On the composition ideals of Lipschitz mappings

Abstract

Abstract. In this paper, we study some property of Lipschitz mappings which admit factorization through an operator ideal. Lipschitz cross-norms has been established from known tensor norms in order to represent certain classes of Lipschitz mappings. Inspired by the definition of p-summing linear operators, we derive a new class of Lipschitz mappings that is called strictly Lipschitz p-summing.

Further results on strictly Lipschitz summing operators

“…For every u ∈ X ⊠ E we have µ SL p,r,s (u) = µ L p,r,s (u) , where µ L p,r,s is the Lipschitz cross-norms corresponding to the Lapreste tensor norm µ p,r,s . Then, by [16,Corollary 3.2] we get the following coincidence result.…”

“…with n 2 = max n 1 i=1 k i and the terms between k i and n 2 are zero. Now, Let α be a tensor norm defined on two Banach spaces, by [16,Theorem 3.1], there is a Lipschitz cross-norm α L which is defined on Lipschitz tensor product X ⊠ E.…”

“…We denote by Π SL p (X, E) the Banach space of all strictly Lipschitz p-summing operators from X into E which its norm π SL p (T ) is the smallest constant C verifying (2.4). If we consider linear operators defined on Banach spaces, we have shown in [16,Proposition 3.8] that the three notions: p-summing, Lipschitz p-summing and strictly Lipschitz p-summing are coincide. The following characterization is the main result of this section.…”

“…(1) =⇒ ( 2) : Theorem 3.5 in [16]. ( 2) =⇒ (3) : We apply Pietsch Domination Theorem for p-summing linear operators [7, Theorem 2.12], then there is a Radon probability µ on B X # such that for any m ∈ F (X) we have…”

“…Then, the above representation, that we consider interesting, expresses good relation between Lipschitz operators and their linearizations. To make the relation (1.1) attainable, we have improved in [16] the definition of Lipschitz p-summing by introducing the strictly Lipschitz p-summing operators whose original ideal is Π p , the Banach space of p-summing operators, and admits a similar representation of (1.1). The goal of this paper is to explore more properties of the class of strictly Lipschitz p-summing by showing some characterizations of those operators by means of fundamental inequalities and the domination theorem of Pietsch.…”