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1 Document downloaded from: htt://hdl.handle.net/10251/49170 This aer must be cited as: Achour, D.; Dahia, E.; Rueda, P.; D. Achour; Sánchez Pérez, EA. (2013). Factorization of absolutely continuous olynomials. Journal of Mathematical Analysis and Alications. 405: doi: /j.jmaa The final ublication is available at htt://dx.doi.org/ /j.jmaa Coyright Elsevier

2 FACTORIZATION OF ABSOLUTELY CONTINUOUS POLYNOMIALS D. ACHOUR, E. DAHIA, P. RUEDA AND E. A. SÁNCHEZ PÉREZ1 Abstract. In this aer we study the ideal of dominated (; σ)-continuous olynomials, that extend the nowadays well known ideal of -dominated olynomials to the more general setting of the interolated ideals of olynomials. We give the olynomial version of Pietsch s factorization Theorem for this new ideal. Although based in [11], our factorization theorem requires new techniques insired in the theory of Banach lattices. Introduction The oerator ideal of (, σ)-absolutely continuous oerators was introduced in 1987 in order to analyze suer-reflexivity and some other roerties of Banach saces ([20]). This new ideal was created by means of a general interolation rocedure due to Jarchow and Matter ([16]), and must be understood as an ideal located in between absolutely -summing oerators and continuous oerators. Matter [20] alied (, σ)-absolutely continuous oerators to obtain a descrition of oerators factoring through suer-reflexive Banach saces. Later, several authors studied factorization roerties of this new class of oerators, the tensor roduct reresentation and found more alications (see for examle [1, 17, 18]). The multi-ideal of (; 1,..., m ; σ)-absolutely continuous multilinear oerators on Banach saces has been recently defined and characterized by Dahia et al. in [13] as a natural multilinear extension of the classical ideal of (; σ)-absolutely continuous linear oerators. This multi-ideal has many good roerties and extends almost all the ones that are satisfied by the ideals of absolutely -summing and -dominated multilinear oerators, as inclusion theorems, Pietsch domination theorems, factorization theorems and tensor roduct reresentations. On the other hand, in the last ten years a considerable effort has been made in order to increase the knowledge on the olynomials that belong to some oerator ideal (see for instance [2, 3, 4, 12, 22] and the references therein). The case of the -dominated olynomials is articularly relevant and has been intensively studied ([6, 7, 8, 9, 10]). In this aer we introduce and study the olynomial version of (; 1,..., m ; σ)-absolutely continuous multilinear oerators, that will be called (; q; σ)-absolutely continuous olynomials. An esecial attention is given to the articular case of dominated (; σ)- continuous olynomials. Insired by the factorization theorem for dominated olynomials 2000 Mathematics Subject Classification. Primary 46A32, Secondary 47B10. Key words and hrases. Absolutely continuous olynomials, Pietsch domination theorem. 1 Corresonding Author. Instituto Universitario de Matemática Pura y Alicada, Universidad Politécnica de Valencia. Tel easance@mat.uv.es. 1

3 2 D. ACHOUR, E. DAHIA, P. RUEDA AND E. A. SÁNCHEZ PÉREZ [8], we rove a more general factorization scheme for dominated (; σ)-continuous olynomials. The factorization in [8] is based in finding a rototye of a -dominated olynomial with values in a linear subsace of an L sace, endowed with a suitable norm, through which any -dominated olynomial factors. The imossibility of keeing the L -norm on such a subsace obligates to consider again another suitable norm when working on a factorization diagram for dominated (; σ)-continuous olynomials that, far from being a mere adatation of the revious one, is built by means of convexification techniques. Although the main stes come from [11], the factorization theorem we resent here requires new techniques adated to (; σ)-dominated olynomials. To be more recise, as in the above aer, adequate renormed subsaces of L saces are constructed. However, the interolating nature of this new class of olynomials yields to build them by means of methods that have their roots on the convexification of Banach functions saces and the interolation theory. We also retend to suggest that the basic lines of the rocedure shown in [11] could be actually extended, with extra work, to the broad class of ideals P of olynomials so that the underlying ideal I of linear oerators is characterized by a domination theorem and that any P P can be decomosed as P = Q u, where Q is a olynomial and u a linear oerator in I. This aer is organized as follows. In Section 1, we recall some notation and basic facts on sequences saces and olynomials on Banach saces. In Section 2, we study and characterize the ideal of (; q; σ)-absolutely continuous olynomials. In Section 3 we resent a articular case: the dominated (; σ)-continuous olynomials, where a factorization should aly. Far from being trivial, the exectations are met. First we establish a domination theorem for such oerators similar to one that holds in the m-linear case, comaring also dominated (; σ)-continuous and -dominated olynomials and, in Section 4, we show our main result: the factorization theorem for dominated (; σ)-continuous olynomials. 1. Definitions and general results The definitions and notations used in the aer are, in general, standard. Let m N and X 1,..., X m, X, Y, F, G be Banach saces over K ( either R or C). The sace of all continuous m-linear maings T : X 1... X m Y will be denoted by L(X 1,..., X m ; Y ). It becomes a Banach sace with the natural norm T = su { T (x 1,..., x m ) : x j 1, j = 1,..., m }. In the case X 1 =... = X m = X, we will simly write L ( m X; Y ). As usual, L(X; Y ) := L( 1 X; Y ) is the sace of bounded linear oerators from X to Y. Let 1 <. We will write l n (X) for the sace of all sequences (x i ) n in X with the norm (x i ) n = ( n x i ) 1, and l n,ω (X) for the sace of all sequences (x i ) n in X with the norm (x i ) n,ω = su φ X 1 ( n φ(x i ) ) 1, where X denotes the toological dual of X. The closed unit ball of X will be denoted by B X. Let l (X) be the Banach sace of all absolutely -summable sequences (x i ) in X

