PRODUCT FACTORABLE MULTILINEAR OPERATORS DEFINED ON SEQUENCE SPACES

We prove a factorization theorem for multilinear operators acting in topological products of spaces of (scalar) p-summable sequences through a product. It is shown that this class of multilinear operators called product factorable maps coincides with the well-known class of the zero product preserving operators. Due to the factorization, we obtain compactness and summability properties by using classical functional analysis tools. Besides, we give some isomorphisms between spaces of linear and multilinear operators, and representations of some classes of multilinear maps as n-homogeneous orthogonally additive polynomials.


Introduction
The objective of the paper is to present a factorization theorem for multilinear operators de…ned on the topological product of spaces of p-summable sequences through the product of (multiple) scalar sequences. Such a factorization has been studied for multilinear operators de…ned on Banach algebras and vector lattices, and in the last years it has been studied for Banach spaces (see [1,6,12] and references therein).
Factorization through a product is closely related to a property that is called zero product preservation, or orthosymmetry in the case of vector lattices, for which orthogonality is used to generalize the notion of having product equal to 0, that is just given for the case of function lattices. This property states that a multilinear map B : X 1 :::: X n ! Y is 0-valued whenever x i~xj = 0 for some x i 2 X i , x j 2 X j (i; j 2 f1; 2; :::; ng), where~: X 1 :::: X n ! G is a speci…c map called product. For multilinear operators acting in Banach algebras, this factorization gives useful results for the weighted homomorphisms and derivations, where algebraic multiplication is considered as the speci…c map (see [1,2] and references therein). For Riesz spaces, such a factorization is used to obtain interesting results regarding powers of vector lattices, in which orthogonality is involved (see [4,6,7] and references therein).
Recently, the author together with other mathematicians have investigated the class of multilinear operators acting in the topological product of Banach function spaces and integrable functions factoring through the pointwise product and the convolution operation, respectively (see [12][13][14]). Motivated by these ideas, in this paper we introduce the notion of product factorability for multilinear operators de…ned on topological products of spaces of (scalar) p-summable sequences, and we prove that this class coincides with the class of zero product preserving multilinear maps.
This paper is organised as follows: after some preliminaries and notations, in Section 2 we give the de…nitions of the speci…c map product and product factorability for multilinear operators with a necessary and su¢ cient requirement. Section 3 includes the main result of the paper, which as we said above, states that for a particular product and multilinear operators de…ned on the topological product of spaces of p-summable sequences, the class of product factorable maps is the same as the class of zero product preserving maps. In the sequel, some isometries between multilinear operators and linear operators are presented. Section 4 concerns compactness and summability properties based on classical functional analysis properties and theorems of product factorable maps. Section 5 is devoted to give a generalization of the main factorization theorem by using isomorphism between Banach spaces and`p spaces. In the last section, some isometries between product factorable multilinear maps and orthogonally additive n-homogeneous polynomials are given as an application, and the paper is …nished with an example related to diagonal forms.
Throughout the paper, the standard notations from the Banach space theory are used. Nevertheless, before going any further let us describe some of them. The capital letters X; Y; Z will denote the Banach spaces over the scalar …eld K = R or C. We write B X for the unit ball of a Banach space X. X denotes the topological dual of the Banach space X. The notations E = Y and E = Y mean E and Y are isometric and isomorphic, respectively.
Operator (linear, multilinear or polynomial) indicates continuous operator. L n (X 1 ::: X n ; Y ) denotes the Banach space of n-linear maps endowed with the norm kT k = supfkT (x 1 ; :::; x n )k : x i 2 B Xi ; 1 i ng: It will be denoted by L n (X 1 ::: X n ), respectively, L(X; Y ) if Y = R, respectively, n = 1.

