Diagonal Multilinear Operators on K\" othe Sequence Spaces

DIAGONAL MULTILINEAR OPERATORS ON KÖTHE SEQUENCE SPACES

arXiv:1706.04901v1 [math.FA] 15 Jun 2017

VERÓNICA DIMANT AND ROMÁN VILLAFAÑE Abstract. We analyze the interplay between maximal/minimal/adjoint ideals of multilinear operators (between sequence spaces) and their associated Köthe sequence spaces. We establish relationships with spaces of multipliers and apply these results to describe diagonal multilinear operators from Lorentz sequence spaces. We also deﬁne and study some properties of the ideal of (E, p)-dominated multilinear mappings, a natural extension of the linear ideal of absolutely (E, p)-summing operators.

Introduction Trying to describe the connections between different ideals of linear operators (and the internal structure of them), it began, in the 70’s, the study of diagonal linear operators on ℓp spaces with the work of Carl [6], König [20] and Pietsch [23]. By means of limit orders they have compared different ideals of linear operators and described their diagonal elements. Next, this research continued in the context of Köthe sequence spaces, leading to the so called multipliers. The concept of ideal of multilinear operators was also introduced by Pietsch in [24] and it has been developed by several authors since then. Even if the multilinear theory has its roots in the linear one, it had its own development that led to different situations involving new interesting techniques. In [14, 15], general results about ideals of multilinear mappings are presented. The concept of limit order for multilinear forms was introduced in [3] and was extended for general sequence spaces in [4]. There, it was defined the Köthe sequence space associated to an ideal of multilinear forms acting on Köthe sequence spaces. Later, in [5], this kind of study reached vector valued multilinear ideals between ℓp -spaces. Here, we propose a more general approach of the relationship between ideals of multilinear operators (acting on Köthe sequence spaces) and their respective associated sequence spaces. In particular, we analyze if for a maximal (minimal) ideal its associated sequence space is also maximal (minimal). In addition, we relate the sequence spaces associated to an ideal and its adjoint. 2010 Mathematics Subject Classification. 46A45, 47L22,47H60. Key words and phrases. Multilinear ideals, Köthe sequence spaces, Diagonal multilinear operators. Both authors were partially supported by CONICET PIP 2014-0483 and ANPCyT PICT 2015-2299. The second author was also partially supported by UBACyT 20020130100474 BA. 1

2

VERÓNICA DIMANT AND ROMÁN VILLAFAÑE

The spaces of multipliers appear to give us new descriptions of our sequence spaces associated to multilinear ideals. As an application, we can characterize diagonal multilinear operators from Lorentz sequence spaces. In the final section, we define the ideal of (E, p)-dominated multilinear mappings, as a natural extension of the linear ideal of absolutely (E, p)-summing operators. We obtain some properties of this multilinear ideal by means of our previous results on associated sequence spaces.

1. Preliminaries Throughout the paper we will use standard notation of the Banach space theory. We will consider complex Banach spaces E, F, . . . and their duals will be denoted by E ′ , F ′ , . . .. We 1 will write E = F if they are topologically isomorphic and E = F if they are isometrically ≤1

1

isomorphic. The symbol ֒→ means an isometric injection and the symbol ֒→ means a norm one inclusion (not necessarily isometric). Sequences of complex numbers will be denoted by x = (x(k))∞ k=1 , where each x(k) ∈ C. By a Köthe sequence space (also known as Banach sequence space) we mean a Banach ≤1

≤1

space E ⊆ CN such that ℓ1 ֒→ E ֒→ ℓ∞ and with the normal property: if x ∈ CN and y ∈ E satisfy |x(k)| ≤ |y(k)| for all k ∈ N then x ∈ E and kxk ≤ kyk. Note that in a Köthe sequence space E, given x ∈ E and a sequence of complex numbers s with |s(k)| = 1 for all k ∈ N, we should have s · x ∈ E and ks · xk = kxk (where the product is coordinatewise). For each N ∈ N, we consider the N-dimensional truncation EN := span{e1 , . . . , eN } (where en denotes the n-th canonical unit vector: en (k) = δn,k for all k) and we denote by E0 the space of sequences in E that are all 0 except for a finite number of coordinates. The canonical inclusion and projection will be denoted by 1 iN : EN ֒→ E and πN : E ։ EN . The Köthe dual of a Köthe sequence space E, defined as ( ) X E × := z ∈ CN : |z(j) · x(j)| < ∞ for all x ∈ E , j∈N

is a Köthe sequence space with the norm kzkE × := sup

X

kxkE ≤1 j∈N

|z(j) · x(j)| = sup kz · xkℓ1 . kxkE ≤1

It is well known (see, for example, [2, Lemma 2.8]) that z ∈ E × if and only if x(j) converges for all x ∈ E. Also, X × z(j) · x(j) . kzkE = sup kxkE ≤1

j∈N

P

j∈N

z(j) ·


3

1

Note that (EN )′ = (E × )N holds for every N. In the same way, we can considerate 1 (E × )× = E ×× and we say that E is Köthe reflexive if E ×× = E. Following [21, 1.d], a Köthe sequence space E is said to be r-convex (with 1 ≤ r < ∞) if there exists a constant κ > 0 such that for any choice x1 , . . . , xm ∈ E we have

m X 1/r m 1/r ∞ X

r r

≤κ

kxj kE . |xj (k)|

j=1

k=1 E

j=1

We denote by M(r) (E) the smallest constant which satisfies the inequality. The minimal kernel of a Köthe sequence space E is defined as the set E min := x ∈ ℓ∞ : x = y · z with y ∈ E and z ∈ c0

which is also a Köthe sequence space if we endow it with the norm kxkE min = inf kykE · kzkℓ∞ : x = y · z with y ∈ E and z ∈ c0 .

The maximal hull of a Köthe sequence space E is defined as the set E max := x ∈ ℓ∞ : x · z ∈ E for all z ∈ c0 ,

which results a Köthe sequence space if the norm is given by kxkE max = sup kx · ykE . y∈Bc0

1

1

A Köthe sequence space E is said to be maximal if E = E max and minimal if E = E min . For example, a Köthe dual E × is always maximal. For a detailed study and general facts about Köthe sequence spaces, see [22, 21]. The space of continuous linear operators between two Banach spaces E and F will be denoted by L(E; F ) and the space of continuous n-linear mappings from E1 × · · · × En to F by L(E1 , . . . , En ; F ). This is a Banach space with the usual sup norm, given by kT k := sup {kT (x1 , . . . , xn )kF : kxi kEi ≤ 1 , i = 1, . . . n}. If E1 = · · · = En = E we will write L(n E; F ) and whenever F = C we will simply write L(E1 , . . . , En ) or L(n E). Ideals of multilinear forms and multilinear operators were introduced by Pietsch in [24]. Let us recall the definition. An ideal of multilinear operators A is a subclass of L, the class continuous multilinear operators, such that, for any Banach spaces E1 , . . . , En and F the set A(E1 , . . . , En ; F ) = A ∩ L(E1 , . . . , En ; F ) satisfies (1) If S, T ∈ A(E1 , . . . , En ; F ), then S + T ∈ A(E1 , . . . , En ; F ). (2) If T ∈ A(E1 , . . . , En ; F ) and Bi ∈ L(Gi , Ei ) for i = 1, . . . , n and A ∈ L(F ; H), then A ◦ T ◦ (B1 , . . . Bn ) ∈ A(G1 , . . . , Gn ; H).

