Weak compactness and strongly summing multilinear operators
aa r X i v : . [ m a t h . F A ] N ov WEAK COMPACTNESS AND STRONGLY SUMMING MULTILINEAROPERATORS
DANIEL PELLEGRINO, PILAR RUEDA, ENRIQUE A. S ´ANCHEZ-P´EREZ
Abstract.
Every absolutely summing linear operator is weakly compact. However,for strongly summing multilinear operators and polynomials – one of the most naturalextensions of the linear case to the non linear framework – weak compactness does nothold in general. We show that a subclass of the class of strongly summing multilinearoperators/polynomials, sharing its main properties such as Grothendieck’s Theorem,Pietsch Domination Theorem and Dvoretzky–Rogers Theorem, has even better propertieslike weak compactness and a natural factorization theorem. Introduction
The theory of absolutely summing linear operators has its roots in the 1950s with A.Grothendieck’s pioneer ideas; in its modern presentation, it appeared in 1966-67 in theworks of A. Pietsch [40] and B. Mitiagin and A. Pe lczy´nski [30]. A cornerstone in thetheory is the remarkable paper of J. Lindenstrauss and A. Pe lczy´nski [25], which clarifiedGrothendieck’s ideas, without the use of tensor products. Lindenstrauss and Pe lczy´nskiwere also responsible for the reformulation of Grothendieck’s inequality, which is still afundamental result of Banach Space Theory and Mathematical Analysis in general (see[42]). Nowadays absolutely summing operators is a current subject in books of BanachSpace Theory (see, for instance, [1, 20]). For a detailed approach to the linear theory ofabsolutely summing operators we refer to the excellent book of J. Diestel, H. Jarchow andA. Tonge [17].It is then comprehensive that a big effort has been made, since Pietsch’s proposal [41],to try and generalize the linear theory to non linear operators. Many families of non linearoperators have been considered such as multilinear operators, homogeneous polynomials,holomorphic mappings, α -homogeneous mappings, Lipschitz mappings among others. How-ever, extending summability properties to non linear operators has been proved difficultand intriguing. For instance, there are several extensions of absolutely p -summing linearoperators to the multilinear setting that have been considered in the literature. Besides itsintrinsic interest, the multilinear theory of absolutely summing operators has shown impor-tant connections, including applications to Quantum Information Theory (see [31]). Thisproliferation of classes of summing multilinear maps has lead to the appearance of worksthat compare different approaches (see [14, 39]). The first challenging task when dealingwith multilinear operators is probably to identify the class of multilinear operators that bestinherits the spirit of the absolutely summing linear operators. According to [36, 38] oneof the most natural extensions of the notion of absolutely p -summing linear operators to Mathematics Subject Classification.
Primary 46A32, Secondary 47B10.
Key words and phrases. absolutely summing operators, strongly summing multilinear mappings, stronglysumming polynomials, composition ideals.D. Pellegrino acknowledges with thanks the support of CNPq Grant 313797/2013-7 (Brazil). P. Ruedaacknowledges with thanks the support of the Ministerio de Econom´ıa y Competitividad (Spain) MTM2011-22417. E.A. S´anchez P´erez acknowledges with thanks the support of the Ministerio de Econom´ıa y Com-petitividad (Spain) MTM2012-36740-C02-02. the multilinear setting is the notion of strongly p -summing multilinear operators, due to V.Dimant ([18]). This class lifts to the multilinear framework most of the main propertiesof absolutely p -summing linear operators: Grothendieck’s Theorem, Pietsch DominationTheorem, Inclusion Theorem. However, as we will see, a natural version of the PietschFactorization Theorem does not hold for this class.The good behavior of multilinear extensions has found no echo when considering exten-sions of absolutely summing operators to polynomials. In this non linear setting, severalattempts have been made but all of them have found rough edges to succeed in. This isthe case of p -dominated homogeneous polynomials, for which a Pietsch type factorizationtheorem has been pursuit (see [26, 29, 8, 13]) and succeeded just when the domain is sep-arable. Recently, the second and third authors [43] have isolate the class of p -dominatedpolynomials that satisfy a Pietsch type factorization theorem: the factorable p -dominatedpolynomials. However, even if this makes a big difference with p -dominated polynomials,they still lack good properties as evidenced by the fact that factorable p -dominated poly-nomial do not define a composition ideal or, equivalently, the linearization of a factorable p -dominated polynomial may not be absolutely p -summing.Our aim in this paper is to introduce factorable strongly p -summing multilinear operatorsand homogeneous polynomials to the full extent of absolutely p -summing linear operators.These new classes of summing polynomials/multilinear operators stand apart from previousgeneralizations as they keep a big amount of the fundamental properties that are satisfied inthe linear theory and are not satisfied by former non linear classes. Factorable strongly p -summing multilinear operators is a subclass of strongly p -summing multilinear operators thathas in addition a quite natural Pietsch Factorization type theorem and weak compactness.Factorable strongly p -summing homogeneous polynomials also fulfills a factorization theoremin the spirit of Pietsch, are weakly compact and a polynomial belongs to the class if andonly if its second adjoint (in the sense of Aron and Schottenloher) is in the class. Actually,an homogeneous polynomial is factorable strongly p -summing if and only if its associatedmultilinear map is factorable p -summing or, equivalently, its linearization is absolutely p -summing. This brings deep strengths that are not shared by former classes of summingpolynomials as dominated or strongly summing polynomials.This paper is organized as follows. The next section contains the basics (definitions andmain results) on linear and non linear summability that are in order for our purposes. InSection 3 we show that a slight modification of the notion of strongly p -summing operator(inspired in a recent paper of the second and third author) generates