4 with the norm FACTORIZATION OF ABSOLUTELY CONTINUOUS POLYNOMIALS 3 (x i ) = ( x i ) 1 We denote by l ω (X) the Banach sace of all weakly -summable sequences (x i ) in X with the norm (x i ),ω = su ξ X 1 (ξ(x i )). Note that l,ω (X) = l (X) for some 1 < if, and only if, X is finite dimensional. The sace l σ (X) of (; σ)-weakly summable sequences was introduced in [17] in order to give a characterization of the class of (; σ)-absolutely continuous oerators (see [17, Theorem 1.7]). Now we recall some roerties of this sace. Let 1 < and 0 σ < 1. Define and It is clear that δ σ ((x i ) ) = su φ B X ( ) ( φ(x i ) x i σ) H ;σ (X) = {(x i ) X : δ σ((x i ) ) < }. (x i ),ω δ σ((x i ) ) (x i), (x i ) H ;σ(x). (1) A sequence (x i ) in X is (; σ)-weakly summable if it belongs to the vector normed sace l σ (X) sanned by H ;σ (X) with the norm (x i ) ;σ = inf k ) δ σ ((x l i) where the infimum is taken over all reresentations of (x i ) of the form l=1 (x i ) = k ( ) x l i, with ( x l i) H ;σ(x), k N. In addition, we have the inclusions with l=1 l (X) lσ (X) l,ω (X), (x i ),ω (x i) ;σ (x i) for all (x i ) l (X). The comletion of l σ (X) is denoted by ˆl σ (X). A ma P : X Y is an m-homogeneous olynomial if there exists a unique symmetric m-linear oerator ˇP : X (m)... X Y such that P (x) = ˇP ( ) x, (m)..., x for every x X. Both are related by the olarization formula [23, Theorem 1.10] ˇP ( x 1,..., x m) = 1 2 m ε 1 ε m P (ε 1 x ε m x m ), (x j ) m X. m! ε i =±1

5 4 D. ACHOUR, E. DAHIA, P. RUEDA AND E. A. SÁNCHEZ PÉREZ We denote by P( m X; Y ) the Banach sace of all continuous m -homogeneous olynomials from X into Y endowed with the norm P = su { P (x) : x 1} = inf {C : P (x) C x m, x X}. The (finite) linear combinations of the m-homogeneous olynomials x φ (x) m y, where φ X and y Y, are called olynomials of finite tye. For the general theory of homogeneous olynomials we refer to [14]. X denote the m-fold symmetric tensor roduct of X endowed with the rojective s-tensor norm, and m,s X stands for its comletion. We use P L to denote the linearization of the olynomial P P( m X; Y ), that is, P L is a linear oerator from m,s X into Y such that P (x) = P L (x x) for every x X. The corresondence between a olynomial and its linearization establishes an isometric isomorhism between P( m X; Y ) and L( m,s X; Y ). For definitions and basic roerties of symmetric tensor roducts, the s-tensor norm and the interlay with homogeneous olynomials we refer to [15]. In this aer we follow the standard definition of ideal of olynomials which can be found for examle in [5]. For a fixed ideal of olynomials Q and m N, the class Q m := E,F Q( m E; F ) is called ideal of m-homogeneous olynomials. Let m N and let X 1,..., X m and Y be Banach saces. Let 1, 1,..., m < with m and 0 σ < 1. The definition of (; 1,..., m ; σ)-absolutely continuous multilinear oerator below was firstly given in [13]. Let m,s Definition 1.1. A maing T L(X 1,..., X m ; Y ) is (; 1,..., m ; σ)-absolutely continuous if there is a constant C > 0 such that ( T ( x 1 i,..., x m )) n m ( ) i C δ j σ (x j i )n (2) for all choices of m N and x j 1,..., xj n X j, (1 j m). The sace of all such m-linear oerators is denoted by L σ as(; 1,..., m) (X 1,..., X m ; Y ) and is endowed with the norm given by T L σ = inf {C > 0 : C satisfies (2)}. With this notation, (L σ as(; 1,..., m),. L σ as(; 1,...,m)) is a Banach multi-ideal that generalizes the corresonding ideal of linear oerators. Indeed, when m = 1 and 1 =, (L σ as(;),. L σ ) coincides with the ideal of all (; σ)-absolutely continuous linear oerators. This multi-ideal has to be thought as an intermediate ideal in between the ideal of all as(;) absolutely (; 1,..., m )-summing m-linear oerators and the whole class of continuous m-linear maings. Indeed, both classes are attained for σ = 0 and σ = 1 resectively. In the case that 1 =... = m = q and 1 = m = m q we say that T is (q, σ)-dominated continuous and we denote the corresonding vector sace and norm by L σ d,q (X 1,..., X m ; Y ) and. L σ resectively. In this case, the inequality (2) can be written d,q as ( T ( x 1 i,..., x m i )) n q m() m C δ qσ ((x j i )n ). (3)

