; F ) ⊂ Lf as,1 (E1 , . . . , En ; F ) is also sometimes strict, since L(l2 , l1 ; K) = Lf as,1 (l2 , l1 ; K) [20] and L(l2 , l1 ; K) = Lsi,1 (l2 , l1 ; K). In fact, if we had L(l2 , l1 ; K) = Lsi,1 (l2 , l1 ; K), by Theorem 3 (ii) we would have L(l2 , l∞ ) = Las,1 (l2 , l∞ ), and it is a contradiction. The inclusion Lf as,1 (E1 , . . . , En ; F ) ⊂ Las,1 (E1 , . . . , En ; F ) is also strict, see [12]. In general Lsas,p (E1 , . . . , En ; F ) is not contained in Las,p (E1 , . . . , En ; F ) and Las,p (E1 , . . . , En ; F ) is not contained in Lsas,p (E1 , . . . , En ; F ). In fact, Las,1 (2 l1 ; l1 ) = L(2 l1 ; l1 ) and Lsas,1 (2 l1 ; l1 ) = L(2 l1 ; l1 ). On the other hand, Lsas,2 (2 l2 ; K) = L(2 l2 ; K) and Las,2 (2 l2 ; K) = L(2 l2 ; K). n ∞ In [4], Botelho proves that Pn : l1 → l1 : Pn ((αi )∞ i=1 ) = ((αi ) )i=1 is n-dominated and is not weakly compact. The same occurs with the symmetric n-linear mapping associated to P .

The question, “Is every strongly p-summing n-linear mapping weakly compact?” appears in [8] and was recently answered by CarandoDimant in [6]. However, by Theorem 3, since Ld,p (E1 , . . . , En ; F ) ⊂ Lsas,p (E1 , . . . , En ; F ), one can see that Botelho’s counterexample is a direct answer to this question. The next result shows that the spaces of semi-integral and dominated mappings coincide in some situations. Firstly, let us recall the concept of cotype: Let 2 ≤ q ≤ ∞ and (rj )∞ j=1 be Rademacher functions. We say that E has cotype q if there exists C ≥ 0 such that for any choice of k ∈ N and x1 , . . . , xk ∈ E we have k

1/q xj q

≤C

0

j=1

In the case q = ∞, we replace (

k

1

k

rj (t)xj 2 dt

1/2 .

j=1

q 1/q j=1 xj )

by max{xj ; 1 ≤ j ≤ k}.

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Theorem 4. If E has cotype 2, then Lsi,1 (2 E; F ) = Ld,1 (2 E; F ) for every F . Proof. If E has cotype 2, we know that Las,1 (E; F ) = Las,2 (E; F ) for every Banach space F . Thus, if T ∈ Ld,2 (2 E; F ), then T = R ◦ (u1 , u2 ), with R ∈ L(2 G; F ) and u1 , u2 ∈ Las,2 (E; G) = Las,1 (E; G). Hence, T ∈ Ld,1 (2 E; F ) and thus Ld,1 (2 E; F ) = Ld,2 (2 E; F ) for every F . Since Ld,1 (2 E; F ) ⊂ Lsi,1 (2 E; F ) ⊂ Ld,2 (2 E; F ), we thus have Lsi,1 (2 E; F ) = Ld,1 (2 E; F ). Remark 2. We can find Banach spaces E and F so that Ld,1 (2 E; F ) [Πas(1) ](2 E; F ). The following example is suggested by M.C. Matos. ∞ Consider T : l2 × l2 → K given by T (x, y) = j=1 (1/j α )xj yj with α = 1/2 + ε and ε ∈ ]0, 1/2[. We will show that T ∈ [Πas(1) ](2 l2 ; K) \ Ld,1 (2 E; F ). Since ⎤ ⎡ ⎡ ⎤ 2 1/2 2 2 m m m 1 1 ⎦ ≥⎣ ⎦ = m2−α , T (ej , ej )1/2 =⎣ α α/2 j m j=1 j=1 j=1 if we had

m

T (ej , ej )1/2

2

2 ≤ C (ej )m j=1 w,1 ,

j=1

for every m, we would obtain m2−α ≤ C(m1/2 )2 = Cm, a contradiction. (2)

In order to prove that Ψ1 (T ) ∈ Las,1 (l2 ; l2 ), observe that ∞ 1 (2) Ψ1 (T )((xj )∞ ) = x . j j=1 jα j=1 (2)

Now it suffices to show that Ψ1 (T ) is a Hilbert-Schmidt operator, see [15]. But it is easy to check, since 2 ∞ ∞ 2 1 (2) < ∞. Ψ1 (T )(ek ) = kα l2 k=1

k=1


1153

Hence the inclusion is strict. Since l2 has cotype 2, Theorem 4 yields that Lsi,1 (2 l2 ; K) = Ld,1 (2 l2 ; K) [Πas(1) ](2 l2 ; K).

Acknowledgments. The authors thank Professor Matos for introducing the subject and for valuable suggestions. The authors also thank the referee for detecting several misprints and mistakes on the original version. Note added in proof. A recent paper [21] due to David Pérez-Garc´ıa presents nice information comparing the classes of multiple (fully) summing, dominated and absolutely summing multilinear mappings. REFERENCES 1. R. Alencar and M.C. Matos, Some classes of multilinear mappings between Banach spaces, Publ. Dep. An´ al. Mat. Univ. Complutense Madrid, Sect. 1, no. 12, 1989. 2. F. Bombal, D. Pérez-Garc´ıa and I. Villanueva, Multilinear extensions of a Grothendieck’s theorem, Quart. J. Math. 55 (2004), 441 450. 3. G. Botelho, Cotype and absolutely summing multilinear mappings and homogeneous polynomials, Proc. Roy. Irish Acad. Sect. A 97 (1997), 145 153. 4. , Weakly compact and absolutely summing polynomials, J. Math. Anal. Appl. 265 (2002), 458 462. 5. G. Botelho and D.M. Pellegrino, Dominated polynomials in Lp -spaces, Arch. Math. 83 (2004), 364 370. 6. D. Carando and V. Dimant, On summability of bilinear operators, Math. Nachr. 259 (2003), 3 11. 7. J. Diestel, H. Jarchow and A. Tonge, Absolutely summing operators, Cambridge Stud. Adv. Math., vol. 43, Cambridge Univ. Press, Cambridge 1995. 8. V. Dimant, Strongly p-summing multilinear mappings, J. Math. Anal. Appl. 278 (2003), 182 193. 9. J. Lindenstrauss and A. Pelczy´ nski, Absolutely summing operators in Lp -spaces and their applications, Studia Math. 29 (1968), 275 324. 10. M.C. Matos, On multilinear mappings of nuclear type, Rev. Mat. Univ. Complut. Madrid 6 (1993), 61 81. , Absolutely summing holomorphic mappings, An. Acad. Brasil. Ciênc. 11. 68 (1996), 1 13. 12. , Fully absolutely summing mappings and Hilbert Schmidt operators, Collect. Math. 54 (2003), 111 136.

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Yild´ız Technical University, Faculty of Science and Letters, Department of Mathematics, 34210 Davutpasa, Esenler, Ístanbul, Turkey E-mail address: [email protected] ´ tica, UFPB, Joa ˜ o Pessoa, PB, 58.051-900, Brazil Depto. de Matema E-mail address: [email protected]