a subclass that keepsits main properties and also has a factorization theorem in the lines of the approach above.These are the factorable strongly p -summing multilinear operators. As a consequence wehave weak compactness, as in the linear case. In Section 4 we deal with homogeneouspolynomials, proving that a polynomial is factorable strongly p -summing if and only if itslinearization is absolutely p -summing. The connection between m -homogeneous polynomialsand m -linear operators is established: an m -homogeneous polynomial is factorable strongly p -summing if and only if its associated symmetric m -linear map is factorable strongly p -summing. These results yield to obtain in Section 5 proper generalizations of fundamentalproperties related to summability for linear operators to multilinear maps and homoge-neous polynomials. Among other results, we show that a Dvoretzky-Rogers type theorem, aLindenstrauss–Pe lczy´nski type theorem or a Grothendieck type theorem work for factorablestrongly summability. Finally, in Section 6 we show that the sequence formed by the idealsof factorable strongly summing homogeneous polynomials and factorable strongly summingmultilinear operators is coherent and compatible with the ideal of absolutely summing linearoperators. EAK COMPACTNESS AND STRONGLY SUMMING MULTILINEAR OPERATORS 3 Background: linear and multilinear summability
If 1 ≤ p < ∞ and X, Y are Banach spaces, a continuous linear operator u : X → Y isabsolutely p -summing ( u ∈ Π p ( X ; Y )) if there is a constant C ≥ m X j =1 k u ( x j ) k p /p ≤ C sup ϕ ∈ B X ∗ m X j =1 | ϕ ( x j ) | p /p for all x , ..., x m ∈ X and all positive integers m . The infimum of all C that satisfy theabove inequality defines a norm, denoted by π p ( u ) , and (Π p ( X, Y ) , π p ) is a Banach space.The cornerstones of the theory of absolutely summing linear operators are the followingtheorems: • (Dvoretzky-Rogers theorem) If p ≥ , then Π p ( X ; X ) = L ( X ; X ) if and only ifdim X < ∞ . • (Grothendieck’s theorem ) Every continuous linear operator from ℓ to ℓ is abso-lutely 1-summing. • (Lindenstrauss–Pe lczy´nski theorem) If X and Y are infinite-dimensional Banachspaces, X has an unconditional Schauder basis and Π ( X ; Y ) = L ( X ; Y ) then X = ℓ and Y is a Hilbert space. • (Pietsch Domination theorem) If X and Y are Banach spaces, a continuous linearoperator u : X → Y is absolutely p -summing if and only if there exist a constant C ≥ µ on the closed unit ball of the dual of X, ( B X ∗ , σ ( X ∗ , X )) , such that(1) k u ( x ) k ≤ C (cid:18)Z B X ∗ | ϕ ( x ) | p dµ (cid:19) p for all x ∈ X. • (Inclusion theorem) If 1 ≤ p ≤ q < ∞ , then every absolutely p -summing operator isabsolutely q -summing. • (Pietsch Factorization theorem) A continuous linear operator u : X → Y is abso-lutely p -summing if, and only if, there exist a regular Borel probability measure µ on B X ∗ , a closed subspace X p of L p ( µ ) and a continuous linear operator b u : X p → Y such that j p ◦ i X ( X ) ⊂ X p and b u ◦ j p ◦ i X = u, where i X : X → C ( B X ∗ ) and j p : C ( B X ∗ ) → L p ( µ ) are the canonical inclusions.Moreover, every absolutely p -summing linear operator is weakly compact.From now on p ∈ [1 , ∞ ) and X, X , ..., X n , Y are Banach spaces over the same scalar field K = R or C . A continuous n -linear operator T : X × · · · × X n → Y is p -dominated if thereis a constant C ≥ m X j =1 k T ( x j , ..., x nj ) k pn n/p ≤ C sup ϕ ∈ B X ∗ m X j =1 (cid:12)(cid:12) ϕ ( x j ) (cid:12)(cid:12) p /p · · · sup ϕ ∈ B X ∗ n m X j =1 (cid:12)(cid:12) ϕ ( x nj ) (cid:12)(cid:12) p /p for all x kj ∈ X j , all m ∈ N and ( j, k ) ∈ { , ..., m } × { , ..., n } . This concept is essentiallydue to Pietsch (see [2, 26]) and lifts several important properties of the original linearideal of absolutely summing operators to the multilinear framework. The terminology “ p -dominated”, coined by M.C. Matos, is motivated by the following Pietsch-Domination typetheorem: Theorem 2.1 (Pietsch, Geiss, 1985) . ( [22] ) A continuous n -linear operator T : X × · · · × X n → Y is p -dominated if and only if there exist C ≥ and regular probability measures µ j DANIEL PELLEGRINO, PILAR RUEDA, ENRIQUE A. S ´ANCHEZ-P´EREZ on the Borel σ -algebras of B X ∗ j endowed with the weak star topologies such that k T ( x , ..., x n ) k ≤ C n Y j =1 Z B X ∗ j | ϕ ( x j ) | p dµ j ( ϕ ) /p for every x j ∈ X j and j = 1 , ..., n . Corollary 2.2. If ≤ p ≤ q < ∞ , then every p -dominated multilinear operator is q -dominated . The notion of p -semi-integral operator is another possible multilinear generalization ofthe class of absolutely summing linear operators. If p ≥ , a continuous n -linear operator T : X × · · · × X n → Y is p -semi-integral if there exists a C ≥ m X j =1 k T ( x j , ..., x nj ) k p /p ≤ C sup ( ϕ ,..,ϕ n ) ∈ B X ∗ ×···× B X ∗ n m X j =1 | ϕ ( x j ) ...ϕ n ( x nj ) | p /p for every m ∈ N , x kj ∈ X k with k = 1 , ..., n and j = 1 , ..., m. This class dates back to the research report [2] of R. Alencar and M.C. Matos. As in thecase of p -dominated multilinear operators, a Pietsch Domination theorem is valid in thiscontext: Theorem 2.3.
A continuous n -linear operator T : X × · · · × X n → Y is p -semi-integralif and only if there exist C ≥ and a regular probability measure µ on the Borel σ − algebra B ( B X ∗ ×· · · × B X ∗ n ) of B X ∗ ×· · · × B X ∗ n endowed with the product of the weak star topologies σ ( X ∗ l , X l ) , l = 1 , ..., n, such that k T ( x , ..., x n ) k≤ C Z B X ∗ ×···× B X ∗ n | ϕ ( x ) ...ϕ n ( x n ) | p dµ ( ϕ , ..., ϕ n ) ! /p for all x j ∈ X j , j = 1 , ..., n. Corollary 2.4. If ≤ p ≤ q < ∞ , every p -semi-integral multilinear operator is q -semi-integral. This class is strongly connected to the class of p -dominated multilinear operators. Forexample, in [14] it is shown that every p -semi integral n -linear operator is np -dominated.The following result shows that we cannot expect to lift coincidence results of the linearcase to dominated multilinear operators: Theorem 2.5 (Jarchow, Palazuelos, P´erez-Garc´ıa and Villanueva, 2007) . ( [24] ) For every n ≥ and every p ≥ and every infinite dimensional Banach space X there exists acontinuous n -linear operator T : X × · · · × X → K that fails to be p -dominated. Since p -semi-integral n -linear operators are np -dominated, we have: Corollary 2.6.