6 FACTORIZATION OF ABSOLUTELY CONTINUOUS POLYNOMIALS 5 2. (;q;σ)-absolutely continuous m-homogeneous olynomials In this section we define and characterize the notion of (; q; σ)-absolutely continuous m-homogeneous olynomials, according to the definition of (; 1,..., m ; σ)-absolutely continuous multilinear oerators. We start by resenting the following result which characterizes (; 1,..., m ; σ)-absolutely continuous m-linear oerators as those which take adequate (; σ)-weakly summable sequences into adequate -summable sequences as exected. This result will be useful to relate (; 1,..., m ; σ)-absolutely continuous multilinear oerators and the corresonding homogeneous olynomials. Definition 2.1. Let m N, 1, q < + such that m q and 0 σ < 1. A olynomial P P( m X; Y ) is called (; q; σ)-absolutely continuous if there exists a constant C > 0 such that for every (x i ) n X, (P (x i )) n C. (δ qσ ((x i ) n ))m. (4) The sace of all such olynomials is denoted by Pas(,q) σ (m X; Y ). It is equied with the comlete norm. P σ, which is comuted as the infimum of all constants C such that as(,q) the inequality (4) holds. For σ = 0 we have P 0 as(,q) (m X; Y ) = P,q ( m X; Y ), the sace of absolutely (; q)- summing olynomials (see [19]). Remark 2.2. Let us show some basic ways of constructing olynomials belonging to our new class. Let X, Y and Z be Banach saces and let m N, 1, q < + such that m q and 0 σ < 1. (a) Every m-homogeneous olynomial of finite tye from X into Y is (; q; σ)-absolutely continuous. A simle calculation shows this result. (b) Let us show a articular examle of the case mentioned above. Let P = id K m be the olynomial id K m : K K given by id K m(x) = x m. The following calculations show that it is (; q; σ)-absolutely continuous and that id K m P σ = 1. Let (x i ) n as(,q) K. By the inequality (1) we can write (id K m(x i )) n = (x i ) n m m (x i ) n m q,ω (δ qσ ((x i ) n ))m. It follows that id K m Pas(,q) σ (K; K) and id K m Pas(,q) σ 1. In fact, it can be easily shown that id K m P σ id K m = 1. as(,q) (c) Let Q P( m X; Y ) and let u : Z X be a (; σ)-absolutely continuous linear oerator. Then the olynomial P = Q u is ( m ; ; σ)-absolutely continuous and P P σ m Q (π ;σ (u)) m., In order to see this, note that if (z i ) n Z, then (P (z i )) n Q. u(z i ) n m Q. u m L σ. (δ σ((z i ) n )) m. as(;) m()

7 6 D. ACHOUR, E. DAHIA, P. RUEDA AND E. A. SÁNCHEZ PÉREZ As a consequence of arts (a) and (b) of the remark above and the next result which ( roof is straightforward ) using calculation as in Remark 2.2(c) we obtain that Pas(,q) σ,. Pas(,q) σ is a normed olynomial ideal. Proosition 2.3. (Ideal roerty). Let u L(X, G) and v L(F, Y ). If P P( m G, F ) is (; q; σ)-absolutely continuous, then v P u is (; q; σ)-absolutely continuous and v P u P σ as(,q) v P P σ as(,q) u m. Although (; q; σ)-absolutely continuous olynomials have been introduced indeendently of (; 1,..., m )-absolutely continuous multilinear maings, in order to relate both classes we characterize first these classes of non linear oerators by means of their summability roerties. As in the classical cases, the natural way of resenting the summability roerties of our m-linear oerators is by defining the corresonding oerator between adequate sequence saces. An m-linear oerator T L(X 1,..., X m ; Y ) induces an m-linear oerator T maing ˆl 1σ (X 1 )... ˆl mσ (X m ) into Y N that is given by T ( (x 1 i ),..., (x m i ) ) = (T (x 1 i,..., x m i )). Proosition 2.4. For T L(X 1,..., X m ; Y ) the following conditions are equivalent: (a) T is (; 1,..., m ; σ) -absolutely continuous. (b) If (x j i ) ljσ (X j ), for j = 1,..., m, then (T (x 1 i,..., xm i )) l (Y ). (c) The maing T : ˆl 1 σ (X 1 )... ˆl mσ (X m ) l (Y ) is well-defined and continuous. In this case T L σ = T. Proof. It is clear that (c) imlies (b) and that (c) imlies (a) with T L σ T. Assume that T L σ as(; 1,..., m) (X 1,..., X m ; Y ). Note first that if (x j i ) H j,σ(x j ), j = 1,..., m we have ( T ( x 1 i,..., xm i )) n for all n N. Then ( T ( x 1 i,..., xm i )) T L σ T L σ m δ j σ((x j i )n ) m δ j σ((x j i ) ). T L σ m δ j σ((x j i ) ). Now let (x j i ) ljσ (X j ), j = 1,..., m. For each j = 1,..., m and ε > 0, there exists ) H j,σ(x j ) such that (x j,l i k j (x j i ) = l=1 (x j,l i ) and k j l=1 δ j σ ( ) (x j,l i ) ε + (x j i ). j,σ