E. ERDOG AN
For a positive real number p 1,`p is the Banach space of all scalar valued absolutely p-summable sequences with the norm k(x i )k p = ( P 1 i=1 jx i j p ) 1=p and`1 shows the Banach space of all bounded sequences endowed with the norm k(x i )k 1 = sup i2N jx i j. f1;2;:::;mg will denote the sequence f1; m :::1; 0; 0; 0::::g and fjg shows the elements of standard basis of the space`p whose coordinates are all zero, except j th that equals 1.
For brevity we will write n i=1 X i for the Cartesian product space X 1 :::: X n and n X for the n-fold Cartesian product of the Banach space X.
A linear operator T : X ! Y is called (p,q)-summing if there exists a constant c > 0 such that for every choice of the elements x 1 ; :::; x m 2 X and for all positive integers m, The space of (p; q)-summing operators from X to Y is denoted by p;q (X; Y ) Recall that a Banach space E is said to have the Schur property whenever weak convergent and norm convergent sequences coincide in it. A Banach space E has the Dunford-Pettis property if every linear operator from E into a Banach space F maps weakly compact sets to norm compact ones.
Recall that an (linear, multilinear or polynomial) operator is called (weakly) compact if it maps the unit ball to a relatively (weakly) compact set.
A multilinear operator B : X 1 ::: X n ! Y is called~-factorable for the n.p. product~if it can be factored through the product~: X 1 ::: X n ! Z and a linear operator T : Z ! Y such that B(x 1 ; x 2 ; :::; x n ) = T ~(x 1 ; x 2 ; :::; x n ) = T (x 1~x2~: ::~x n ) for all x i 2 X i (i = 1; :::; n) (see [12, Thus, for a certain continuous linear operator T : Z ! Y , the map B admits a factorization as the form; The author proved in [12,Lemma 2.3.] that a necessary and su¢ cient condition for the~-factorability of a multilinear operator B : X 1 X 2 ::: X n ! Y is given by the existence of a constant k > 0 satisfying the following inequality for every …nite sets of vectors fx j i g m i=1 X j (j = 1; 2; :::; n). A multilinear map B : X 1 X 2 ::: X n ! Y is called zero product preserving (or zero~-preserving) if B(x 1 ; x 2 ; :::; x n ) = 0 if x k~xl = 0 for some x k 2 X k ; x l 2 X l where k; l 2 f1; 2; :::; ng and k 6 = l.
The class of zero~-preserving multilinear operators is a Banach space endowed with the usual operator norm. The Banach space of n-linear zero~-preserving operators de…ned on the X 1 X 2 ::: X n to Y will be denoted by L n 0 (X 1 X 2 ::: X n ; Y ).

Product Factorability of Multilinear Maps acting in Sequence Spaces
Now, we will give the main theorem of the paper that states the class of zero product preserving maps de…ned on n i=1`p i to the Banach space Y are equal to the class of the product factorable operators.
1 pi = 1 r for 1 r; p i < 1 (i = 1; :::; n). The product : n i=1`p i !`r de…ned by x 1 :::: x n = fx 1 (k) :::: we get fjx(k)j r=pi sign(x(k))g 1 k=1 2`p i and B`r (B`p1 B`p2 ::: B`pn ). Now, let us show the equality given in the de…nition of the n.p. product. Take into account sequences x i = fx i (k)g 1 k=1 2`p i for i = 1; 2; :::; n such that x 1 x 2 ::: x n = x. By the generalization of Hölder's inequlity it is easily seen that kx 1 x 2 ::: x n k r kx 1 k p1 kx 2 k p2 :::kx n k pn : Now, let us show the inverse. Since for all k, we can write : Therefore Thus, we get kxk r = inffkx 1 k p1 kx 2 k p2 :::kx n k pn g and is an n.p. product from p1 `p 2 :::: `p n to`r. (1) The operator B is zero -preserving.
(3) There is a constant k > 0 such that for every …nite sets of sequences fx i 1 ; ::::; x i m g `p i (i = 1; 2; :::; n), the following inequality holds; Thus, B admits the following factorization for a unique linear operator T :`r ! Y ; Proof. (1) ) (2) Assume that B is zero -preserving. Let us write the sequences x i 2`p i (i = 1; 2; :::; n) in the form x 1 :::: x n = fx 1 (k) :::: Since fkg flg = 0 whenever k 6 = l, the following equality is obtained B(x 1 ; ::::; for all x i 2`p i (i = 1; 2; :::; n): Now, for all natural number m and every x = x 1 ::: x n , de…ne the map T m :`r ! Y by T m (x) = T m (x 1 ::: x n ) = B m (x 1 ; :::; x n ). Then, it is seen that for all m, the map T m is well-de…ned, linear and continuous operator. Indeed, the linearity is seen by the linearity in the …rst variable of the map B m . Let us show the continuity of the map T m ; f1;:::;mg ; :::; x n f1;:::;mg )k Y kBkkx 1 f1;:::;mg k:::kx n f1;:::;mg k kBkkx 1 k:::kx n k; since this holds for all representations of the sequence x, it is seen that kT m (x)k Y kBkkxk r by the de…nition of n.p. product. For all m, the operator T m is independent of the representation of the sequence x. Indeed, let us assume x = x 1 ::: x n = x 0 1 ::: x 0 n , then it is seen that On the other hand, the set of operators fT m g 1 m=1 is pointwise convergent for each x = x 1 ::: x n 2`r. By the separate continuity of the multilinear map B, this is seen as follows; Thus, fT m (x)g 1 m=1 converges to B(x 1 ; :::; x n ) for all x 2`r such that x = x 1 ::: x n for the elements x i 2`p i (i = 1; :::; n). Let us de…ne the pointwise limit T (x) = lim m!1 T m (x). It is clear that the limit map T is well-de…ned and linear. Besides it is continuous by the uniform boundedness theorem.
Summing up, the linear bounded map T :`r ! Y de…ned by T (x 1 ::: x n ) = B(x 1 ; :::; x n ) is the desired map.