For every n ≥ , every p ≥ and every infinite dimensional Banach space X there exists a continuous n -linear operator T : X × · · · × X → K that fails to be p -semi-integral. So, in view of the previous result, it is obvious that we cannot expect a Grothendiecktype theorem for dominated or semi-integral operators. In this direction, the classes ofmultiple summing multilinear operators ([3, 27]), strongly multiple summing multilinearoperators ([7]) and strongly summing multilinear operators ([18]) are other possible gener-alizations, with a quite better behavior if we are interested in lifting coincidence theorems,
EAK COMPACTNESS AND STRONGLY SUMMING MULTILINEAR OPERATORS 5 like Grothendieck’s theorem. But, as a matter of fact, none of these classes lifts all the mainproperties of absolutely summing linear operators to the multilinear setting.In [43], a variant of the notion of p -dominated polynomials which satisfy (in a very naturalform) a Pietsch factorization type theorem, is introduced. A continuous n -homogeneouspolynomial P : X → Y is factorable p -dominated if there is a C ≥ x ij ∈ X , and scalars λ ij , 1 ≤ j ≤ m , 1 ≤ i ≤ m and all positive integers m , m , we have m X j =1 (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) m X i =1 λ ij P (cid:0) x ij (cid:1)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) p p ≤ C sup ϕ ∈ B X ∗ m X j =1 (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) m X i =1 λ ij ϕ (cid:0) x ij (cid:1) n (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) p p . The natural multilinear version of the notion of “factorable p -dominated polynomials” seemsto be: Definition 2.7.
A continuous n -linear operator T : X × · · · × X n → Y is factorable p -dominated if there is a constant C ≥ such that for every x ik,j ∈ X k , and scalars λ ij , ≤ j ≤ m , ≤ i ≤ m and all positive integers m , m , we have m X j =1 (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) m X i =1 λ ij T (cid:0) x i ,j , ..., x in,j (cid:1)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) p p ≤ C sup ϕ k ∈ B X ∗ k k =1 ,...,n m X j =1 (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) m X i =1 λ ij ϕ (cid:0) x i ,j (cid:1) · · · ϕ n (cid:0) x in,j (cid:1)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) p p . These notions have some connection with the idea of weighted summability, sketched in[37]. It is likely that this class has a nice factorization theorem (like its polynomial version)but a simple calculation shows that any factorable p -dominated multilinear operator is p -semi-integral and thus we have: Proposition 2.8.
For every n ≥ and every p ≥ and every infinite dimensional Banachspace X there exists a continuous n -linear operator T : X × · · · × X → K that fails tobe factorable p -dominated. A fortiori, regardless of the Banach space Y , there exists acontinuous n -linear operator T : X × · · · × X → Y that fails to be factorable p -dominated. So, since we are looking for classes that also lift coincidence results to the multilinear set-ting, the class of factorable p -dominated multilinear operators is not what we are searching.A continuous n -linear operator T : X × · · · × X n → Y is strongly p -summing if thereexists a constant C ≥ m X j =1 k T ( x j , ..., x nj ) k p /p ≤ C sup φ ∈ B L ( X ,...,Xn ; K ) m X j =1 | φ ( x j , ..., x nj ) | p /p . for every m ∈ N , x kj ∈ X k with k = 1 , ..., n and j = 1 , ..., m. The class of strongly p -summing multilinear operators is due to V. Dimant [18] andaccording to [36, 38] it is perhaps the class that best translates to the multilinear setting theproperties of the original linear concept. For example, a Grothendieck type theorem and aPietsch-Domination type theorem are valid: Theorem 2.9 (Grothendieck-type theorem) . ( [18] ) Every continuous n -linear operator T : ℓ × · · · × ℓ → ℓ is strongly -summing. Theorem 2.10 (Pietsch Domination type theorem) . ( [18] ) A continuous n -linear operator T : X × · · · × X n → Y is strongly p -summing if, and only if, there are a probability measure DANIEL PELLEGRINO, PILAR RUEDA, ENRIQUE A. S ´ANCHEZ-P´EREZ µ on B ( X b ⊗ π ··· b ⊗ π X n ) ∗ , with the weak-star topology, and a constant C ≥ so that (3) k T ( x , ..., x n ) k ≤ C Z B ( X b ⊗ π ··· b ⊗ πXn ) ∗ | ϕ ( x ⊗ · · · ⊗ x n ) | p dµ ( ϕ ) ! p for all ( x , ..., x n ) ∈ X × · · · × X n . Corollary 2.11. If p ≤ q then every strongly p -summing multilinear operator is strongly q -summing. It is not hard to prove that a Dvoretzky-Rogers Theorem is also valid for this class:
Theorem 2.12 (Dvoretzky-Rogers type theorem) . Every continuous n -linear operator T : X × · · · × X → X is strongly p -summing if, and only if, dim X < ∞ . A property fulfilled by the class of absolutely summing operators which is not lifted to themultilinear framework by the notion of strong summability is the weak compactness. In fact,it is well known that every absolutely p -summing linear operator is weakly compact, butCarando and Dimant have shown that there exist strongly p -summing multilinear operatorsthat fail to be weakly compact [15]. This result implies that a natural version of the PietschFactorization Theorem is not valid for strongly summing multilinear operators, as we willsee below.Suppose that the following factorization theorem holds: T : X ×· · ·× X n → Y is strongly p -summing if and only if there is a regular Borel probability measure µ on B ( X b ⊗ π ··· b ⊗ π X n ) ∗ ,with the weak-star topology, a closed subspace Z p of L p (cid:16) B ( X b ⊗ π ··· b ⊗ π X n ) ∗ , µ (cid:17) and a contin-uous linear operator b T : Z p → Y such that j p ◦ i X ×···× X n ( X × · · · × X n ) ⊂ Z p and b T ◦ j p ◦ i X ×···× X n = T, where i X ×···× X n : X × · · · × X n → C (cid:16) B ( X b ⊗ π ··· b ⊗ π X n ) ∗ (cid:17) is the canonical n -linear map i X ×···× X n ( x , ..., x n ) ( ϕ ) = ϕ ( x ⊗ · · · ⊗ x n ) and j p : C (cid:16) B ( X b ⊗ π ··· b ⊗ π X n ) ∗ (cid:17) → L p (cid:16) B ( X b ⊗ π ··· b ⊗ π X n ) ∗ , µ (cid:17) is the canonical linear inclusion.Since j p is absolutely p -summing (and thus weakly compact), then we conclude that theset j p ( i X ×···× X n ( B X × · · · × B X n )) is relatively weakly compact in Z p . Since b T is contin-uous and linear, then T ( B X , ..., B X n ) = b T ( j p ( i X ×···× X n ( B X × · · · × B X n ))) is relativelyweakly compact in Y and thus T is weakly compact, but this is not true in general ([15]).In this paper we combine the idea of factorable summability from [43] with the notionof strongly p -summing multilinear operators and we show that the new class we introducerecovers all these lacks suffered by the former multilinear extensions.3. Factorable strongly p -summing multilinear operators The following definition is inspired in ideas from [43], adapted to the notion of stronglysumming multilinear operators:
Definition 3.1.