8 So we have ( T ( x 1 i,..., x m i FACTORIZATION OF ABSOLUTELY CONTINUOUS POLYNOMIALS 7 T L σ )) k 1 l 1 =1 k 1 l 1 =1 δ 1 σ ( T L σ ε + (x 1 as(;1 i ),...,m) Since this holds for all ε > 0, we obtain ( T ( x 1 i,..., x m i )) k m l m=1 ( ( T x 1,l 1 i ( ) (x 1,l 1 i ) 1,σ )),..., x m,lm i k m δ mσ l m=1 ) ( ) ε + (x m i ) m,σ. T L σ m (x j i ) ( ) (x m,lm i ). j,σ Then it follows that T : l 1σ (X 1 )... l mσ (X m ) l (Y ) is well-defined and continuous with norm T L σ. Its continuous extension to ˆl 1σ (X 1 )... ˆl mσ (X m ) coincides with the maing T : ˆl 1σ (X 1 )... ˆl mσ (X m ) l (Y ) already defined and we see that (a) imlies (b) and (a) imlies (c) with T T L σ. An aeal to the Closed Grah Theorem shows that (b) imlies (c). Actually the Closed Grah Theorem is used to show that T is searately continuous, hence continuous (see [17, Theorem 1.7 (2 1)]). Adating the roof of Proosition 2.4, we easily get the characterization of (; q; σ)- absolutely continuous olynomials by means of transformations of vector valued sequence saces. Proosition 2.5. Let m N, 1, q < + such that m q and 0 σ < 1. Let P P( m X; Y ). Then the olynomial P is (; q; σ)-absolutely continuous if and only if (P (x i )) l (Y ) for every (x i) lqσ (X). The above characterization allows to relate the roerties of the (; q; σ)-absolutely continuous olynomials with the ones of their corresonding symmetric multilinear mas. Corollary 2.6. Let P P( m X; Y ). Then P P σ as(,q) (m X; Y ) if and only if ˇP L σ as(;q,...,q) (m X; Y ). Proof. Assume that ˇP L σ as(;q,...,q) (m X; Y ). For each sequence (x i ) n in X we have (P (x i )) n ( ) = ˇP n (xi, (m)..., x i ˇP L σ. [δ qσ ((x i ) n as(;q,...,q) )] m. It follows that P Pas(,q) σ (m X; Y ). Conversely, let P Pas(,q) σ (m X; Y ) and (x j i ) l qσ (X), j = 1,..., m. By the olarization formula we have ( ( )) ˇP x 1 i, (m)..., x m i = 1 ( 2 m ε 1... ε m P (ε1 x 1 i ε m x m i ) ) m!. (5) ε j =±1

9 8 D. ACHOUR, E. DAHIA, P. RUEDA AND E. A. SÁNCHEZ PÉREZ Now, for every choice ε 1,..., ε m = ±1 of signs we get ( ε 1 x 1 i ε mx m ) i lqσ (X). As in the roof of Proosition 2.4, (a) (b), we obtain ( P (ε 1 x 1 i ε m x m i ) ) P P σ. (ε 1 x 1 as(,q) i ε m x m i ) m q,σ <. This imlies ( P (ε 1 x 1 i ε mx m i )) l (Y ). It follows from (5) that ( ( )) ˇP x 1 i, (m)..., x m i l (Y ), and by Proosition 2.4 this shows that ˇP L σ as(; 1,..., m) (m X; Y ). Let us see that this large class of summing olynomials also satisfies an inclusion theorem. Proosition 2.7. (Inclusion theorem). Let 1 q < and 1 1 q 1 < be such that m 1 1 m q 1 1 q. Then Pσ, 1 ( m X; Y ) Pq,q σ 1 ( m X; Y ). Moreover, we have. P σ q,q1. P σ,1. Proof. By the monotonicity of the l s -norms we may assume that m 1 1 = m q 1 1 q. Considering 1 r, r 1 < with 1 r + 1 q = 1, 1 r q 1 = 1 1 it follows that m r 1 = 1 r. Take P in P, σ 1 ( m X; Y ) and a sequence (x i ) n in X. Observe that for λ i = P (x i ) q r 1, (i = 1,..., n) we have Using Hölder s inequality we obtain P (λ i x i ) = P (xi ) q. P (x i ) q () ) = P P σ,1 P (λ i x i ) su φ B X P P σ,1 ) λ i r 1 (λ i φ(x i ) x i σ) 1 ) m() r 1. (δ q1 σ((x i ) n )m ) 1 m = P P σ,1 P (x i ) q ) r. (δ q1 σ((x i ) n )m. Since r = q, we obtain the following inequality, which roves the result. (P (x i )) n q P P σ,1. [δ q1 σ((x i ) n ]m.