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(3) The mapping (x1 , . . . , xn ) 7→ γ1 (x1 ) · · · γn (xn ) · f belongs to A(E1 , . . . , En ; F ) for any γ1 ∈ E1′ , . . . , γn ∈ En′ and f ∈ F . An ideal of multilinear operators is called normed if for each E1 , . . . , En and F there is a norm k · kA(E1 ,...,En;F ) in A(E1 , . . . , En ; F ) such that (1) k(x1 , . . . , xn ) 7→ γ1 (x1 ) · · · γn (xn ) · f kA(E1 ,...,En ;F ) = kγ1k · · · kγn k · kf k. (2) kA ◦ T ◦ (B1 , . . . , Bn )kA(G1 ,...,Gn ;H) ≤ kAk · kT kA(E1 ,...,En ;F ) · kB1 k · · · kBn k. If A(E1 , . . . , En ; F ) is complete for every Banach spaces E1 , . . . , En , F we say that A is a Banach ideal of multilinear operators (or just, a Banach multilinear ideal). The minimal kernel of A is defined as the composition ideal Amin := F ◦ A ◦ (F, . . . , F), where F stands for the ideal of approximable operators (i. e. the closure of the ideal of finite rank linear operators). In other words, a multilinear operator T belongs to Amin (E1 , . . . , En ; F ) if it admits a factorization (1)

E1 × · · · × En

T

/

FO

(B1 ,...,Bn )

A

X1 × · · · × Xn

S

/

Y

where A, B1 , . . . , Bn ∈ F and S ∈ A(X1 , . . . , Xn ; Y ). The Amin -norm of T is given by kT kAmin := inf{kAk · kSkA · kB1 k · · · kBn k}, where the infimum runs over all possible factorizations as in (1). Amin is the smallest Banach multilinear ideal whose norm coincides with k·kA over finite dimensional spaces. This and other properties of Amin can be found in 1 [12]. An ideal of multilinear operators is said to be minimal if (A, k·kA ) = (Amin , k·kAmin ). If A is a normed ideal of n-linear operators, the maximal hull Amax of A is defined as the class of all n-linear operators T such that

X1 Xn ) : M ∈ F IN(X ), L ∈ COF IN(Y ) kT kAmax := sup QYL ◦ T ◦ (IM , . . . , I i i Mn 1 A X is finite, where IM : M → X is the inclusion from M into X, QYL : Y → Y /L is the projection of Y over Y /L and F IN(X) (COF IN(X)) represents the class of subspaces of X of finite dimension (codimension). Amax is always complete and it is the largest ideal whose norm coincides with k · kA over finite dimensional spaces. A normed ideal A 1 is called maximal if (A, k · kA ) = (Amax , k · kAmax ). If A is an ideal of multilinear operators, its associated tensor norm is the unique finitely generated tensor norm α, of order n + 1, satisfying 1

A(M1 , . . . , Mn ; N) = (M1′ ⊗ · · · ⊗ Mn′ ⊗ N; α) for every finite dimensional spaces M1 , . . . , Mn , N. In that case we write A ∼ α. A detailed study of tensor norms and their relationship with linear/multilinear ideals can be found in [8, 12, 13, 14, 15]. Note that A, Amax and Amin have the same associated tensor norm since they coincide isometrically on finite dimensional spaces.


5

Given a normed ideal A associated to a finitely generated tensor norm α, its adjoint ideal A∗ is defined by ′ A∗ (E1 , . . . , En ; F ) := E1 ⊗ · · · ⊗ En ⊗ F ′ , α ∩ L(E1 , . . . , En ; F ).

The adjoint ideal is called dual ideal in [12]. The tensor norm associated to A∗ is denoted by α′ . It is well known, by the representation theorem for maximal ideals [15, Section 1 4.5], that A∗ is always maximal and A∗∗ = Amax . Recall that a multilinear operator T ∈ L(E1 , . . . , En ; F ) is (Grothendieck) integral if there exists a regular F ′′ -valued Borel measure µ, of bounded variation on (BE1′ × · · · × BEn′ , w ∗ ) such that Z T (x1 , . . . , xn ) = (x′1 (x1 )) · · · (x′n (xn )) dµ(x′1 , . . . , x′n ) BE ′ ×···×BE ′ 1

n

for every xk ∈ Ek . The space of Grothendieck integral n-linear operators is denoted by I(E1 , . . . , En ; F ) and the integral norm of a multilinear operator T is defined as inf{kµk}, where the infimum runs over all the measures µ representing T . This ideal is maximal and its adjoint is the ideal L of all continuous multilinear mappings. 2. Interplay between an ideal and its associated sequence space For Köthe sequence spaces E and F , an n-linear operator T ∈ L(n E; F ) is said to be diagonal if there exists a bounded sequence α = (α(k))k ∈ CN such that for all x1 , . . . , xn ∈ E we can write X α(k) · x1 (k) · · · xn (k) · ek . T (x1 , . . . , xn ) = α · x1 · · · xn = k∈N

In this case, we say that T is the diagonal multilinear operator associated with α and we note it Tα . Given A, an ideal of multilinear operators, we define the sequence space associated with A as ℓn (A; E, F ) := α ∈ ℓ∞ : Tα ∈ A(n E; F ) .

This is a sequence space endowed with the norm kαkℓn(A;E,F ) = kTα kA(n E;F ). When F = C we simply write ℓn (A; E). The following finite-dimensional identifications are easy to check. They will enable us to prove a duality result next. 1

1

1

(2)

×× ℓn (A; E, F )N = ℓn (A; EN , FN ) = ℓn (A; EN , FN×× ) = ℓn (A; E ×× , F ×× )N .

(3)

ℓn (A; EN , FN ) = ℓn (Amax ; EN , FN ) = ℓn (Amin ; EN , FN ).

1

1

Our aim is to analyze first the relationship between minimal or maximal ideals with their respective associated sequence spaces and later the interplay between the sequence space associated with an ideal and its adjoint.