A continuous n -linear operator T : X × · · · × X n → Y is factorablestrongly p -summing if there is a constant C ≥ such that for every x ik,j ∈ X k , and scalars λ ik,j , ≤ j ≤ m , ≤ i ≤ m and all positive integers m , m , we have m X j =1 (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) m X i =1 λ ij T (cid:0) x i ,j , ..., x in,j (cid:1)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) p p ≤ C sup k ϕ k≤ m X j =1 (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) m X i =1 λ ij ϕ (cid:0) x i ,j , ..., x in,j (cid:1)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) p p . EAK COMPACTNESS AND STRONGLY SUMMING MULTILINEAR OPERATORS 7 where the supremum is taken over all the continuous n -linear functionals ϕ : X ×· · ·× X n → K of norm less or equal than . The class of all factorable strongly p -summing n -linearoperators T : X × · · · × X n → Y is denoted by Π F St,p ( X , . . . , X n ; Y ) and endowed withthe norm k · k F St,p , where k T k F St,p is given by the infimum of all constant C fulfilling theabove inequality. Note that if T is factorable strongly p -summing then making m = 1 and λ j = 1 for all j = 1 , . . . , m , we have (cid:13)(cid:13)(cid:13)(cid:0) T (cid:0) x ,j , ..., x n,j (cid:1)(cid:1) m j =1 (cid:13)(cid:13)(cid:13) p ≤ C sup k ϕ k≤ m X j =1 (cid:12)(cid:12) ϕ (cid:0) x ,j , ..., x n,j (cid:1)(cid:12)(cid:12) p p , i.e., T is strongly p -summing. In particular, whenever n = 1, Π F St,p ( X ; Y ) = Π p ( X ; Y ) isthe class of all absolutely p -summing operators from X to Y .The ideal property is straightforward. It is also trivial that every scalar-valued n -linearoperator is factorable strongly p -summing. Straightforward calculations show that this classforms a Banach multi-ideal.As we will see in Section 5, this class preserves the nice properties of the class of stronglysumming multilinear operators and has extra desirable properties: weak compactness and afactorization theorem. Theorem 3.2 (Pietsch-Domination type theorem) . A continuous n -linear operator T : X ×· · ·× X m → Y is factorable strongly p -summing if and only if there is a regular probabilitymeasure µ on B ( X b ⊗ π ··· b ⊗ π X n ) ∗ , endowed with the weak-star topology, and a constant C ≥ ,such that (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) m X i =1 λ i T (cid:0) x i , ..., x in (cid:1)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) ≤ C Z B ( X b ⊗ π ··· b ⊗ πXn ) ∗ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) m X i =1 λ i ϕ (cid:0) x i , ..., x in (cid:1)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) p dµ ( ϕ ) ! p . Proof.
The notion of factorable strongly p -summing multilinear operator is precisely theconcept of RS -abstract p -summing (see [10, 35, 38]) for R : B ( X b ⊗ π ··· b ⊗ π X n ) ∗ × ( K × X × · · · × X n ) N × { } → [0 , ∞ )given by R (cid:0) ϕ, (cid:0) λ , x , ..., x n (cid:1) , ..., ( λ m , x m , ..., x mn ) , (cid:1) = (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) m X i =1 λ i ϕ (cid:0) x i ⊗ · · · ⊗ x in (cid:1)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) and S : L ( X , ..., X n ; Y ) × ( K × X × · · · × X n ) N × { } → [0 , ∞ )given by S (cid:0) T, (cid:0) λ , x , ..., x n (cid:1) , ..., ( λ m , x m , ..., x mn ) , (cid:1) = (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) m X i =1 λ i T (cid:0) x i , ..., x in (cid:1)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) . Since R and S satisfy the hypotheses of the general Pietsch Domination Theorem, the resultfollows straightforwardly. (cid:3) Theorem 3.3 (Pietsch-Factorization type theorem) . A continuous n -linear operator T : X × · · · × X n → Y is factorable strongly p -summing if and only if there exist a regularprobability measure µ on B ( X b ⊗ π ··· b ⊗ π X n ) ∗ , endowed with the weak-star topology, a constant DANIEL PELLEGRINO, PILAR RUEDA, ENRIQUE A. S ´ANCHEZ-P´EREZ C ≥ , a closed subspace Z p of L p (cid:16) B ( X b ⊗ π ··· b ⊗ π X n ) ∗ , µ (cid:17) and a continuous linear operator b T : Z p → Y such that j p ◦ i X ×···× X n ( X × · · · × X n ) ⊂ Z p and b T ◦ j p ◦ i X ×···× X n = T, where i X ×···× X n : X × · · · × X m → C (cid:16) B ( X b ⊗ π ··· b ⊗ π X n ) ∗ (cid:17) is the canonical n -linear map i X ×···× X n ( x , ..., x n ) ( ϕ ) = ϕ ( x ⊗ · · · ⊗ x n ) and j p : C (cid:16) B ( X b ⊗ π ··· b ⊗ π X n ) ∗ (cid:17) → L p (cid:16) B ( X b ⊗ π ··· b ⊗ π X n ) ∗ , µ (cid:17) is the canonical linear