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Lip-Linear operators and their connection to Lipschitz tensor products

Athmane Ferradi, Khalil Saadi

TL;DR

This work introduces Lip-Linear operators, mappings $T:X×E→F$ that are Lipschitz in the metric space variable and linear in the Banach-space variable, and develops a rich operator-ideal theory around them. It establishes core identifications: $LipL_0(X×E;F)\cong L(X⊗π E;F)$ and $LipL_0(X×E;F)\cong B(\mathcal{F}(X)×E;F)$, linking Lip-Linear operators to linear and bilinear structures on Lipschitz tensor products and Lipschitz-free spaces. The paper then extends integral and dominated summability to Lip-Linear operators, proving that $LipL_{0}^{int}(X×E;F)\cong BI(\mathcal{F}(X)×E;F)\cong I(X⊗ε E;F)$ and providing measure-based and factorization characterizations via $L_1(μ)$. A parallel dominated $(p,q)$-summing theory is developed, yielding equalities $\Delta_{(p,q)}^{LipL}(X×E;F)=Π_p^L(X;Π_q(E;F))=Π_q(E;Π_p^L(X;F))$ and a Pietsch-type domination principle, with a Dvoretzky–Rogers dichotomy at the finite-dimensional boundary. Overall, the work unifies Lipschitz and linear/bilinear frameworks and extends summability concepts to Lip-Linear operators, broadening the toolbox for Lipschitz operator theory and its tensorial dualities.

Abstract

The linear operators defined on the Lipschitz projective tensor product of X and E motivate the study of a distinct class of operators acting on the cartesian produc X E. This class, denoted by LipL(X E;F), combines Lipschitz and linear properties, forming an intermediate framework between bilinear operators and two-Lipschitz operators. We establish an identification between this space and L(X E;F), which also links it to the space of bilinear operators B(AE(X) E;F). Furthermore, we extend summability concepts within this category, with a particular focus on integral and dominated (p;q)-summing operators

Lip-Linear operators and their connection to Lipschitz tensor products

TL;DR

This work introduces Lip-Linear operators, mappings that are Lipschitz in the metric space variable and linear in the Banach-space variable, and develops a rich operator-ideal theory around them. It establishes core identifications: and , linking Lip-Linear operators to linear and bilinear structures on Lipschitz tensor products and Lipschitz-free spaces. The paper then extends integral and dominated summability to Lip-Linear operators, proving that and providing measure-based and factorization characterizations via . A parallel dominated -summing theory is developed, yielding equalities and a Pietsch-type domination principle, with a Dvoretzky–Rogers dichotomy at the finite-dimensional boundary. Overall, the work unifies Lipschitz and linear/bilinear frameworks and extends summability concepts to Lip-Linear operators, broadening the toolbox for Lipschitz operator theory and its tensorial dualities.

Abstract

The linear operators defined on the Lipschitz projective tensor product of X and E motivate the study of a distinct class of operators acting on the cartesian produc X E. This class, denoted by LipL(X E;F), combines Lipschitz and linear properties, forming an intermediate framework between bilinear operators and two-Lipschitz operators. We establish an identification between this space and L(X E;F), which also links it to the space of bilinear operators B(AE(X) E;F). Furthermore, we extend summability concepts within this category, with a particular focus on integral and dominated (p;q)-summing operators
Paper Structure (4 sections, 22 theorems, 172 equations)

This paper contains 4 sections, 22 theorems, 172 equations.

Table of Contents

  1. Introduction and preliminaries
  2. The Class of Lip-Linear operators
  3. Integral Lip-Linear operators and the dual of $X\widehat{\boxtimes }_{\varepsilon }E$
  4. Dominated (p,q)-summing Lip-Linear operators

Key Result

Proposition 2.2

Let $R:X\rightarrow G$ be a Lipschitz operator and $v:E\rightarrow F$ a linear operator. For any bilinear operator $B:G\times F\rightarrow H$, the operator $T=B\circ (R,v)$ is Lip-Linear, and This means that $\mathcal{B}\circ (Lip_{0},\mathcal{L})\left( X\times E;H\right) \subset LipL_{0}\left( X\times E;H\right)$.

Theorems & Definitions (44)

  • Definition 2.1
  • Proposition 2.2
  • proof
  • Proposition 2.3
  • proof
  • Corollary 2.4
  • Proposition 2.5
  • Example 2.6
  • Lemma 2.7
  • proof
  • ...and 34 more