6


In [4, Proposition 5.5 and 5.6] it is proved that if A is a maximal ideal of multilinear 1 1 forms (scalar valued multilinear operators), then ℓn (A; E) = ℓn (A; E ×× ) and ℓn (A; E) = ℓn (A∗ ; E × )× . In both cases the key of the proofs is the use of [4, Lemma 5.4], which is a version of the Density Lemma [8, 13.4] for diagonal multilinear forms. So, we begin by proving a new version of this Lemma in our vector-valued context and then we can establish some similar results to those given above. Lemma 2.1. Let A be a maximal ideal of n-linear operators and let E and F be Köthe sequence spaces. For a sequence α, suppose that there exists a constant C > 0 such that the projection πN (α) satisfies kπN (α)kℓn (A;EN ,F ) ≤ C for all n ∈ N. Then, α ∈ ℓn (A; E, F ×× ) and kαkℓn (A;E,F ××) ≤ C. In other words, if TπN (α) ∈ A(n EN ; F ) has A-norm less than or equal to C for all n ∈ N then Tα ∈ A(n E; F ×× ) with A-norm less than or equal to C. Proof. Since A is maximal, by [15, Theorem 4.5] there exists a finitely generated tensor 1 norm ν of order n + 1 such that A(n E; F ′′ ) = (⊗n E ⊗ F ′ ; ν)′ . Then, the ball BA(n E;F ′′) is weak-star compact. Thus, the set JF ◦ TπN (α) ◦ (πN , . . . , πN ) N , which is contained in the ball C · BA(n E;F ′′) , has a weak-star accumulation point Φ ∈ C · BA(n E;F ′′) . This mapping should satisfy Φ(x1 , . . . , xn )(e′k ) = α(k) · x1 (k) · · · xn (k), for all x1 , . . . , xn ∈ E, e′k ∈ F ′ . On the other hand, the canonical mapping ξ : F ′′ → F ×× is well defined and has norm less than or equal to 1. Hence, ξ ◦ Φ belongs to C · BA(n E;F ×× ) and ξ ◦ Φ(x1 , . . . , xn ) = (Φ(x1 , . . . , xn )(e′k ))k = (α(k) · x1 (k) · · · xn (k))k . This says that ξ ◦ Φ coincides with the mapping Tα . In consequence, Tα ∈ A(n E; F ×× ) with kTα kA(n E,F ××) ≤ C. In particular, if F is maximal, given a diagonal multilinear operator Tα : E0 ×· · ·×E0 → F such that their truncated operators satisfy kTπN (α) kA(n EN ;F ) ≤ C for all N ∈ N, it follows that Tα ∈ A(n E; F ) with kTα kA(n E;F ) ≤ C. In order to prove the following result, recall the well known characterization of the maximal hull of a sequence space: x ∈ E max iff supN ∈N kπN (x)kE is finite, and the norm is given by this supremum. In other words, to ensure that a sequence space E is maximal it is enough to show that if kπN (x)kE ≤ C for all N ∈ N, then x ∈ E with kxkE ≤ C. Proposition 2.2. Let A be an ideal of n-linear operators and let E and F be Köthe sequence spaces. 1

(i) If A is maximal, then ℓn (A; E, F ×× ) is a maximal sequence space and ℓn (A; E, F ×× ) = ℓn (A; E ×× , F ×× ). 1 (ii) ℓn (A; E, F )max = ℓn (Amax ; E, F ×× ). Proof. (i) Suppose that kπN (α)kℓn (A;E,F ×× ) ≤ C for all N ∈ N. By identity (2), we have that kπN (α)kℓn (A;EN ,F ×× ) = kπN (α)kℓn (A;E,F ××) ≤ C. Then, by Lemma 2.1, α ∈


7

ℓn (A; E, F ×× ) with kαkℓn (A;E,F ××) ≤ C. So, ℓn (A; E, F ×× ) is maximal and the same is true for ℓn (A; E ×× , F ×× ). Again, identity (2), assures, for each sequence α, that 1 kπN (α)kℓn (A;E,F ××) = kπN (α)kℓn (A;E ×× ,F ×× ) , for all N ∈ N. Therefore, ℓn (A; E, F ×× ) = ℓn (A; E ×× , F ×× ). (ii) For a sequence α, identities (2) and (3) say that kπN (α)kℓn (A;E,F ) = kπN (α)kℓn (Amax ;E,F ××) . 1

1

Then, ℓn (A; E, F )max = ℓn (Amax ; E, F ×× )max . Also, by (i), ℓn (Amax ; E, F ×× )max = ℓn (Amax ; E, F ×× ), which completes the proof. Remark 2.3. By the previous proposition, if both the ideal A and the sequence space F are maximal, then the sequence space ℓn (A; E, F ) is maximal. Note that the condition over F is necessary for ℓn (A; E, F ) to be maximal. Indeed, if A = L, E = ℓ∞ and F = c0 , it follows that ℓn (L; ℓ∞ , c0 ) = c0 , which obviously is not a maximal sequence space. Now, we turn to look into the minimal hull. Recall that a sequence space E is minimal E iff for all x ∈ E, k(πNE − IE )(x)kE → 0, or equivalently, πN tends to IE over compact sets. Proposition 2.4. Let A be an ideal of n-linear operators and let E and F be Köthe sequence spaces. (i) If A is a minimal ideal and F is a minimal sequence space, then ℓn (A; E, F ) is a minimal sequence space. 1 (ii) ℓn (A; E, F )min = ℓn (Amin ; E, F min ). Proof. (i) Let α ∈ ℓn (A; E, F ). Since A is minimal, there exist mappings A1 , . . . , An , B ∈ F and S ∈ A(X1 , . . . , Xn ; Y ) such that Tα = B ◦ S ◦ (A1 , . . . , An ). Then, (πNF − IF ) ◦ Tα = (πNF − IF ) ◦ B ◦ S ◦ (A1 , . . . , An ) ∈ A(n E; F ), and k(πNF − IF ) ◦ Tα kA(n E;F ) ≤ k(πNF − IF ) ◦ Bk · kSkA(X1 ,...,Xn ;Y ) · kA1 k · · · kAn k. Now, the mapping B ∈ F(Y ; F ) is compact and πNF tends to IF over compact sets (because F is minimal), therefore k(πNF −IF )◦Bk tends to zero. In consequence, k(πNF −IF )◦Tα kA(n E;F ) → 0 and ℓn (A; E, F ) is minimal. (ii) For α ∈ ℓn (A; E, F )min , the norm kπN (α) − αkℓn (A;E,F ) tends to zero. By the identities (2) and (3) we have kπN (α)kℓn (A;E,F ) = kπN (α)kℓn (Amin ;E,F min ) . Then, (πN (α))N is a Cauchy sequence in the Banach sequence space ℓn (Amin ; E, F min ) and hence, it converges to a sequence in ℓn (Amin ; E, F min ), that coincides with α coordinate by coordinate. In other words, this says that α ∈ ℓn (Amin ; E, F min ) and kαkℓn (Amin ;E,F min) ≤ kπN (α) − αkℓn (Amin ;E,F min) + kπN (α)kℓn (Amin ;E,F min ) = kπN (α) − αkℓn (Amin ;E,F min) + kπN (α)kℓn (A;E,F ) → kαkℓn (A;E,F )min . ≤1

In conclusion, ℓn (A; E, F )min ֒→ ℓn (Amin ; E, F min ). The reverse inclusion holds by item (i).