inclusion.Proof. Suppose that T is factorable strongly p -summing. Let µ be the measure given bythe Pietsch Domination Theorem (Theorem 3.2) applied to T . Let W p be the subspace of L p (cid:16) B ( X b ⊗ π ··· b ⊗ π X n ) ∗ , µ (cid:17) given by the linear span of j p ◦ i X ×···× X m ( X × · · · × X m ) . Definethe linear operator b T : W p → Y by b T ( z ) = n X i =1 λ i T (cid:0) x i , ..., x in (cid:1) for z = n X i =1 λ i h· , (cid:0) x i ⊗ · · · ⊗ x in (cid:1) i ∈ W p . Note that b T is well-defined. In fact, if z = m X i =1 λ i h· , (cid:0) x i ⊗ · · · ⊗ x in (cid:1) i and z = m X i =1 α i h· , (cid:0) y i ⊗ · · · ⊗ y in (cid:1) i coincide in W p , then considering w := m X i =1 λ i h· , (cid:0) x i ⊗ · · · ⊗ x in (cid:1) i − m X i =1 α i h· , (cid:0) y i ⊗ · · · ⊗ y in (cid:1) i , we have w = 0 almost everywhere in W p , i.e., Z B ( X b ⊗ π ··· b ⊗ πXn ) ∗ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) m X i =1 λ i ϕ (cid:0) x i ⊗ · · · ⊗ x in (cid:1) − m X i =1 α i ϕ (cid:0) y i ⊗ · · · ⊗ y in (cid:1)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) p dµ ( ϕ ) = 0 . Thus, from the domination theorem, (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) m X i =1 λ i T (cid:0) x i , ..., x in (cid:1) − m X i =1 α i T (cid:0) y i , ..., y in (cid:1)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) ≤ C Z B ( X b ⊗ π ··· b ⊗ πXn ) ∗ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) m X i =1 λ i ϕ (cid:0) x i ⊗ · · · ⊗ x in (cid:1) − m X i =1 α i ϕ (cid:0) y i ⊗ · · · ⊗ y in (cid:1)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) p dµ ( ϕ ) ! p = 0and we conclude that b T ( z ) − b T ( z ) = m X i =1 λ i T (cid:0) x i , ..., x in (cid:1) − m X i =1 α i T (cid:0) y i , ..., y in (cid:1) = 0 . EAK COMPACTNESS AND STRONGLY SUMMING MULTILINEAR OPERATORS 9
Note also that for z = m X i =1 λ i h· , (cid:0) x i ⊗ · · · ⊗ x in (cid:1) i ∈ W p we have (cid:13)(cid:13)(cid:13) b T ( z ) (cid:13)(cid:13)(cid:13) = (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) m X i =1 λ i T (cid:0) x i , ..., x in (cid:1)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) ≤ C Z B ( X b ⊗ π ··· b ⊗ πXn ) ∗ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) m X i =1 λ i ϕ (cid:0) x i ⊗ · · · ⊗ x in (cid:1)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) p dµ ( ϕ ) ! p = C k z k L p ( µ ) and b T is continuous. It is obvious that from the very definition of b T we have b T ◦ j p ◦ i X ×···× X n = T . Now we extend b T to Z p = W p . The converse is immediate. (cid:3) Factorable strongly p -summing polynomials The m -fold symmetric tensor product of X is the linear span of all tensors of the form x ⊗ · · · ⊗ x , x ∈ X , and is denoted by ⊗ m,s X . This space is endowed with the s -projectivetensor norm, defined as π s ( z ) = inf { k X j =1 | λ j |k x j k n : k ∈ N , z = k X j =1 λ j x j ⊗ · · · ⊗ x j } , for z ∈ ⊗ m,s X . Let ˆ ⊗ m,sπ s X denote the completion of ⊗ m,sπ s X .Given P ∈ P ( m X ; Y ), the linearization of P is the unique linear operator P L,s : ˆ ⊗ m,sπ s X → Y such that P L,s ( x ⊗· · ·⊗ x ) = P ( x ) for all x ∈ X . Ryan [44] proved that the correspondence P ↔ P L,s establishes a isometric isomorphism between the space P ( m X ), endowed with theusual sup norm, and the strong dual of ˆ ⊗ m,sπ s X . Another map associated to P ∈ P ( m X ; Y )is the unique continuous symmetric m -linear mapping ˇ P that satisfies ˇ P ( x, . . . , x ) = P ( x ),for all x ∈ X . It is well known that k ˇ P k ≤ c ( m, X ) k P k for all P ∈ P ( m X ), where c ( m, X )is the m -th polarization constant of X . For the general theory of homogeneous polynomialswe refer to [19] and [32].Concomitantly to multilinear mappings, factorable strongly p -summing homogeneouspolynomials can be introduced. Our aim is to prove that both classes coincide in the sensethat a polynomial is factorable strongly p -summing if and only if its associated symmetricmultilinear mapping is factorable strongly p -summing. Moreover, we will see the deep rela-tionship between factorable strong summability and absolute summability by proving that,for an homogeneous polynomial, it is equivalent that the polynomial is factorable stronglysumming to that its linearization is an absolutely summing operator. To attain this purpose,we will show that both, factorable strongly p -summing polynomials and factorable strongly p -summing multilinear operators, form composition ideals. Definition 4.1.