159
The above theorem gives an isometry between the spaces L n 0 ( n i=1`p i ; Y ) and L(`r; Y ).
Theorem 4. The correspondence B ! T is an onto isometry between the Banach Proof. It is easily seen that the map L n  where x i = fx i (k)g 1 k=1 = fjx(k)j r=pi sgn(x(k))g 1 k=1 for all i = 1; :::; n. It is easily seen that the map B ! T is onto, since an n-linear map B T is obtained for every linear map T by de…ning T (x) = B(x 1 ; :::; x n ) for all x = x 1 ::: x n 2`r for the n.p product : n i=1`p i !`r.
Corollary 5. As a result of the above isometry, the following isometries are given for particular p i values.

Compactness and Summability Inquiries for -Factorable Maps
In this section, we investigate compactness and summability for -factorable multilinear operator that are based on the classical analysis properties and theorems like Dunford Pettis property, well-known Grothendieck's theorem or cotype related properties.

Compactness of -Factorable operators.
By the de…nition of norm preserving product, it is seen that a -factorable multilinear map B : n i=1`p i ! Y is (weakly) compact if and only if the linear operator T :`r ! Y appearing in the factorization is (weakly) compact. Now, we will give more speci…c compactness implications for -factorable maps. (1) For r > 1, the map B is weakly compact.
(2) If r = 1 and Y is re ‡exive, then the map B is compact.
(3) For 1 s < r < 1 and Y =`s, the map B is compact.
Proof. (1) This is easily seen by the weakly compactness of the factorization operator T :`r ! Y which is de…ned on the re ‡exive space`r.
(2) B factors through the linear map T :`r ! Y that is weakly compact due to re ‡exivity of the space Y . In addition, T -hence B-is compact by the Dunford-Pettis property of the space`1.
(3) Since the linear operator T :`r !`s is compact whenever 1 s < r < 1 by the Pitt's theorem, the map B is so also (see [9,Chapter 12]).  we obtain T is a 1-summing operator and thus, B satis…es the inequality given in statement (ii).
ii) ) iii) The integral domination given in the third statement is clearly obtained by Pietsch Domination Theorem (see [9,Theorem 2.12]).
iii) ) i) If the map B has the integral domination then it is seen that B(x 1 ; :::; x n ) = 0 whenever x k x l = 0 for some di¤erent k; l 2 f1; :::; ng. Thus, B is zeropreserving and it is -factorable by the main theorem of the paper.