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Now, we analyze the relationship between the sequence space associated to an ideal with the associated to its adjoint. Note that when we have a finite dimensional space, its Köthe dual and its classical dual coincide. Moreover, if we call ν the tensor norm associated to the ideal A, we have, !′ n O 1 1 1 × ×× × A∗ (n EN ; FN×× )′ = A(n EN ; FN )′ . ; FN× ) = EN ⊗ FN×× , ν = A(n EN × The duality is given in the following way: if T ∈ A(n EN ; FN ) and S ∈ A∗ (n EN ; FN× ), we can represent them as P × × T = i x× ⊗ · · · ⊗ x× i,n ⊗ yi , with xi,k ∈ EN and yi ∈ FN , P i,1 S = j xj,1 ⊗ · · · ⊗ xj,n ⊗ yj× , with xj,k ∈ EN and yj× ∈ FN× .

Then,

hS, T i =

X

× × x× i,1 (xj,1 ) · · · xi,n (xj,n ) · yj (yi )

i,j

=

X

× S(x× i,1 , . . . , xi,n )(yi ) =

i

X

yj× (T (xj,1 , . . . , xj,n )).

j

From the duality, |hS, T i| ≤ kSkA∗ (n E × ;F × ) ·kT kA(n EN ;FN ) . Moreover, if S is diagonal, there N N P × exists a sequence β such that S = Sβ = N j=1 β(j) · ej ⊗ · · · ⊗ ej ⊗ ej . Then, hSβ , T i = PN PN i=1 α(i) · β(i) = hSβ , D(T )i, where α(i) = T (ei , . . . , ei )(i) j=1 β(j) · T (ej , . . . , ej )(j) = PN × and D(T ) = i=1 α(i) · ei ⊗ · · · ⊗ e× i ⊗ ei .

Lemma 2.5. Let A be an ideal of n-linear operators and let E and F be Köthe sequence 1 × spaces. Then, ℓn (A; EN , FN )× = ℓn (A∗ ; EN , FN× ). Proof. For any sequence β, kβkℓn(A;EN ,FN )× =

sup kαkℓn (A;EN ,FN

=

sup

N X sup α(i) · β(i) = kαkℓn (A;E ,F ) ≤1 i=1

N

|hSβ , Tα i| ≤

kTα kA(n EN ;FN ) ≤1

sup


|hSβ , Tα i| ≤1 N)

kSβ kA∗ (n E × ;F × ) · kTα kA(n EN ;FN ) N

N

= kSβ kA∗ (n E × ;F × ) = kβkℓn (A∗ ;E × ,F × ) . N

N

N

N

Conversely, kβkℓn (A∗ ;E × ,F × ) = N

N

≤

sup

|hSβ , T i| =

kT kA(n EN ;FN ) ≤1

sup kD(T )kA(n EN ;FN ) ≤1

sup

|hSβ , D(T )i|

kT kA(n EN ;FN ) ≤1

|hSβ , D(T )i| =

sup

|hSβ , Tα i|


= kβkℓn(A;EN ,FN )× .


9

As a consequence of the preceding lemma and the identity (2) we have (4)

1

1

1

× ∗ × × ∗ × × ℓn (A; E, F )× N = ℓn (A; EN , FN ) = ℓn (A ; EN , FN ) = ℓn (A ; E , F )N .

This equality allows us to give a general result that relates the sequence space associated to an ideal with the corresponding sequence space associated to its adjoint ideal. Proposition 2.6. Let A be an ideal of n-linear operators and let E and F be Köthe 1 sequence spaces. Then, ℓn (A; E, F )× = ℓn (A∗ ; E × , F × ). Proof. For a sequence α, by identity (4), we have kπN (α)kℓn (A;E,F )× = kπN (α)kℓn (A∗ ;E × ,F × ) . Then, through Proposition 2.2 we derive max 1 1 1 = ℓn (A∗ ; E × , F × )max = ℓn (A∗ ; E × , F × ). ℓn (A; E, F )× = ℓn (A; E, F )×

1

Finally, as a consequence of the Propositions 2.6 and 2.2 and the fact that A∗∗ = Amax , we obtain the following equalities: (5)

1

1

ℓn (A∗ ; E × , F × )× = ℓn (A∗∗ ; E ×× , F ×× ) = ℓn (Amax ; E, F ×× ). 1

In particular, if A and F are maximal, ℓn (A; E, F ) = ℓn (A∗ ; E × , F × )× . 3. Some applications: Multipliers and Lorentz sequence spaces An example of a sequence space associated to a set of operators is the space of multipliers from E into F , M(E, F ) [9], which is defined, in our notation, as M(E; F ) = ℓ1 (L; E, F ). We begin by showing that our sequence space associated to a multilinear ideal can be seen inside a suitable space of multipliers. Proposition 3.1. Let A be an ideal of n-linear operators and let E and F be sequence ≤1

spaces. Then, ℓn (A; E, F ) ֒→ M(F × , ℓn (A; E)). Proof. Let α ∈ ℓn (A; E, F ) and let β ∈ F × . Consider ϕβ ∈ F ′ given by ϕβ (x) = P k∈N β(k) · x(k). If we compose ϕβ with Tα , we obtain φα·β the diagonal n-linear form associated to α · β. Then, by the ideal property, φα·β belongs to A(n E) and kφα·β kA(n E) ≤ kϕβ kF ′ · kTα kA(n E;F ) = kβkF × · kαkℓn (A;E,F ). In consequence, α belongs to M(F × , ℓn (A; E)) and kαkM(F × ,ℓn (A;E)) ≤ kαkℓn (A;E,F ).

In general, the inclusion given in Proposition 3.1 is not an equality. For example, in [5] it is proved that ℓn (E; ℓ 3 , ℓ4 ) = ℓ4 6= M(ℓ× 4 , ℓn (E; ℓ 23 )) = M(ℓ 34 , ℓ 23 ) = ℓ∞ , where E is the 2 ideal of extendible multilinear operators. Another example from the same article is the × following: ℓn (I; ℓ1 , ℓq ) = ℓq 6= M(ℓ× q , ℓn (I; ℓ1 )) = M(ℓq , ℓ∞ ) = ℓ∞ .