10 FACTORIZATION OF ABSOLUTELY CONTINUOUS POLYNOMIALS 9 3. Dominated (,σ)-continuous Polynomials A relevant secial case of (; q; σ)-absolutely continuous olynomial is when we have m = q. In this situation and following the standard notations in similar cases, we will call the maings dominated (; σ)-continuous olynomials, and we will denote the corresonding vector sace and norm by P σ d, (m X; Y ) and. P σ d,, resectively, for m. Actually, we have Pd, σ (m X; Y ) = P σ m,(m X; Y ), i.e. a olynomial P P( m X; Y ) is dominated (; σ)-continuous if there is a constant C > 0 such that for every (x i ) n X we have P (x i ) n C. [δ σ ((x i ) n )] m. (6) m() Notice that for m = 1 we recover also the ideal of (, σ)-absolutely continuous linear oerators. When σ = 0, Pd, 0 (m X; Y ) is the sace of all -dominated m-homogeneous olynomials, which is denoted simly by P d, ( m X; Y ). The definition and some fundamental results on -dominated homogeneous olynomials between Banach saces can be found in [5], [19] or [22]. Remark 3.1. From Corollary 2.6, [13, Theorem 3.6] and [5, Proosition 9] it follows that the decomosition in art (c) of Remark 2.2 actually characterizes dominated (; σ)- continuous olynomials. Indeed, P P σ d, (m X; Y ) if and only if there is a Banach sace Z, a (; σ)-absolutely continuous linear oerator u : X Z and a olynomial Q P( m Z; Y ) such that P = Q u. This factorization will be used several times and is essential for our uroses of getting a Pietsch tye factorization theorem for dominated (; σ)-continuous olynomials. It is well known that (P d,,. d, ) is a Banach ideal of olynomials if m. Although a domination theorem for dominated olynomials follows easily as in the linear case, to get the corresonding factorization theorem requires new techniques that use mainly symmetric tensor roducts and an adequate reresentation of these saces. This has been done in [8] and [11], where it is shown that any -dominated olynomial factors through a canonical rototye of a -dominated olynomial in the sirit of Pietsch s classical result. Our aim is to obtain a domination/factorization result for the larger class of dominated (; σ)-continuous olynomials. We will see that new constructions of renormed subsaces of L saces different from the ones used in [8] and [11] are required. Theorem 3.2. Let m N and 1 <. An m-homogeneous olynomial P P( m X; Y ) is dominated (; σ)-continuous if and only if there is a regular Borel robability measure µ on B X (with the weak* toology) and a constant C > 0 such that for all x X ( ) m() P (x) C x mσ φ(x) dµ(φ). (7) B X Moreover, in this case P P σ d, = inf {C > 0 : C satisfies (7)}. Proof. First let us assume that (7) holds with C and µ as described. A direct calculation shows easily that (P (x i )) n C. [δ σ ((x i ) n )] m m()

11 10 D. ACHOUR, E. DAHIA, P. RUEDA AND E. A. SÁNCHEZ PÉREZ for every n N and (x i ) n X. Hence P Pσ d, (m X; Y ) and P P σ d, C. For the converse, take P Pd, σ (m X; Y ). By Remark 3.1 there is a Banach sace Z, a (; σ)-absolutely continuous linear oerator u : X Z and a olynomial Q P( m Z; Y ) such that P = Q u. The domination theorem for (; σ)-absolutely continuous linear oerators [20] ensures that there exists a regular Borel robability measure µ on B X and a constant C > 0 such that ( ) u(x) C x σ φ(x) d µ for all x X. Then B X P (x) = Q u(x) Q u(x) m Q C m x mσ ( for all x X. B X ) m() φ(x) d µ Any regular Borel robability measure µ on B X, with the weak* toology that satisfies (7) is called a Pietsch measure for P. An inclusion between the classes of the dominated (; σ)-continuous olynomials and the -dominated olynomials follows easily from the definitions. Proosition 3.3. Let 1 < and 0 σ < 1. Then P d, (m X; Y ) P σ () d, (m X; Y ). Consequently, P d, ( m X; Y ) Pd, σ (m X; Y ). Proof. Let P P d, (m X; Y ). Let (x i ) n () be a sequence in X. Using inequality (1) we have ) m() [ ] m P (x i ) m() P d, (x i ) n P (),ω d, [δ σ ((x i ) n ]m. () Then P P σ d, (m X; Y ) and P P σ d, P d, () Since it follows that P d,( m X; Y ) P d, inclusion P d, ( m X; Y ) Pd, σ (m X; Y ) is roved.. Hence P d, (m X; Y ) P σ () d, (m X; Y ). (m X; Y ) (see [22]). Hence the () By [4, Examle 1] there is a m-dominated olynomial P P( m X; Y ), m 2, which is not weakly comact. Then Proosition 3.3 gives the existence of a dominated (m, σ)- continuous olynomial which is not weakly comact. Remark 3.4. In general, P σ d, P d,. (). () Let us show an examle of a olynomial belonging to Pd, σ that is not in P d, Let L1 := L 1 [0, 1] or L 1 := l 1, and L 2 the corresonding Hilbert sace. We know by [13, Examle 3.8 ] that there is a symmetric bilinear oerator T : L 2 L 2 L 1 such that T L σ as(1;2,2) (2 L 2 ; L 1 ) but T / L as( 1 () ; 2 (), 2 () )(2 L 2 ; L 1 ). Then, by Corollary 2.6 and [22, Theorem 6], the olynomial ˆT P( 2 L 2 ; L 1 ) associated to T satisfies that ˆT Pd,2 σ (2 L 2 ; L 1 ), but ˆT / P d, 2 ( 2 L 2 ; L 1 ).