A continuous n -homogeneous polynomial P : X → Y is factorable strongly p -summing if there is a C ≥ such that for every x ij ∈ X , and scalars λ ij , ≤ j ≤ m , ≤ i ≤ m and all positive integers m , m , we have that k ( m X i =1 λ ij P ( x ij )) j k p ≤ C sup k q k≤ ,q ∈P ( m X ) ( m X j =1 | m X i =1 λ ij q ( x ij ) | p ) /p . The class of all factorable strongly p -summing m -homogeneous polynomials from X to Y isdenoted by P F St,p ( m X ; Y ) and endowed with the norm k · k F St,p given by the infimum of allconstants C fulfilling the above inequality. It is clear that factorable p -dominated polynomials are factorable strongly p -summing.An easy calculation shows the following ideal property: Proposition 4.2. If P ∈ P F St,p ( m X ; Y ) and u : G → X , v : Y → Z are continuous linearoperators then v ◦ P ◦ u ∈ P F St,p ( m G ; Z ) and k v ◦ P ◦ u k F St,p ≤ k v k · k P k F St,p k u k m . It is not difficult to complete Proposition 4.2 and show that factorable strongly n -homogeneous polynomials form an ideal of polynomials (for the definition of ideal of poly-nomials we refer to [4]).Dimant [18] introduced the class of strongly p -summing m -homogeneous polynomialsfrom X to Y as those m -homogeneous polynomials P : X → Y that satisfy that there exists K > n ∈ N and any x , . . . , x n ∈ X ,( n X j =1 k P ( x j ) k p ) /p ≤ K sup k q k≤ ,q ∈P ( m X ) ( n X j =1 | q ( x j )) | p ) /p . In [18, Proposition 3.2] it is proved that if the linearization P L,s of P ∈ P ( m X ; Y ) is ab-solutely p -summing then p is strongly p -summing. However, the converse is not true (see[15, Example 3.3]). The reason, as for p -dominated polynomials, is that not every strongly p -summing polynomial is weakly compact. So, once again, the lack of connection with weakcompactness turns out to be a deep inconvenience in the way that strongly p -summingpolynomials generalize absolutely p -summing linear operators. Even if a domination holdsalso for strongly p -summing polynomials [18, Proposition 3.2], no factorization theorem isexpected. Let us prove a factorization theorem for factorable strongly p -summing polyno-mials. We first need a domination theorem, that is obtained as a particular case of [10,Theorem 2.2]. We denote by δ : X → C ( B P ( m X ) ) the m -homogeneous polynomial given by δ ( x ) := δ x : B P ( m X ) → K , where δ x ( P ) := P ( x ). Considering that the space of continu-ous m -homogeneous polynomials is a dual space (see [44]), its closed unit ball B P ( m X ) is aweak-star compact set. Theorem 4.3 (Pietsch-Domination type theorem) . Let P ∈ P ( m X ; Y ) . Then P is fac-torable strongly p -summing if and only if there exists a regular Borel probability measure µ on B P ( m X ) , endowed with the weak-star topology, such that k k X i =1 λ i P ( x i ) k ≤ C ( Z B P ( mX ) | k X i =1 λ i q ( x i ) | p dµ ) /p for all x , . . . , x k ∈ X and λ , . . . , λ k ∈ K .Proof. It is a particular case of [10, Theorem 2.2] analogous to the proof of Theorem 3.2. (cid:3)
We shall need the following result to prove the sufficiency of the Factorization Theorem.Besides, Proposition 4.4 will be the key for our purposes to obtain that factorable strongly p -summing homogeneous polynomials form a composition ideal. Proposition 4.4. If Q ∈ P ( m G ; X ) and u : X → Y is an absolutely p -summing linearoperator, then u ◦ Q ∈ P F St,p ( m G ; Y ) and k u ◦ Q k F St,p ≤ π p ( u ) k Q k . EAK COMPACTNESS AND STRONGLY SUMMING MULTILINEAR OPERATORS 11
Proof.
Let m , m be positive integers, x ij ∈ X , and scalars λ ij , 1 ≤ j ≤ m , 1 ≤ i ≤ m .Then,( m X j =1 k ( m X i =1 λ ij u ◦ Q ( x ij ) k p ) /p = ( m X j =1 k u ( m X i =1 λ ij Q ( x ij ) k p ) /p ≤ π p ( u ) sup k x ∗ k≤ ,x ∗ ∈ X ∗ ( m X j =1 |h x ∗ , m X i =1 λ ij Q ( x ij ) i| p ) /p ≤ π p ( u ) k Q k sup k x ∗ k≤ ,x ∗ ∈ X ∗ ( m X j =1 | m X i =1 λ ij h x ∗ , Q/ k Q k ( x ij ) i| p ) /p ≤ π p ( u ) k Q k sup k q k≤ ,q ∈P ( m G ) ( m X j =1 | m X i =1 λ ij q ( x ij ) | p ) /p . (cid:3) Theorem 4.5 (Pietsch-Factorization type theorem) . Let P ∈ P ( m X ; Y ) . Then P is fac-torable strongly p -summing if and only if there exists a regular Borel probability measure µ on B P ( m X ) , a closed subspace G p of L p ( µ ) and a continuous linear operator v : G p → Y such that j p ◦ δ ( X ) ⊂ G p and v ◦ j p ◦ δ = P , where j p : C ( B P ( m X ) ) → L p ( B P ( m X ) , µ ) is thecanonical inclusion.Proof. Assume first that P is factorable strongly p -summing. Let µ be given by Theo-rem 4.3. Take G p the completion of the image by j p of the linear span of δ ( X ). De-fine v ( j p ( P ki =1 λ i δ x i )) := P ki =1 λ i P ( x i ). To see that v is well defined, consider that j p ( P ki =1 λ i δ x i ) = j p ( P li =1 η i δ y i ). Then w := P ki =1 λ i δ x i − P li =1 η i δ y i = 0 a.e. on B P ( m X ) .Hence, k k X i =1 λ i P ( x i ) − l X i =1 η i P ( y i ) k ≤ k P k F St,p ( Z B P ( mX ) | k X i =1 λ i q ( x i ) − l X i =1 η i q ( y i ) | p dµ ) /p = 0 . Thus, v ( w ) = 0. That proves that v is well defined. The continuity of v follows from thecalculations: k v ( z ) k = k k X i =1 λ i P ( x i ) k ≤ k P k F St,p ( Z B P ( mX ) | k X i =1 λ i q ( x i ) | p dµ ) /p = k P k F St,p k k X i =1 λ i δ x i k L p ( µ ) = k P k F St,p k z k L p ( µ ) for any z = j p ( P ki =1 λ i δ x i ). The desired linear operator is just the continuous extension of v to G p . The converse follows from Proposition 4.4. (cid:3) Corollary 4.6.