10


However, for the very particular case of the ideal of continuous multilinear operators the (isometric) equality holds when the target set is a Köthe dual. 1

Proposition 3.2. Let E and F be Köthe sequence spaces. Then, ℓn (L; E, F × ) = M(F, ℓn (L; E)). 1 In particular, if F is maximal we have that ℓn (L; E, F ) = M(F × , ℓn (L; E)). ≤1

≤1

Proof. By Proposition 3.1, ℓn (L; E, F × ) ֒→ M(F ×× , ℓn (L; E)) ֒→ M(F, ℓn (L; E)). Conversely, let α ∈ M(F, ℓn (L, E)). For any x1 , . . . , xn ∈ E and β ∈ F , we have X α(k) · x1 (k) · · · xn (k) · β(k) = |φα·β (x1 , . . . xn )| ≤ kφα·β kL(n E) · kx1 kE · · · kxn kE . k∈N

So, Tα ∈ L(n E; F × ) and

kαkℓn (L;E,F ×) = kTα kL(n E;F × ) ≤ sup kφα·β kL(n E) = sup kα · βkℓn (L;E) = kαkM(F,ℓn (L;E)) . β∈BF

β∈BF

1

1

Last, if F is maximal, then ℓn (L; E, F ) = ℓn (L; E, F ×× ) = M(F × , ℓn (L; E)).

1

Note that if F is not maximal, the equality ℓn (L; E, F ) = M(F × , ℓn (L; E)) might not be true. For instance, take E = ℓ∞ and F = c0 , then ℓn (L; ℓ∞ , c0 ) = c0 6= M(ℓ1 ; ℓn (L; ℓ∞ )) = M(ℓ1 , ℓ1 ) = ℓ∞ . 1

Corollary 3.3. Let E and F be Köthe sequence spaces. Then, ℓn (I; E, F × ) = ℓ1 (I; F, ℓn (I; E)) . 1 In particular, if F is maximal, ℓn (I; E, F ) = ℓ1 (I; F × , ℓn (I; E)) . 1

1

Proof. Being I ∗ = L and L∗ = I, we have ℓn (I; E, F × )

1

=

(5) 1

= ℓ1 Prop 2.6 1

=

× 1 = ℓ1 L; F × , ℓn (L; E × ) Prop 3.2 1 I; F ×× , ℓn (L; E × )× = ℓ1 I; F, ℓn (L; E × )× Prop 2.2 1 I; F, ℓn (I; E ×× ) = ℓ1 (I; F, ℓn (I; E)) ,

ℓn (L; E × , F ×× )×

ℓ1

where the last two equalities hold by [4, Prop 5.5, Prop 5.6].

Recall the definition of powers of sequence spaces. Let E be a Köthe sequence space and 0 < r < ∞ such that M(max(1,r)) (E) = 1. Then, E r := {x ∈ ℓ∞ : |x|1/r = (|x(k)|1/r )k∈N ∈ E} endowed with the norm kxkE r := k |x|1/r krE results a Köthe sequence space which is 1 -convex. And, the sequence space E r is maximal if E is maximal. min(1,r) Observe that since E is normal, we can use x1/r instead of |x|1/r in the definition of E r and its norm. We will use the fact that whenever x1 , . . . , xn ∈ BE , then its prodn| , implies that uct x1 · · · xn ∈ BE n . Indeed, the inequality (|x1 · · · xn |)1/n ≤ |x1 |+···+|x n


11

(x1 · · · xn )1/n ∈ E. Also, kx1 · · · xn kE n

|x1 | + · · · + |xn | n

= ≤

n E n kx1 kE + · · · + kxn kE ≤1 ≤ n k(|x1 · · · xn |)1/n knE

In the case that E is n-convex, there is an alternative description of ℓn (L; E, F ) as a space of multipliers: Proposition 3.4. Let E and F be an Köthe sequence spaces such that E is n-convex with 1 M(n) (E) = 1. Then, ℓn (L; E, F ) = M(E n , F ). Proof. Let α ∈ ℓn (L; E, F ) and take x ∈ E n . Then x1/n ∈ E and Tα (x1/n , . . . , x1/n ) = α · x1/n · · · x1/n = α · x ∈ F. Thus, α ∈ M(E n , F ) and kαkM(E n ,F ) = sup kα · xkF = sup kTα (x1/n , . . . , x1/n )kF x∈BE n

x∈BE n

≤ sup kTα kL(n E;F ) · kx1/n knE = sup kαkℓn (L;E,F ) · kxkE n = kαkℓn (L;E,F ). x∈BE n

x∈BE n

Conversely, let α ∈ M(E n , F ). Then, Tα (x1 , . . . , xn ) = α · x1 · · · xn ∈ F , for all x1 , . . . , xn ∈ E. In consequence, Tα is well defined from E × · · · × E to F and kTα kL(n E;F ) =

sup kTα (x1 , . . . , xn )kF = sup kα · x1 · · · xn kF

xi ∈BE

≤

xi ∈BE

sup kα · xkF = kαkM(E n ,F ).

x∈BE n

As a direct consequence of Propositions 3.2 and 3.4 we have: Corollary 3.5. Let E and F be an Köthe sequence spaces such that E is n-convex with 1 M(n) (E) = 1. Then, M(E n , F × ) = M(F, ℓn (L; E)). Recall the definition of Lorentz sequence spaces. For each element x ∈ E its decreasing rearrangement (x⋆ (k))k∈N is given by ( ) x⋆ (k) := inf

sup |x(j)| : J ⊆ N, card(J) < k

.

j∈N\J

Let (w(k))∞ be a decreasing sequence of positive numbers such that w(1) = 1, limk w(k) = P∞k=1 0 and k=1 w(k) = ∞ and let 1 ≤ p < ∞. Then the corresponding Lorentz sequence space, denoted by d(w, p) is defined as the set of all sequences (x(k))k such that X 1/p X 1/p ∞ ∞ p ⋆ p |x(σ(k))| · w(k) kxk = sup = |x (k)| · w(k) < ∞, σ∈ΣN

k=1

k=1

where ΣN denotes the group of permutations of the natural numbers.

12


P The sequence w is said to be α-regular (0 < α < ∞) if w(k)α ≍ k1 kj=1 w(j)α and regular if it is α-regular for some α. In [25] it can be found that the Köthe sequence space d(w, p) is r-convex (and with M(r) (d(w, p)) = 1) whenever 1 ≤ r ≤ p. In [17] and [22] a description of d(w, p)′, the dual of d(w, p), is given as the space of those sequences x such that there exists a decreasing y ∈ Bℓp′ with PN ⋆ x (k) sup PN k=1 1. The norm in d(w, p)′ is the infimum of the expression above over all possible decreasing y ∈ Bℓp′ . For p = 1, PN ⋆ n o x (k) ′ d(w, 1) = x : kxk = sup Pk=1 < ∞ . N N k=1 w(k)

If w is regular, an easier description of d(w, p)′ with p > 1 can be given. In this case we have in [1] and [26] that n x⋆ (k) ∞ o ′ ′ . d(w, p) = x : ∈ ℓ p w(k)1/p k=1