12 FACTORIZATION OF ABSOLUTELY CONTINUOUS POLYNOMIALS The factorization theorem As in the case of the -dominated multilinear oerators, we will show that there is a factorization theorem that characterizes when a olynomial is dominated (; σ)-continuous. In fact, this theorem resents the rototye of dominated (; σ)-continuous olynomial, i.e. the olynomial belonging to this class through which each olynomial of the class factors. The ideas for roving the factorization follows the lines of the one that are used in [11] but there are some meaningful differences based on the fact that the domination for the dominated (; σ)-continuous olynomials is not based in a norm but in some interolated exression between the norm of X and the one of L (µ). To deal with, we use techniques insired on the convexification of Banach lattices. If X is a Banach sace and m N, we define the m-homogeneous olynomial : X C(B X ) ; (x)(ϕ) = ϕ(x) m. We consider the restriction δ of its linearization to the m-fold symmetric tensor roduct m,s X. So defined, δ is a linear oerator δ : m,s X C(B X ) given by δ(x x)(ϕ) = ϕ(x) m, x X, ϕ B X. By Lemma 4.1. in [8], this ma is injective. To simlify the notation, sometimes we shall write m x := x x. Let δ m stand for the canonical m-homogeneous olynomial from X to m,s X defined by δ m (x) = m x. Let i X : X C(B X ) be the canonical isometric inclusion given by the evaluation. Given µ a Borel measure on B X, j : C(B X ) L (µ) denotes the canonical ma. On j i X (X) consider the seminorm j i X (x) := inf x j σ j i X (x j ) L (µ) : x = x j, x j X, n N Consider the relation j i X (x) j i X (y) if and only if i X (x y) = 0 and denote by L,σ (µ) the quotient sace and by j,σ : i X (X) L,σ (µ) the quotient ma. Then becomes a norm on L,σ (µ) that we shall call L,σ. Its comletion can be identified with the real interolation sace (X, L (µ)),1 (see [13, Section 3] for more details on this sace). Following a general construction that is well-known for the case of Banach function saces (see for examle [24, Ch.2]), we can define what we call the m-th ower L,σ (µ) [m] of L,σ (µ). However, notice that in this case this new sace is not a Banach function sace. It is a linear sace that is defined as the linear san of all olynomials of the form (j i X(x)) m for x being an element of X, i.e. m L,σ (µ) [m] := = h L m (µ) : h = h L m (µ) : h = λ j j (x j), x j X, λ j K, n N m λ j j (i X(x j ) m ), x j X, λ j K, n N m. Consider the m-homogeneous olynomial Q : j,σ i X (X) L,σ (µ) [m] given by Q(j,σ i X (x)) := j m (x),

13 12 D. ACHOUR, E. DAHIA, P. RUEDA AND E. A. SÁNCHEZ PÉREZ and let Q L be its linearization. Given h = n λ jj (x j) L,σ (µ) m [m] we will denote by θ the tensor θ := n λ j m x j. Let T : n,s E n,s j i X (E) be the linear oerator given by T ( m x) = m j (i X (x)) for every x E. For each h L,σ (µ) [m] define π ;σ,m (h) := inf λ j j,σ i X (x j ) m L,σ : T (θ) = λ i m j i X (x i ) { = inf λ i The equalities ( ni x i k σ j i X (x i k ) L Q L T (θ) = j δ(θ) = j δ m m yield that π ;σ,m (h) is well defined. ) m : T (θ) = n j λ j m ( x j k ) = Proosition 4.1. If m then π ;σ,m is a norm on L,σ (µ) [m]. } n i λ i m (j i X ( x i k )) nj λ j j ( x j m k ) = h Proof. Easy calculations rove that π ;σ,m satisfies the axioms of norm. The only one that requires some attention is that π ;σ,m (h) = 0 imlies h = 0. Following [11], on the sace X m := j m i X (X) L (µ) a norm is defined by m m { } π ((j δ)(θ)) := inf λ i (j δ) m x i ) m m m L m, π ((j δ)(θ)) = inf m m { where the infimum is taken over all reresentations of T (θ) m,s j i X (X) of the form T (θ) = n λ i m (j i X (x i )) with n N, λ i K and x i X. We have that { } λ i m (j i X (x i )) = inf { inf λ i { inf λ i = π ;σ,m (h) λ i (j i X (x i ) m L : T (θ) = ( ni λ i (j m δ)( mx i ) L m : T (θ) = } λ i m (j i X (x i )) ) m j i X (x i k ) L : T (θ) = ( ni x i k σ j i X (x i k ) L } n i λ i m (j i X ( x i k )) ) m : T (θ) = } n i λ i m (j i X ( x i k )) Therefore, if π ;σ,m (h) = 0 then π ((j δ)(θ)) = 0. Hence (j m m m h = 0. δ)(θ) = 0 and so

