Journal of Mathematics

Journal of Mathematics / 2016 / Article

Research Article | Open Access

Volume 2016 |Article ID 3492934 | 5 pages | https://doi.org/10.1155/2016/3492934

On the Composition Ideals of Schatten Class Type Mappings

Academic Editor: Alfred Peris
Received23 Jun 2016
Accepted25 Jul 2016
Published14 Aug 2016

Abstract

We study the composition ideals of multilinear and polynomial mappings generated by Schatten classes. We give some coincidence theorems for Cohen strongly 2-summing multilinear operators and factorization results like that given by Lindenstrauss-Pełczński for Hilbert Schmidt linear operators.

1. Introduction and Preliminaries

Let , be two Hilbert spaces and the th Schatten class, that is, the space of compact operators such that . This space is an ideal of linear operators and coincides with the space of Hilbert Schmidt operators for . Pietsch in [1] has introduced and studied multilinear ideals, by introducing two methods to construct multilinear ideals from a given linear ideal. It is well known that some classes of multilinear operators can be interpreted via these methods of Pietsch, namely, the classes of -dominated, weakly compact, and compact multilinear operators. If we consider the Schatten class , Braunss and Junek [2] have studied the multi-ideals generated by this linear ideal, where the authors have used the factorization method of Pietsch. Andrade Mendes [3] has given some factorization results of these mappings and showed that the resulting space by this procedure coincides with the class of 2-dominated multilinear operators. The results in this paper are concentrated to study the multilinear operators generated by the Schatten class. We give another class of -linear mappings, noted by , whose operators can be written as with belonging to and is a multilinear operator (this technique of factorization known as composition ideals; see [4]). We will show that the class coincides with , the class of Cohen strongly -summing multilinear operators. We also present some factorization results for multilinear mappings of type like that given by Lindenstrauss-Pełczński for Hilbert Schmidt linear operators and Mendes for multilinear mappings of type . The same results are given for the polynomial case.

This paper is organized as follows. In the rest of Section 1, we recall some standard notations and definitions concerning polynomial and multilinear mappings. Section 2 is devoted to prove our main results; we study the class of multilinear operators which admit a factorization where is in . In particular, we show that the class coincides with the class of Cohen strongly 2-summing multilinear operators. The same results are true for polynomial mappings. We end this paper by giving some factorization theorems concerning this new class.

We begin by recalling briefly some basic notations and terminology. Let be a Banach space; then is its closed unit ball and is its topological dual. Let now and let be Banach spaces over ( or ). We will denote by the space of all continuous -linear operators from into . In the case , we will simply write . As usual, stands for the (complete) projective tensor product of the Banach spaces . If , we denote by the linearization of , which is the linear map given by , for all (, , ). The canonical multilinear mapping is defined by . We have the following factorization:Let ; is symmetric if it is invariant for any permutation of its components. For any we will denote by its associated symmetric multilinear operator, that is, defined by where is a permutation of the set . If is symmetric then . A map is -homogeneous polynomial if there exists a unique symmetric -linear operator such that for every . The polynomial is continuous if and only if is continuous. We denote, by , the Banach space of all continuous -homogeneous polynomials from to endowed with the norm .

Let us recall the definitions of concept of -summing [5, p. 31] and Cohen strongly -summing operators [6], which will be used in the sequel.

Definition 1. Let and be a linear operator.(1)is -summing if it takes weakly -summable sequences in to strongly -summable sequences in . We denote by the class of -summing linear operators from to , which is a Banach space.(2) is Cohen strongly -summing if its adjoint operator is -summing. We denote by the class of Cohen strongly -summing linear operators from to , which is a Banach space.If , are Hilbert spaces, we have [6, Theorem 4.1.1]for and . If , we have

Definition 2 (see [7]). Let . An -linear operator is Cohen strongly -summing if its linearization is Cohen -summing linear operator. The class of Cohen strongly -summing -linear operators from to is denoted by , which is a Banach space.

Definition 3 (see [8]). Let . -homogeneous polynomial is Cohen strongly -summing if its symmetric -linear operator is Cohen strongly -summing multilinear operator. The class of Cohen strongly -summing -homogeneous polynomials from to is denoted by , which is a Banach space.

2. Main Results

2.1. Multilinear Mappings Generated by

Let . Let be a Banach space and a Hilbert space. The linear operator is said to be of type , in symbols , if it factors through a Hilbert space where . The space is a Banach space with the following norm: where the infimum is taken over all possible factorizations of the form (5). If is a Hilbert space, we have, for , .

Proposition 4. Let be a Banach space and a Hilbert space. Then(1)if , we have ;(2)If , we have

Proof. (1) The result follows from (4).
(2) Let ; then , with . The second adjoint of is given by then and consequently .
Reciprocally, we use the elementary identity , where is the isometric embedding defined by Then , and consequently .

Before the presentation of the next result, we need the following lemma.

Lemma 5. Let , be Banach spaces such that is reflexive. Let be a Cohen strongly 2-summing linear operator. Then, factors through a Hilbert space, that is, a Hilbert space and two linear operators , such thatwith .

Proof. Let . By [6, Theorem 2.2.2], its adjoint is 2-summing. Then, factors through a Hilbert space (see the Pietsch Factorization Theorem [5, Corollary 2.16]); that is, , where is 2-summing (i.e., its adjoint is Cohen strongly 2-summing). The second adjoint of is given by On the other hand, we have with being bijective ( is reflexive). Consequently, we obtain . This completes the proof of Lemma 5.