The ℓp′ norm of this sequence is a positive homogeneous function of x which, although not a norm, is equivalent to the norm in d(w, p)′ (see [26, Theorem 1]). Lorentz spaces d(w, p) are reflexive whenever p > 1 [22, Section 4.e]. In the case p = 1, the predual of d(w, 1) can be described as (see [27, 16]) ) ( PN ⋆ x (k) =0 d∗ (w, 1) = x ∈ c0 : lim Pk=1 N N →∞ k=1 w(k) PN

x⋆ (k)

. with the norm kxk = supN Pk=1 N k=1 w(k) Let us recall also that, given a strictly positive, increasing sequence Ψ such that Ψ(0) = 0, the associated Marcinkiewicz sequence space mΨ (see [19, Definition 4.1], or [7, 18]) consists of all sequences (x(k))k such that PN ⋆ x (k) < ∞. kxkmΨ = sup k=1 Ψ(N) N The results of the previous section combined with the scalar-valued case for Lorentz spaces studied in [4, Section 5] allow us to give a description of diagonal multilinear mappings from Lorentz sequence spaces (or their duals). Proposition 3.2 along with [4] produce ( M(F, d(w, p/n)×) if n ≤ p; 1 1 ℓn (L; d(w, p), F ×) = M(F, ℓn (L; d(w, p))) = M(F, mΨ ) if n > p,


where Ψ(N) =

P N

k=1 w(k)

n/p

. Moreover, if n > p and w is

n -regular, n−p (n)

13

then ℓn (L; d(w, p), F ×) =

M(F, ℓ∞) = ℓ∞ . For n ≤ p, since d(w, p) is n-convex with M (d(w, p)) = 1, Proposition 3.4 gives an alternative description: ℓn (L; d(w, p), F ) = M(d(w, p)n , F ) = M(d(w, p/n), F ). Proposition 3.2 combined with some results of [4], also imply  if n′ ≤ p;   M(F, ℓ∞ ) =′ ℓ∞ n 1 1 ′ ℓn (L; d(w, p)×, F × ) = M(F, ℓn (L; d(w, p)×)) = M(F, d(w n′ −p , p′p−n )) if 1 < p < n′   M(F, d(w n, 1)) if p = 1 To complete this description it remains to calculate the space of multipliers from F to a Lorentz sequence space. We can give an explicit characterization when F = ℓq . We affirm that ( q pq si p < q; d w q−p , q−p M (ℓq , d(w, p)) = ℓ∞ si p ≥ q. Indeed, when p ≥ q, the equality is clear from the inclusions ℓq ⊂ ℓp ⊂ d(w, p). When p < q, kαkM(ℓq ,d(w,p)) =

=

sup kα · βkd(w,p) = sup sup

β∈Bℓq σ∈ΣN

β∈Bℓq

sup sup σ∈ΣN γ∈Bℓ q

p

=

sup σ∈ΣN

X k∈N

X

X

|ασ (k)|p · |βσ (k)|p · w(k)

k∈N

|ασ (k)|p · |γ(k)| · w(k)

k∈N

|ασ (k)|

p( pq )′

· w(k)

( pq )′

!

!1/p

!1/p

1/p = sup kασp · wk( pq )′ σ∈ΣN

1 q p( p )′

= kαk

. q pq d w q−p , q−p

We can obtain, applying Theorem 2.6 and taking into account that I ∗ = L, similar results for the ideal of integral multilinear operators. 4. Dominated multilinear operators In [3] diagonal r-dominated n-linear forms are studied and related to diagonal absolutely r-summing operators. Specifically, it is proved that the diagonal n-linear form φα ∈ L(n ℓp ) is r-dominated iff the diagonal linear operator Dα1/n ∈ L(ℓp ; ℓn ) is absolutely r-summing. After next lemma, we can rewrite this result as follows ℓn (Dr ; ℓp ) = [ℓ1 (Πr ; ℓp , ℓn )]n . Lemma 4.1. Let F be an n-convex Köthe sequence space with M(n) (F ) = 1. Then, for any sequence space E and numbers r ≥ n, the sequence space ℓ1 (Πr ; E, F ) is n-convex with M(n) (ℓ1 (Πr ; E, F )) = 1.

14


Proof. Let α1 , . . . , αN ∈ ℓ1 (Πr ; E, F ), we have to show that

!1/n !1/n

N N X

X n n

kαk kℓ1 (Πr ;E,F ) . ≤ |αk (j)|

k=1 k=1 ℓ1 (Πr ;E,F )

Equivalently, if we call β(j) =

P N

k=1

|αk (j)|

kDβ kΠr (E;F ) ≤

n

N X

1/n

, the condition to be checked is

kDαk knΠr (E;F )

k=1

!1/n

.

Now, let x1 , . . . , xm ∈ E. Since F is n-convex with M (n) (F ) = 1 and r ≥ n, we obtain  !1/n !1/r  m !1/r

r 1/r

N m m X X X X



|αk · xi |n = = kβ · xi krF kDβ (xi )krF

i=1 i=1 i=1 k=1 F    1/r ! !r/n 1/r r/n m N m N X X X X  =  ≤  kαk · xi knF kDαk (xi )knF i=1

i=1

k=1

 !n/r 1/n m N X X  ≤ kDαk (xi )krF (by Minkowski) ≤  k=1

Then, kDβ kΠr (E;F ) ≤

i=1

P N

n k=1 kDαk kΠr (E;F )

1/n

.

N X k=1

k=1

kDαk knΠr (E;F )

!1/n

· wr (xi ).

Now we see that the relationship between r-dominated diagonal multilinear forms and absolutely r-summing diagonal operators mentioned above can be extended to the vector valued case. Proposition 4.2. Let E and F be Köthe sequence spaces and let Tα ∈ L(n E; F ) be a diagonal n-linear operator. Then, Tα is r-dominated iff the diagonal linear operator Dα1/n ∈ n L(E; F 1/n ) is absolutely r-summing. In other words, ℓn (Dr ; E, F ) = ℓ1 (Πr ; E, F 1/n ) . Proof. First note that since F 1/n is n-convex with M(n) (F 1/n ) = 1, by Lemma 4.1 it n makes sense to consider the space ℓ1 (Πr ; E, F 1/n ) . Let α ∈ ℓn (Dr ; E, F ) and let x1 , . . . , xm ∈ E, then !1/r !1/r m m X X r/n 1/n = kDα1/n (xi )krF 1/n kTα (xi , . . . , xi )kF ≤ kαkℓn (Dr ;E,F ) · wr (xi ). i=1

i=1

n

Thus, α ∈ ℓ1 (Πr ; E, F 1/n ) and kαk[ℓ1 (Πr ;E,F 1/n)]n = kα1/n knℓ1 (Πr ;E,F 1/n) ≤ kαkℓn (Dr ;E,F ). n Conversely, if α ∈ ℓ1 (Πr ; E, F 1/n ) , then Dα1/n ∈ Πr (E; F 1/n ). Through the factorization Tα = Ψ ◦ (Dα1/n , . . . , Dα1/n ), where the operator Ψ ∈ L(n F 1/n ; F ) is given