14 FACTORIZATION OF ABSOLUTELY CONTINUOUS POLYNOMIALS 13 Next roosition shows that Q L is an isometric isomorhism between ˆ m,s j,σ i X (X) and the comletion of the sace (L,σ (µ) [m], π ;σ,m ). Proosition 4.2. The comletion of the sace (L,σ (µ) [m], π ;σ,m ) is isometrically isomorhic to ˆ m,s j,σ i X (X). Proof. Consider Q L restricted to m,s j,σ i X (X), that is Q L : m,s j,σ i X (X) (L,σ (µ) [m], π ;σ,m ). Let us see first that Q L is onto. Given h = n λ jj (x j) m L,σ (µ) [m], let θ = n λ jx j x j. Then Q L (T (θ)) = λ j Q L ( m j,σ i X (x j )) = λ j Q(j,σ i X (x j )) = λ j j m (x j) = h From the definitions of the norms it follows that π ;σ,m (j i X(x)) = (T (θ)) and so Q L m is an isometry. Therefore, the extension of Q L to the comletions is the required isometric isomorhism. To simlify the notation we shall use (L,σ (µ) [m], π ;σ,m ) for its comletion too. Let us define a olynomial which shall lay the role of the canonical rototye of dominated (; σ)-continuous m-homogeneous olynomial through which any other olynomial of the class must factor. Define j ;σ,m := Q j,σ : i X (X) L,σ (µ) [m]. For each x X, j ;σ,m (i X (x)) = Q j,σ (i X (x)) = j m (x) = (j m i X(x)) m and so j ;σ,m can be identified with the restriction of j m m L,σ (µ) [m]. to i X (X) and with values in Proosition 4.3. The m-homogeneous olynomial j ;σ,m is dominated (; σ)-continuous. Proof. Given x 1,..., x n X, π ;σ,m (j ;σ,m (i X (x i ))) m() ) m() = = π ;σ,m ((j i X(x i )) m ) m (j,σ i X (x i ) (j i X (x i ) C[δ σ ((i X (x i ) n )] m. L,σ L,σ m() ) m() ) m() ) m()

15 14 D. ACHOUR, E. DAHIA, P. RUEDA AND E. A. SÁNCHEZ PÉREZ To get the factorization of dominated (; σ)-continuous olynomials through j ;σ,m we need some reliminary results. Although the following roosition retends to generalize [11, Proosition 3.4], it only alies for some secific Pietsch measures of a dominated (; σ)-continuous m-homogeneous olynomial P and a different roof is required. From Remark 3.1, P can be written as P = Q u, where u is a (; σ)-absolutely continuous linear oerator from X into some Banach sace Z and Q : Z Y is a continuous m-homogeneous olynomial. An easy calculation shows that any Pietsch measure µ for u is a Pietsch measure for P. In the following result we are considering µ such a measure. Proosition 4.4. Let P P σ d, (m X; Y ). If x, y X are such that j,σ i X (x) = j,σ i X (y) then P (x) = P (y). Proof. By Remark 3.1, there exist a Banach sace Z, a (; σ)-absolutely continuous linear oerator u : X Z and a olynomial Q P( m Z; Y ) such that P can be written as P = Q u. Let µ be a Pietsch measure for u with constant C. Therefore, ˇP = ˇQ (u,..., u). Take ɛ > 0. The equality j,σ i X (x) = j,σ i X (y) says that j,σ i X (x y) = 0. Then, there exist x 1,..., x n X such that x y = n x k and It follows that P (x) P (y) = ˇP (x,..., x) ˇP (y,..., y) = m m ˇQ < C ˇQ x k σ j i X (x k ) L (µ) < ɛ. (8) ˇP (y j+1, x k, x n j ) = m m ˇP (y j 1, x y, x n j ) ˇQ u(y) j+1 u(x k ) u(x) n j m u(y) j+1 u(x) n j C m u(y) j+1 u(x) n j ɛ ˇQ(u(y) j+1, u(x k ), u(x) n j ) x k σ j i X (x k ) L (µ) As ɛ > 0 is arbitrary we conclude that P (x) = P (y). We resent now the factorization theorem for dominated (; σ)-continuous olynomials. Theorem 4.5. Let m N and m. A olynomial P P( m X; Y ) is dominated (; σ)- continuous if and only if there exist a regular Borel robability measure µ on B X, with the weak* toology, and a continuous linear oerator v : (L,σ (µ) [m], π ;σ,m ) Y such that

16 FACTORIZATION OF ABSOLUTELY CONTINUOUS POLYNOMIALS 15 the following diagram commutes X i X i X (X) P j ;σ,m Y v L,σ (µ) [m] (9) C(B X ) L m (µ) Proof. Assume that P Pd, σ (m X; Y ). By Remark 3.1, there exist a Banach sace Z, a (; σ)-absolutely continuous linear oerator u : X Z and a olynomial Q P( m Z; Y ) such that P can be written as P = Q u. Let µ be a Pietsch measure for u. For each x X define R(j,σ i X (x)) := P (x). By Proosition 4.4 the ma R : j,σ i X (X) Y j,σ i X (X) L,σ (µ) [m] be the surjective isometric isomorhism given by Proosition 4.2. Taking into account the commutative diagrams is a well-defined m-homogeneous olynomial. Let Q L : m,s P X Y i X (X) v j m R L j ;σ,m i X (X) L,σ (µ) [m] Q j,σ Q L j,σ i X (X) ˆ m,s j,σ i X (X) let us define v := R L (Q L ) 1. It is clear that, so defined, v is continuous and closes the diagram (9). The converse follows from Proosition 4.3 and the ideal roerty. D. Achour acknowledges with thanks the suort of the Ministére de l Enseignament Suérieur et de la Recherche Scientifique (Algeria) under roject PNR 8-U E. Dahia acknowledges with thanks the suort of the Ministére de l Enseignament Suérieur et de la Recherche Scientifique (Algeria) under grant 170/PGRS/C.U.K.M(2012) for short term stage. P. Rueda acknowledges with thanks the suort of the Ministerio de Ciencia e Innovación (Sain) MTM E.A. Sánchez Pérez acknowledges with thanks the suort of the Ministerio de Economía y Cometitividad (Sain) MTM C References [1] G. Arango, J. A. Lóez Molina and M. J. Rivera, Characterization of g,σ-integral oerators. Math. Nach. 278,9 (2005),