There is an interesting relationship between Cohen strongly -summing linear operators and linear operators of type .

Theorem 6. Let be a Banach space and a Hilbert space. For all , we have

Proof. Let . By the factorization (5), with and . By (3) and the ideal property, we have . Reciprocally, let and . By [6, Theorem 2.4.1], is in . According to Lemma 5, factors through a Hilbert space with   (=). Therefore, .

Now, we introduce a similar definition to the category of multilinear mappings. A multilinear operator is said to be of type , in symbols , if there exist a Hilbert space , a linear operator , and such thatThe space is a Banach space with the following norm: where the infimum is taken over all possible factorizations of form (14). Since is compact, the composed operator in (14) is compact by the properties of compact multilinear operators (see [4]), then every operator in the space is compact. Moreover, if we haveOur theorem below deals with the relation between a multilinear operator of type and its linearization.

Theorem 7. Let be Banach spaces and a Hilbert space. Let . The following properties are equivalent.(1)The multilinear operator belongs to .(2)The linearization .

Proof. First, we suppose that is of type . Then, by the factorization (14) we have with . So, by using (1) we obtain , where is the linearization of ; then Now, we suppose that the second assertion is true. We can write , with .

As in the linear case (12), we can establish the relation between Cohen strongly -summing multilinear operators and multilinear operators of type .

Theorem 8. Let be Banach spaces and a Hilbert space. For all , we have

Proof. Let . By the factorization (14), with and . By (3) and [9, Corollary 4.2], we have Reciprocally, let and . By [7, Corollary 2.5], is in and, by [9, Corollary 4.2], we have , where and is a Banach space. According to Lemma 5, factors through a Hilbert space; that is, with (=). Therefore, the desired result follows.

Theorem 9. Let be Hilbert spaces. Then,(1)if , we have ;(2)if , we have .

Proof. (1) Let and . By (18), belongs to and the result follows by (16).
(2) Let and be a multilinear operator of type . Then, , with . It follows by (3) that . Thus [9, Corollary 4.2] completes the proof of Theorem 9.

2.2. Factorization of Schatten Class Type Mappings

In the linear case, it is well known that Hilbert Schmidt operators factor through -space or -space (Lindenstrauss and Pelczynski [10]) and also through infinite dimensional Banach spaces [5, Theorem 19.2]. The converse is also true in both cases. For the multilinear and polynomial cases, every Hilbert Schmidt multilinears or polynomial mappings factor through any Banach spaces, but the converse is not true (see [11, Theorem 2.10. and Example 2.12]). In this section, we consider the particular class for which it is possible to obtain an extension similar to the linear case cited above.

First, we recall the definition of -factorable multilinear operators introduced by Cerna Maguina in [12] as a generalization of the one given by Diestel et al. in the linear case [5].

Definition 10. Let and be Banach spaces. The operator is said to be -factorable if there exist a measure space , , and such that where is the isometric embedding of into . We denote by the space of all -factorable multilinear operators, which is a Banach space.

Theorem 11 (see [12, Proposition 2.3]). Let be Banach spaces. Then, the following assertions are equivalent.(1)The operator belongs to .(2)The operator factors through a Hilbert space; that is, there exist a Hilbert space , and such that .

We now present a factorization result for mappings of Schatten class type .

Theorem 12. Let be Banach spaces and a Hilbert space. Then, the following properties are equivalent.(1)The operator .(2)There exist a linear operator and such that .(3)There exist and such that .

Proof. : Let with . The operator factors through -space; that is, with and ; thus that is, where .
: Let with and . One can write as By Grothendieck’s Theorem [5, Theorem 3.1], the operator is 2-summing. So it factors through a Hilbert space; that is, with . Therefore, with ; this ends the proof.
: This is a similar argument to the last one.

Now, we give a multilinear version of the Diestel-Jarchow-Tonge result. The proof is similar to that of the last Theorem. So that we omit its proof.

Theorem 13. Let be Banach spaces and a Hilbert space. The following properties are equivalent.(1)The operator .(2)For every Banach space , there exist and such that .

To close this paper, we introduce the defintion of polynomial of type . A polynomial mapping is said to be of type , in symbols , if its -linear symmetric is of type . The space is a Banach space with the following norm: . By applying similar argument of that given for the multilinear case, we have the following result.

Theorem 14. Let be a Banach space and a Hilbert space. Then(1)if , we have ;(2)if , we have .

Proof. (1) If then its -linear symmetric belongs to . By Theorems 8 and 9, , and thus .
(2) Let ; then is of type . The result follows from Theorems 8 and 9.

As consequence of the last Theorems, we have the following corollary.

Corollary 15. Let be a Banach space and a Hilbert space. The following properties are equivalent.(1)The polynomial .(2)There exist a linear operator and such that with .(3)There exist a linear operator and such that with .(4)For every Banach space , there exist and such that with .

Proof. : If then . So, by Theorem 12, factors through -space; that is, , with and . Now, it is not difficult to show that and . Then where .
Reciprocally, we suppose that the second assertion is true. We have ; by Theorem 12   belongs to ; that is, .
and : The rest are similar to the last one.

Competing Interests

The authors declare that they have no competing interests.

References

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Copyright © 2016 Abdelaziz Belaada et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.


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