15

by Ψ(x1 , . . . , xn ) = x1 · · · xn , and the ideal identity Dr = L ◦ (Πr , . . . , Πr ) [28, Proposition 3.6], we obtain that Tα ∈ Dr (n E; F ) and kTα kDr (n E;F ) ≤ kDα1/n knΠr (E;F 1/n ) = kαk[ℓ1 (Πr ;E,F 1/n)]n . In the particular case of ℓp -spaces, this result reads as ℓn (Dr ; ℓp , ℓq ) = [ℓ1 (Πr ; ℓp , ℓnq )]n . Combining this identity with the characterization of the sequence space associated to absolutely r-summing linear operators done in [23, Section 22.4], the space ℓn (Dr ; ℓp , ℓq ) can be precisely described. Now, we focus on (E, p)-dominated multilinear operators. Having in mind the definitions of p-dominated multilinear operators (see [28]) and absolutely (E, p)-summing linear mappings (see [9]), it is natural to introduce (E, p)-dominated multilinear operators in the following way. Let E be an n-convex Köthe sequence space with M(n) (E) = 1 such that ℓp ֒→ E. Then, an n-linear operator T ∈ L(X1 , . . . , Xn ; Y ) is called (E, p)-dominated if there exists C > 0 m such that for every finite sequences x1 = (x1,i )m i=1 ⊆ X1 ,. . . , xn = (xn,i )i=1 ⊆ Xn it holds

(6)

kT (x1,i , . . . , xn,i )k m ≤ C · cE · wp (x1 ) · · · wp (xn ), p Y i=1 E n

where cE p = ki : ℓp ֒→ Ek. The space of (E, p)-dominated n-linear operators, denoted by D(E,p), is a Banach ideal endowed with the norm D(E,p) (T ) = inf{C > 0 : T verifies (6)}. Clearly, for n = 1, the spaces D(E,p) and Π(E,p) coincide. Based on the article of Defant, Mastyło and Michels [9], where inclusion and composition theorems for absolutely (E, p)summing linear operators are proved, we present here extensions to the multilinear setting of some of those results. As we have seen in Lemma 4.1, for some ideals the n-convexity of the target space F is transferred to the associated sequence space. Proposition 4.3. Let F be an n-convex Köthe sequence space with M(n) (F ) = 1. Then, for any Köthe sequence space E, the space ℓn (L; E, F ) is n-convex with M(n) (ℓn (L; E, F )) = 1. Proof. We have to show that for a given finite sequence (αi )m i=1 ⊆ ℓn (L; E, F ), it holds

!1/n

m

X

n

|α | i

i=1

ℓn (L;E,F )

≤

m X i=1

kαi knℓn (L;E,F )

!1/n

.

16


Pm n 1/n Call β = ∈ ℓn (L; E, F ). Then, i=1 |αi |

!1/n

m

X

n

= kTβ kL(n E;F ) = sup kTβ (x1 , . . . , xn )kF |α | i

xi ∈BE

i=1

ℓn (L;E,F )

!1/n

m

X

n

= sup kβ · x1 · · · xn kF = sup |αi · x1 · · · xn |

xj ∈BE xj ∈BE

i=1 F ! ! 1/n 1/n m m X X ≤ sup kαi · x1 · · · xn knF = kαi knℓn (L;E,F ) . xj ∈BE

i=1

i=1

Note that if ℓp ֒→ E and (7)

n r

=

1 p

− 1q , we have

1

ℓq = ℓn (L; ℓr , ℓp ) ֒→ ℓn (L; ℓr , E), ℓ (L;ℓr ,E)

n with kαkℓn (L;ℓr ,E) ≤ cE p · kαkℓn (L;ℓr ,ℓp ) . In other words, cq

≤ cE p.

Proposition 4.4. Let E an n-convex Köthe sequence space with M(n) (E) = 1 such that ℓp ֒→ E. For a mapping T ∈ L(X1 , . . . , Xn ; Y ) the following are equivalent: (1) T ∈ D(E,p)(X1 , . . . , Xn ; Y ) with D(E,p) (T ) ≤ C. (2) D(E,p) (T ◦ (A1 , . . . , An )) ≤ C for all m ∈ N and for all Aj ∈ L(ℓm p′ ; Xj ) with kAj k ≤ 1. Proof. The implication (1)⇒(2) follows by the ideal property of D(E,p) . Conversely, suppose that (2) holds. For finite sequences xj = (xj,i )m i=1 ⊆ Xj (for P m 1 ≤ j ≤ n), we consider linear operators Aj ∈ L(ℓp′ ; Xj ) given by Aj (α) = m i=1 αi · xj,i . m Then, using that kAj k = wp (Aj ei )i=1 , we obtain

(kT (x1,i , . . . , xn,i )kY )m n = k(kT (A1 ei , . . . , An ei )kY )m k n i=1 E i=1 E n ≤ D(E,p) (T ◦ (A1 , . . . , An )) · wp (ei )m i=1 ≤ C · kA1 k · · · kAn k ≤ C · wp (x1 ) · · · wp (xn ).

In consequence, T ∈ D(E,p)(X1 , . . . , Xn ; Y ) with D(E,p) (T ) ≤ C.

Theorem 4.5 (Composition theorem for (E, p)-dominated mappings). Let E be an n-convex Köthe sequence space with M(n) (E) = 1 such that ℓp ֒→ E. If n = 1p − 1q , then r D(ℓn (L;ℓr ,E),q) ◦ (Π nr , . . . , Π nr ) ⊆ D(E,p) . Moreover, D(E,p) (T ◦ (A1 , . . . , An )) ≤ D(ℓn (L;ℓr ,E),q) (T ) · π nr (A1 ) · · · π nr (An ), where T ∈ D(ℓn (L;ℓr ,E),q)(Y1 , . . . , Yn ; Z), A1 ∈ Π nr (X1 ; Y1), . . . , An ∈ Π nr (Xn ; Yn ).


17

Proof. Let Aj ∈ Π nr (Xj ; Yj ) for 1 ≤ j ≤ n and T ∈ D(ℓn (L;ℓr ,E),q)(Y1 , . . . , Yn ; Z). By the Pietsch Domination theorem for absolutely p-summing linear operators, [10, 2.12], for each 1 ≤ j ≤ n, there exists a regular probability measure µj on (BXj′ ; w ∗) such that kAj xkYj

 Z  r ≤ π n (Aj ) ·

BX ′

j

n/r

|x′ (x)|r/n dµj (x′ )

.