A Generalization of the -Factorable Operators
Let P n i=1 1 pi = 1 r for 1 r; p i < 1 (i = 1; :::; n). Consider n Banach spaces X i (i = 1; :::; n) that are isomorphic to`p i by the isomorphisms P i : X i !`p i . Let us de…ne the product n i=1 Pi : n i=1 X i !`r by n i=1 Pi (f 1 ; :::; f n ) = P 1 (f 1 ) :::: P n (f n ); f i 2 X i : This product can be illustrated by the following diagram; We will call a multilinear map B : Pi preserving if B(f 1 ; :::; f n ) = 0 whenever P k P l (f k ; f l ) = P k (f k ) P l (f l ) = 0 for some k; l 2 f1; :::; ng such that k 6 = l. 162 E. ERDOG AN Theorem 9. Let P n i=1 1 pi = 1 r for 1 r; p i < 1 (i = 1; :::; n). Consider the Banach spaces X i (i = 1; :::; n) that are isomorphic to`p i by means of the isomorphisms P i : X i !`p i . For an n-linear map B : n i=1 X i ! Y , the following statements are equivalent.
(1) The operator B is zero n i=1 Pi preserving. If one of the aboves is satis…ed, then B admits the following factorization; Proof. (1) ) (2) Let us assume that B is zero n i=1 Pi preserving and de…ne the map For the sequences x i 2`p i (i 2 f1; :::; ng), it is seen that B(x 1 ; ::; x n ) = B n i=1 P 1 i (P 1 (f 1 ); ::: Pi preserving, it is obtained that B(x 1 ; ::; x n ) = B(f 1 ; :::; f n ) = 0 whenever x k x l = P k (f k ) P l (f l ) = 0 for some k; l 2 f1; :::; ng. This shows zero -preservation of the map B and therefore B is -factorable by Theorem 3. So we have that there is a linear operator T :`r ! Y such that B = T : By the de…nition of B, we obtain For the elements f i 2 X i that are P i (f i ) = x i 2`p i , we get B(x 1 ; :::; 1 (x 1 ); :::; P 1 n (x n )) = T (P 1 P 1 1 (x 1 ) ::: P n P 1 n (x n )) = T (x 1 :::: x n ): This shows, B is -factorable. By Lemma 2.3 given in [12] and Theorem 3, the inequality given in the statement (3) is obtained.
Pi preserving under the assumption of the statement (3).