Let P ∈ P ( m X ; Y ) . Then P ∈ P F St,p ( m X ; Y ) if and only if P = u ◦ Q ,for some continuous m -homogeneous polynomial Q and some absolutely p -summing linearoperator u . In that case k P k F St,p = inf { π p ( u ) k Q k : P = u ◦ Q } .Proof. It follows from Theorem 4.5 and Proposition 4.4. (cid:3)
Corollary 4.6 says that the ideal of all factorable strongly p -summing m -homogeneouspolynomials is the composition ideal with all absolutely p -summing linear operators, thatis, P F St,p = Π p ◦ P (see [9]). An analogous argument for multilinear operators instead ofpolynomials yields to prove that the ideal of all factorable strongly p -summing m -linear operators is the composition ideal with all absolutely p -summing linear operators, that is,Π F St,p = Π p ◦ L . Remark 4.7. In [29] it is shown an example of a continuous m -homogeneous polynomial P : X → Y and φ ∈ Π as,r ( Y ; Z ) such that φ ◦ P : X → Z is not r -dominated. ByProposition 4.4, φ ◦ P is factorable strongly r -summing. Therefore, the class of dominatedpolynomials differs from the class of factorable strongly r -summing polynomials. Theorem 4.8.
Let P ∈ P ( m X ; Y ) . The following are equivalent: (1) P ∈ P F St,p ( m X ; Y ) . (2) P L,s is absolutely p -summing. (3) ˇ P ∈ Π F St,p ( m X ; Y ) .In that case, k P k F St,p = π p ( P L,s ) .Proof. (1) ⇒ (2) Assume first that P is factorable strongly p -summing. Then( m X j =1 k P L,s ( m X i =1 λ ji x ji ⊗ · · · ⊗ x ji ) k p ) /p = ( m X j =1 k m X i =1 λ ji P L,s ( x ji ⊗ · · · ⊗ x ji ) k p ) /p = ( m X j =1 k m X i =1 λ ij P ( x ji ) k p ) /p ≤ k P k F St,p sup k q k≤ ,q ∈P ( m X ) ( m X j =1 | m X i =1 λ ij q ( x ji ) | p ) /p = k P k F St,p sup k q k≤ ,q ∈ ( ˆ ⊗ m,sπs X ) ∗ ( m X j =1 | m X i =1 λ ij q ( ⊗ m x ji ) | p ) /p (2) ⇒ (1) follows from Proposition 4.4.(1) ⇔ (3) As P F St,p ( m X ; Y ) = Π p ◦ P ( m X ; Y ), it follows from [9, Proposition 3.2] that P ∈ P F St,p ( m X ; Y ) if and only if ˇ P ∈ Π p ◦ L ( m X ; Y ) and, similarly to Proposition 4.4 itcan be proved that Π p ◦ L ( m X ; Y ) = Π F St,p ( m X ; Y ). (cid:3) We finish this section with a Grothendieck type theorem:
Theorem 4.9 (Grothendieck type theorem) . If m ≥ is a positive integer, then P ( m ℓ ; ℓ ) = P F St, ( m ℓ ; ℓ ) . Proof.
Let P ∈ P ( m X ; Y ). Then P L,s ∈ L ( ˆ ⊗ m,sπ s ℓ ; ℓ ) = L ( ℓ ; ℓ ) = Π as, ( ℓ ; ℓ ). Theorem4.8 yields the result. (cid:3) The wealth of factorable strong p -summability In this section it is shown that factorable strong p -summability is an excellent non linearframe where linear results for absolute summability are properly generalized to multilinearoperators and polynomials. This evidences the interest of this new class as it really reflectsthe good behavior of absolute summability in the non linear context. Some of these resultsare established for multilinear operators and some for homogeneous polynomials. However,as a consequence of Theorem 4.8 it is clear that one can pass easily from one to each other.The following results are consequences of Theorem 4.8 and their linear analogs (see [17,Theorems 3.15 and 3.17]): Proposition 5.1 (Composition Theorem) . If u ∈ Π p ( X ; Y ) and P ∈ P F St,q ( m G ; X ) then u ◦ P ∈ P F St,r ( m G ; Y ) for /r := min { , /p + 1 /q } . EAK COMPACTNESS AND STRONGLY SUMMING MULTILINEAR OPERATORS 13
Proof.
By Theorem 4.8 P L,s is absolutely q -summing. Then u ◦ P L,s is r -summing for1 /r := min { , /p + 1 /q } (see [17, Theorem 2.22]). Since u ◦ P L,s = ( u ◦ P ) L,s , a secondapplication of Theorem 4.8 yields the result. (cid:3)
Theorem 5.2 (Extrapolation type theorem) . Let < r < p < ∞ , and let X be a Banachspace. If P F St,p ( m X ; ℓ p ) = P F St,r ( m X ; ℓ p ) then P F St,p ( m X ; Y ) = P F St, ( m X ; Y ) for everyBanach space Y . Recall that given 1 ≤ p ≤ ∞ and λ >
1, a Banach space X is said to be an L p,λ -space ifevery finite dimensional subspace E of X is contained in a finite dimensional subspace F of X for which there is an isomorphism v : F → ℓ dim Fp with k v k · k v − k < λ . Theorem 5.3 (Lindenstrauss–Pe lczy´nski type theorem) . Let ≤ p ≤ and < q < ∞ . If X is a Banach space and Y is a subspace of an L p,λ -space, then P F St,q ( m X ; Y ) = P F St, ( m X ; Y ) . We have already proved that Domination/Factorization Theorems are fulfilled in the mul-tilinear and polynomial classes of factorable strongly p -summing maps. As a straightforwardconsequence of the Factorization Theorem 4.5 we get Theorem 5.4.