17 16 D. ACHOUR, E. DAHIA, P. RUEDA AND E. A. SÁNCHEZ PÉREZ [2] R. M. Aron and P. Rueda, -comact homogeneous olynomials from an ideal oint of view. Function saces in modern analysis, 61 71, Contem. Math., 547, Amer. Math. Soc., Providence, RI, [3] R. M. Aron, P. Rueda, Ideals of homogeneous olynomials. Publ. Res. Inst. Math. Sci. To aear. [4] G. Botelho, Weakly comact and absolutely summing olynomials. J. Math. Anal. Al. 265 (2002), no. 2, [5] G. Botelho, Ideals of olynomials generated by weakly comact oerators. Note Mat. 25 (2005/06), no. 1, [6] G. Botelho and D. Pellegrino, Dominated olynomials on L-saces. Arch. Math. (Basel) 83 (2004), no. 4, [7] G. Botelho and D. Pellegrino, Scalar-valued dominated olynomials on Banach saces. Proc. Amer. Math. Soc. 134 (2006), no. 6, [8] G. Botelho, D. Pellegrino and P. Rueda, Pietsch s factorization theorem for dominated olynomials. J. Funct. Anal. 243 (2007), no. 1, [9] G. Botelho, D. Pellegrino and P. Rueda, Dominated bilinear forms and 2-homogeneous olynomials. Publ. Res. Inst. Math. Sci. 46 (2010), no. 1, [10] G. Botelho, D. Pellegrino and P. Rueda, Dominated olynomials on infinite dimensional saces. Proc. Amer. Math. Soc. 138 (2010), no. 1, [11] G. Botelho, D. Pellegrino and P. Rueda, Correction to: Pietsch s factorization theorem for dominated olynomials, Journal of Functional Analysis 243 (2007) [12] D. Carando, V. Dimant, and S. Muro, Holomorhic functions and olynomial ideals on Banach saces. Collect. Math.63 (2012), no. 1, [13] E. Dahia, D. Achour, E.A. Sánchez-Pérez, Absolutely continuous multilinear oerators, J. Math. Anal. Al. 397 (2013) [14] S. Dineen, Comlex Analysis on Infinite Dimensional Saces, Sringer-Verlag, London, [15] K. Floret, Natural norms on symmetric tensor roducts of normed saces, Note Mat. 17 (1997), [16] H. Jarchow and U. Matter, Interolative constructions for oerator ideals, Note Mat. 8 (1988), [17] J.A. Lóez Molina and E.A. Sánchez-Pérez, Ideals of absolutely continuous oerators. (Sanish) Rev. Real Acad. Cienc. Exact. Fís. Natur. Madrid 87 (1993), no. 2-3, [18] J.A. Lóez Molina and E.A. Sánchez-Pérez, On oerator ideals related to (, σ)-absolutely continuous oerators, Studia Math. 138(1) (2000), [19] M. Matos, Absolutely summing holomorhic maings, An. Acad. Brasil. Ciênc. 68 (1996) [20] U. Matter, Absolute continuous oerators and suer-reflexivity, Math. Nachr. 130 (1987) [21] U. Matter, Factoring through interolation saces and suer-reflexive Banach saces. Rev. Roumaine Math. Pures Al. 34 (1989), no. 2, [22] Y. Meléndez and A. Tonge, Polynomials and the Pietsch domination theorem, Proc. Roy. Irish Acad. Sect. A 99 (1999), [23] J. Mujica, Comlex Analysis in Banach Saces, Dover Publications, [24] S. Okada, W. J. Ricker, and E. A. Sánchez-Pérez, Otimal domain and integral extension of oerators, Oerator Theory: Advances and Alications, vol. 180, Birkhäuser Verlag, Basel, 2008, Acting in function saces. [25] A. Pietsch, Oerator ideals. North-Holland Mathematical Library, 20. North-Holland Publishing Co., Amsterdam-New York, D. Achour, University of M sila, Laboratoire d Analyse Fonctionnelle et Géométrie des Esaces, M sila, Algeria. address: dachourdz@yahoo.fr E. Dahia, University of M sila, Laboratoire d Analyse Fonctionnelle et Géométrie des Esaces, M sila, Algeria. address: hajdahia@gmail.com

18 FACTORIZATION OF ABSOLUTELY CONTINUOUS POLYNOMIALS 17 P. Rueda, Deartamento de Análisis Matemático, Universidad de Valencia, Burjasot Valencia. Sain. address: E. A. Sánchez Pérez, Instituto Universitario de Matemática Pura y Alicada, Universidad Politécnica de Valencia, Camino de Vera s/n, Valencia. Sain. address: easance@mat.uv.es