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Building ideals of two-Lipschitz operators between mertic and Banach spaces

Dahmane Achour, Elhadj Dahia

Abstract

In this paper, we present and characterize the injective hull of a two-Lipschitz operator ideals and the definition of two-Lipschitz dual operator ideal. Also we introduce two methods for creating ideals of two-Lipschitz operators from a pair of Lipschitz operator ideals. Namely, Lipschitzization and factorization method. We show the closedness, the injectivity and the symmetry of these two-Lipschitz ideals according to the closedness, injectivity and symmetry of the corresponding Lipschitz operator ideals. Some illustrative examples are given.

Building ideals of two-Lipschitz operators between mertic and Banach spaces

Abstract

In this paper, we present and characterize the injective hull of a two-Lipschitz operator ideals and the definition of two-Lipschitz dual operator ideal. Also we introduce two methods for creating ideals of two-Lipschitz operators from a pair of Lipschitz operator ideals. Namely, Lipschitzization and factorization method. We show the closedness, the injectivity and the symmetry of these two-Lipschitz ideals according to the closedness, injectivity and symmetry of the corresponding Lipschitz operator ideals. Some illustrative examples are given.
Paper Structure (8 sections, 21 theorems, 71 equations)

This paper contains 8 sections, 21 theorems, 71 equations.

Table of Contents

  1. Introduction
  2. Notation and preliminaries
  3. Closed and injective two-Lipschitz operator ideal
  4. Technique to generate two-Lipschitz operators ideals from operator ideals
  5. Techniques to generate two-Lipschitz operators ideals from Lipschitz operator ideals
  6. Lipschitzization method
  7. Factorization method
  8. Strong factorization method

Key Result

Proposition 3.2

If $\mathcal{I}_{BLip}$ is a two-Lipschitz operator ideal, then the class $\overline{\mathcal{I}_{BLip}}$ defined by for all pointed metric spaces $X_{1},X_{2}$ and Banach space $E,$ is a two-Lipschitz operator ideal.

Theorems & Definitions (51)

  • Definition 2.1
  • Definition 2.2
  • Definition 3.1
  • Proposition 3.2
  • proof
  • Example 3.3
  • Definition 3.4
  • Proposition 3.5
  • Proposition 3.6
  • proof
  • ...and 41 more