Any factorable strongly p -summing polynomial is weakly compact. An alternative way to prove it is the following: by Theorem 4.8 the linearization of afactorable strongly p -summing polynomial P is absolutely p -summing and hence weaklycompact. By [44] this is equivalent to the weak compactness of P . The same holds for thecase of multilinear operators.The Domination Theorem 3.2 also yields to the following inclusion theorem. Proposition 5.5 (Inclusion Theorem) . If ≤ p ≤ q < ∞ then every factorable strongly p -summing polynomial is factorable strongly q -summing. The forthcoming lemmas 5.6, 5.9 and its consequences show that, besides its good prop-erties, the classes of factorable strongly p -summing multilinear operators and polynomialshave a coherent size. Lemma 5.6.
If every continuous n -linear operator T : X × · · · × X n → Y is factorablestrongly p -summing, then every continuous linear operator u j : X j → Y is absolutely p -summing for every j = 1 , ..., n .Proof. For the sake of simplicity, let us suppose j = 1 . Let u : X → Y be a continuous linearoperator and ϕ j ∈ X ∗ j , j = 2 , ..., n be non-null linear functionals. Then T ( x , ..., x n ) := u ( x ) ϕ ( x ) ...ϕ n ( x n ) is factorable strongly p -summing. Thus, in particular, there is a C > m X j =1 k T ( x ,j , ..., x n,j ) k p p ≤ C sup k ϕ k≤ m X j =1 | ϕ ( x ,j , ..., x n,j ) | p p . Choose a j ∈ X j such that ϕ j ( a j ) = 1 for all j = 2 , ..., n. Thus m X j =1 k T ( x ,j , a , ..., a n ) k p p ≤ C sup k ϕ k≤ m X j =1 | ϕ ( x ,j , a , ..., a n ) | p p and it follows that u is absolutely p -summing. (cid:3) The following two theorems are immediate consequences of the previous lemma and ofthe respective linear results:
Theorem 5.7 (Dvoretzky-Rogers type theorem) . Let Y be a Banach space. Every contin-uous n -linear operator T : Y × · · · × Y → Y is factorable strongly p -summing if, and onlyif, dim Y < ∞ . Theorem 5.8 (Lindenstrauss–Pe lczy´nski type theorem) . Let m be a positive integer. If X and Y are infinite-dimensional Banach spaces, X has an unconditional Schauder basis and Π F St, ( m X ; Y ) = L ( m X ; Y ) then X = ℓ and Y is a Hilbert space. For polynomials we have a natural version of Lemma 5.6:
Lemma 5.9.
If every continuous n -homogeneous polynomial P : X → Y is factorablestrongly p -summing, then every continuous linear operator u : X → Y is absolutely p -summing.Proof. Let u : X → Y be a continuous linear operator and ϕ ∈ X ∗ be a non-null linearfunctional and a ∈ X be so that ϕ ( a ) = 1. Then P ( x ) := u ( x ) ϕ ( x ) n − is factorablestrongly p -summing. Thus ˇ P is factorable strongly p -summing. From the proof of Lemma5.6 we conclude that the linear operator v : X → Y defined by v ( x ) = ˇ P ( a, ...., a, x )is absolutely p -summing. But v is a linear combination of u ( a ) ϕ and u ; since u ( a ) ϕ isabsolutely p -summing it follows that u is absolutely p -summing. (cid:3) An immediate consequence of the previous lemma is that the analogs of theorems 5.7 and5.8 work for polynomials. For instance:
Theorem 5.10 (Dvoretzky-Rogers type theorem for polynomials) . Let Y be a Banach space.Every continuous n -homogeneous polynomial P : Y → Y is factorable strongly p -summingif, and only if, dim Y < ∞ . Given P ∈ P ( m X ; Y ) let us consider its transpose P t : Y ∗ → P ( m X ) given by P t ( y ∗ ) := y ∗ ◦ P . Note that P t is a continuous linear operator. Let P tt : P ( m X ) ∗ → Y ∗∗ be thetranspose of P t . It is well known (see [17, Theorem 2.21]) that, if Y = H is a Hilbertspace then a continuous linear operator is absolutely 1-summing whenever its transpose isabsolutely p -summing for some 1 ≤ p < ∞ . Let us see that the analogous result is true forpolynomials. Proposition 5.11.
Let H be a Hilbert space and P ∈ P ( m X ; H ) . If P t ∈ Π p ( ˆ ⊗ m,sπ s X ; H ) for some ≤ p < ∞ then P ∈ P F St, ( m X ; H ) .Proof. From the equality P tL,s = δ ◦ P t , where δ : P ( m X ) → (cid:0) ˆ ⊗ m,sπ s X (cid:1) ∗ is the canonicalisomorphism, it follows that P tL,s is absolutely p -summing and then P L,s is absolutely 1-summing. By Theorem 4.8 we conclude that P is factorable strongly 1-summing. (cid:3) Proposition 5.12.
Let P ∈ P ( m X ; Y ) . Then P ∈ P F St,p ( m X ; Y ) if and only if P tt ∈ Π p ( P ( m X ) ∗ ; Y ∗∗ ) .Proof. It is a consequence of Theorem 4.8, the fact that P ttL,s = P tt ◦ δ t and the analogouswell known property for linear operators (see [17, Proposition 2.19]). (cid:3) Coherence and compatibility
Let us denote the ideal of factorable strongly p -summing n -homogeneous polynomials by P nF St,p , whereas Π nF St,p denotes the ideal of factorable strongly p -summing n -linear oper-ators. The notions of coherent and compatible ideals of polynomials were introduced byCarando, Dimant and Muro [16] in order to evaluate what polynomial approaches preservethe spirit of a given operator ideal. Standard calculations show that (cid:0) P nF St,p (cid:1) ∞ n =1 is coherentand compatible with Π p . Very recently, in [34], the notions of coherence and compatibility
EAK COMPACTNESS AND STRONGLY SUMMING MULTILINEAR OPERATORS 15 were extended to pairs of ideals of polynomials and multi-ideals. It is also possible to showthat (cid:0) P nF St,p , Π nF St,p (cid:1) ∞ n =1 is coherent and compatible with Π p . We have shown that P nF St,p coincides with the composition ideal with the absolutely p -summing operators. However, we cannot apply [34, Theorem 5.7] to get the coherence andcompatibility as the topology involved in that result comes from the multilinear operatorsspace norm and it does not coincide with k · k F St,p (see [9]). Despite of this, standardcalculations also allow to get the following:
Theorem 6.1.
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