Application: Representation As n-homogeneous Polynomial
Recall that an n-linear map B : n X ! Y is called symmetric if B(x 1 ; :::; x n ) = B(x (1) ; :::; x (n) ) (x 1 ; :::; x n 2 X) for any permutation of the …rst n natural numbers. L n s ( n X; Y ) denotes the space of symmetric multilinear operators de…ned on X to Y . for all x 1 ; :::; x n 2`p by the commutativity of the product .
In addition to this, a symmetry is obtained for the general version. Let X be isomorphic to the space`p by the isomorphism P : X !`p. Then any n P factorable n-linear map B : n X ! Y is symmetric.
A map P : X ! Y is called n-homogeneous polynomial if it is associated with an n-linear symmetric map B : n X ! Y such that P (x) = B(x; :::; x) for all x 2 X. The class of n-homogeneous polynomials is a Banach space under the norm kP k = sup kxk=1 kP (x)k. It will be denoted by P( n X; Y ). We refer the book [10] for more information about polynomials.
An n-homogeneous polynomial de…ned on the Banach algebra X is called orthogonally additive if P (x + y) = P (x) + P (y) whenever xy = 0 for x; y 2 X. Similarly we will call an n-homogeneous polynomial de…ned on the Banach space X orthogonally additive if P (x + y) = P (x) + P (y) whenever x~y = 0 for x; y 2 X and an n.p. product~. We denote by P 0 ( n X; Y ) the space of n-homogeneous orthogonally additive polynomials from X to Y . We will write P 0 ( n X) for Y = R.
The Banach space of n-homogeneous orthogonally additive polynomials is closely related to the zero product preserving n-linear operators and several papers can be found in this direction in the literature (see [3,5,12,15,16] and references therein). Now we will give a generalization of the isomorphisms between orthogonally additive n-homogeneous polynomial forms and sequences given in the papers [15] and [16]. 164 E. ERDOG AN Theorem 11. Let 1 n p < 1. There is an onto isometry between the spaces L(`p =n ; Y ) and P 0 ( n`p ; Y ). Particularly, P 0 ( n`p ) = (`p =n ) for a scalar …eld range.
Proof. Consider a linear continuous operator T 2 L(`p =n ; Y ). It is seen that T gives a -factorable n-linear map B T : n`p ! Y de…ned by T (x) = T (x 1 ::: x n ) = B(x 1 ; :::; x n ) for all x i 2`p (i = 1; :::; n) such that x 1 ::: x n = x 2`p =n . Due to the symmetry of the -factorable map B T , an n-homogeneous polynomial P B T :`p ! Y is obtained such that it is orthogonally additive. Indeed, for all x; y 2`p P B T (x + y) = B T (x + y; :::; x + y) For the surjectivity, let us consider an orthogonally additive n-homogeneous polynomial P 2 P 0 ( n`p ; Y ). This polynomial de…nes a 1-homogeneous map T by T (x) = P (x 1=n ) for all x = fx(k)g 1 k=1 2`p =n where x 1=n = fjx(k)j 1=n sign(x(k))g 1 k=1 such that x = fjx(k)j 1=n sign(x(k)) n ::: jx(k)j 1=n sign(x(k))g 1 k=1 = x 1=n n ::: x 1=n 2`p =n . The map T is linear. Indeed, to see this consider the sequences x 0 1 = P m k=1 x 1 (k) fkg and x 0 2 = P m k=1 x 2 (k) fkg de…ned by the sequences x 1 ; x 2 2`p =n . Since (x 0 1 + x 0 2 ) 1=n = P m k=1 (x 1 (k) + x 2 (k)) 1=n fkg , by using the n-homogenity and orthogonally additivity of the polynomial P , we get that Since x 1 = lim m!1 P m k=1 x 1 (k) fkg and x 2 = lim m!1 P m k=1 x 2 (k) fkg , it is obtained that Thus, every orthogonally additive n-homogeneous polynomial P de…nes a linear map T 2 L(`p =n ; Y ). We can illustrate this isometry by the following diagram; where n is the canonical embedding called diagonal mapping from`p to n`p used to de…ne the n-homogeneous polynomials.
Particularly, every n-homogeneous polynomial form P in P 0 ( n`p ) is represented by a sequence in the space`p =(p n) .
Corollary 12. L(`1; Y ) = P 0 ( n`n ; Y ) and every orthogonally additive n-homogenous polynomial P :`n ! R is represented by a bounded scalar valued sequence.
From Corollary 5, Theorem 11 and Corollary 12, we get the following isometries; ? L n 0 ( n`p ; Y ) = P 0 ( n`p ; Y ), where p n. ? L n 0 ( n`n ; Y ) = P 0 ( n`n ; Y ) ? L n 0 ( n`p ) = P 0 ( n`p ) ? L n 0 ( n`n ) = P 0 ( n`n ). We can give some isomorphisms for the n P factorable maps as follows; Corollary 13. Let 1 n p < 1 and P : E !`p is an isomorphism. There is an isomorphism between the spaces L(`p =n ; Y ) and P 0 ( n E; Y ). Particularly, P 0 ( n E) = (`p =n ) for a scalar …eld range.
Let us …nish the paper with an example.
Example 14. Let P n i=1 1 pi = 1 r for 1 r; p i < 1 for i = 1; :::; n. Recall that a multilinear form B : n i=1`p i ! C de…ned by B(x 1 ; :::; x n ) = P 1 k=1 k x 1 k :::: x n k is called diagonal operator, where f k g is a bounded sequence. Clearly, it is seen by the de…nition that B(x 1 ; :::; x n ) = 0 whenever x k x l = 0 for some k; l 2 f1; 2; :::; ng. Therefore, it is zero product preserving and there is a linear form T :`r ! C such that B(x 1 ; :::; x n ) = T (x), where x 1 :::: x n = x. Besides, if we consider p i = ::: = p n = p, then we obtain that the zero product preserving map B : n`p ! C 166 E. ERDOG AN has a factorization through the linear form T :`p =n ! C. Since this gives the symmetry of the form B : n`p ! C, we get the diagonal map B is associated with an orthogonally additive n-homogeneous diagonal polynomial form P :`p ! C.