Consider any finite sequence (xj,i )m i=1 ⊆ Xj , with non zero elements and write each one as xj,i For simplicity, we denote xj = (xj,i )m i=1 ,

 Z =

BX ′

x0j = (x0j,i )m i=1

j

n/r

|x′j (xj,i )|p dµj (x′j )

and

 Z  γj = 

BX ′

j

· x0j,i .

n/r m  |x′j (xj,i )|p dµj (x′j )  . i=1

Note that kγj k nr ≤ wp (xj )np/r . In [29] it is proved that, under this assumption, for Aj ∈ Π nr (Xj ; Yj ) it holds that (8)

wq (Aj x0j ) ≤ π nr (Aj ) · sup

x′j ∈BX ′ j

m X

|x′j (xj,i )|p

i=1

!1/q

= π nr (Aj ) · wp (xj )p/q .

Then, from inequality (8) and inclusion (7), we have

m

T (A1 x1,i , . . . , An xn,i ) Z i=1 n E

n

0 0 1/n = T (A1 (γ1 · x1 ), . . . , An (γn · xn )) Z E

n

1/n 1/n 0 0 · kγ1 knr · · · kγn1/n knr ≤ kT (A1 x1 , . . . , An xn )kZ ℓn (L;ℓr ,E)

≤ D(ℓn (L;ℓr ,E),q) (T ) ·

cE p

· π nr (A1 ) · wp (x1 )p/q · · · π nr (An ) · wp (xn )p/q · wp (x1 )np/r · · · wp (xn )np/r

r r = D(ℓn (L;ℓr ,E),q) (T ) · cE p · π n (A1 ) · · · π n (An ) · wp (x1 ) · · · wp (xn ).

In consequence, T (A1 , . . . , An ) ∈ D(E,p) and D(E,p) (T ◦ (A1 , . . . , An )) ≤ D(ℓn (L;ℓr ,E),q)(T ) · π nr (A1 ) · · · π nr (An ). Remark 4.6. If x = (xk )k∈N is a sequence of elements of a Banach space E and y ∈ ℓ nr with nr = 1p − 1q , then, wp (y · x) ≤ kykr/n · wq (x).

18


Theorem 4.7 (Inclusion theorem for (E, p)-dominated multilinear operators). Let E be an n-convex Köthe sequence space with M(n) (E) = 1 such that ℓp ֒→ E. If n = 1p − 1q , then r D(E,p) ⊆ D(ℓn (L;ℓr ,E),q) , −1 ℓn (L;ℓr ,E) with D(ℓn (L;ℓr ,E),q) (T ) ≤ cE · c · D(E,p)(T ), for all T ∈ D(E,p) . q p Moreover, if X1 , . . . , Xn have cotype 2, then for any Banach space Y , (9)

D(ℓn (L;ℓ2n ,E),2) (X1 , . . . , Xn ; Y ) = D(E,1) (X1 , . . . , Xn ; Y ).

Proof. By Proposition 4.3, since E is n-convex with M(n) (E) = 1, the space ℓn (L; ℓr , E) is n-convex with convexity constant equal to 1. In addition, ℓq ֒→ ℓn (L; ℓr , E) (by equation (7)). Then, the ideal D(ℓn (L;ℓr ,E),q) is well defined. Now, given T ∈ D(E,p)(X1 , . . . , Xn ; Y ), take finite sequences x1 = (x1,i )m i=1 ⊆ X1 , . . . , m xn = (xn,i )i=1 ⊆ Xn . Consider the sequence α given by αi = kT (x1,i , . . . , xn,i )kY , then

m

n = sup kα1/n · y1 · · · yn knE kαk(ℓn (L;ℓr ,E))n = |αi |1/n i=1 ℓn (L;ℓr ,E)

=

m sup kα · y1n · · · ynn kE n = sup kT (y1n · x1,i , · · · , ynn · xn,i )kY i=1 E n

yj ∈Bℓr

≤

yj ∈Bℓr yj ∈Bℓr

m sup kT (y1,i · x1,i , . . . , yn,i · xn,i )kY i=1 E n

yj ∈Bℓ r

n

≤

sup D(E,p) (T ) · cE p · wp (y1 · x1 ) · · · wp (yn · xn )

yj ∈Bℓ r

n

(Remark 4.6) ≤ D(E,p) (T ) · cE p · wq (x1 ) · · · wq (xn ), cE p

ℓ (L;ℓ ,E) cq n r

−1

Thus, T ∈ D(ℓn (L;ℓr ,E),q) (X1 , . . . , Xn ; Y ) with D(ℓn (L;ℓr ,E),q)(T ) ≤ · · D(E,p)(T ). Taking p = 1 and q = 2 (and then, r = 2n), this also gives one of the inclusions of identity (9). For the reverse inclusion, let T ∈ D(ℓn (L;ℓ2n ,E),2) (X1 , . . . , Xn ; Y ), where X1 , . . . , Xn have cotype 2. For any Aj ∈ L(ℓm ∞ ; Xj ) with kAj k ≤ 1 for 1 ≤ j ≤ n, by a result of Maurey (see [8, 31.7]), there exists C > 0 such that π2 (Aj ) ≤ C. Now, by Theorem 4.5 for p = 1, q = 2 (and r = 2n), D(E,1) (T ◦ (A1 , . . . , An )) ≤ C n · D(ℓn (L;ℓ2n ,E),2) (T ). Finally, the conclusion follows applying Proposition 4.4

In the same spirit of the definition of (E, p)-dominated multilinear operators and having in mind the concept of strongly p-summing multilinear operators [11], we introduce the class of strongly (E, p)-summing multilinear operators. Let E be a Köthe sequence space such that ℓp ֒→ E. An n-linear operator T ∈ L(X1 , . . . , Xn ; Y ) is said to be strongly (E, p)-summing if exists C > 0 such that for finite


19

m sequences (x1,i )m i=1 ⊆ X1 , . . . ,(xn,i )i=1 ⊆ Xn , it satisfies that

m (10) kT (x1,i , . . . , xn,i )kY i=1 E ≤ C · cE p ·

sup φ∈BL(X1 ,...,Xn )

m X

|φ(x1,i , . . . , xn,i )|p

i=1

!1/p

.

We note S(E,p) the space of strongly (E, p)-summing multilinear operators. It is easy to see that it is an ideal of n-linear operators endowed with the norm S(E,p)(T ) = inf{C : T verifies (10)}. Applying the same arguments that we use to prove the inclusion theorem for (E, p)dominated multilinear operators, it can be proved an analogous inclusion theorem for strongly (E, p)-summing multilinear operators. Theorem 4.8. Let E be a Köthe sequence space such that ℓp ֒→ E. If

n r

=

1 p

− 1q , then

S(E,p) ⊆ S(ℓn (L;ℓr ,E),q). −1 ℓn (L;ℓr ,E) Moreover, if T ∈ S(E,p) , then S(ℓn (L;ℓr ,E),q)(T ) ≤ cE · c · S(E,p)(T ). q p

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