Lipschitz p-integral operators and Lipschitz p-nuclear operators

doi:10.1016/j.na.2012.04.044

Nonlinear Analysis: Theory, Methods & Applications

Volume 75, Issue 13, September 2012, Pages 5270-5282

https://doi.org/10.1016/j.na.2012.04.044 Get rights and content

Abstract

In this paper, we introduce strongly Lipschitz $p$ -integral operators, strongly Lipschitz $p$ -nuclear operators and Lipschitz $p$ -nuclear operators. It is shown that for a linear operator, the Lipschitz $p$ -nuclear norm is the same as its usual $p$ -nuclear norm under certain conditions. We also prove that the Lipschitz $2$ -dominated operators and the strongly Lipschitz $2$ -integral operators are the same with equal norms. Finally, we show that the Lipschitz $p$ -integral norm of a Lipschitz map from a finite metric space into a Banach space is the same as its Lipschitz $p$ -nuclear norm.

Introduction

The linear theory of $p$ -summing operators goes back to the work of Grothendieck in [1]. And the most remarkable result can date its beginning in the paper of Pietsch [2]. Pietsch defined the class of $p$ -summing operators and proved some good properties that made it appear like an interesting class. Among the main results that appear in [2], we can find the Domination/Factorization Theorem, ideal, inclusion and composition properties and good relations with other classes of linear operators, such as the nuclear and Hilbert–Schmidt operators.

At the beginning of the 1980s, mainly due to [3], the idea of generalizing the theory of ideals of linear operators to the multilinear (and polynomial) setting appeared.

Recently, Farmer and Johnson [4] have introduced the notion of Lipschitz $p$ -summing operators and the notion of Lipschitz $p$ -integral operators and proved a nonlinear version of the Pietsch domination theorem. In [5], the authors proved a nonlinear version of Grothendieck’s theorem for Lipschitz $p$ -summing operators. However there is no known nonlinear analogue of $p$ -nuclear operators.

In this paper, we introduce natural notions of Lipschitz $p$ -nuclear operators and strongly Lipschitz $p$ -nuclear operators. We also introduce notions of Lipschitz $p$ -dominated operators and strongly Lipschitz $p$ -integral operators.

In Section 2, it is shown that for a bounded linear operator $T$ from a separable Banach space into a dual space, the Lipschitz $p$ -nuclear norm of $T$ coincides with its $p$ -nuclear norm. So the notion of Lipschitz $p$ -nuclear operators is really a generalization of $p$ -nuclear operators in this setting. In the same section, we also prove a factorization theorem for strongly Lipschitz $p$ -nuclear operators and the result is a nonlinear analogue of the factorization theorem for $p$ -nuclear operators.

In Section 3, we give a characterization of Lipschitz $p$ -dominated operators in terms of domination and factorization. As a consequence, we show that a Lipschitz map is Lipschitz 2-dominated if and only if it is strongly Lipschitz 2-integral. Moreover, the Lipschitz 2-dominated norm is the same as the strongly Lipschitz 2-integral norm.

In the final section, we provide some applications to Lipschitz maps from finite metric spaces to Banach spaces. To be more precise, we show that a Lipschitz $p$ -integral operator $T$ from a finite metric space into a Banach space is automatically Lipschitz $p$ -nuclear and the Lipschitz $p$ -integral norm equals the Lipschitz $p$ -nuclear norm. We also give some estimates of the norms in terms of the Lipschitz constant of $T$ and the cardinality of the finite metric space. We believe that the natural notions introduced together with the results obtained are pioneering and provide foundational contribution to the theory of nonlinear $p$ -integral and nonlinear $p$ -nuclear operators.

We assume that readers are familiar with linear $p$ -summing, $p$ -integral and $p$ -nuclear operators. Standard notations and related results for linear operators can be found in [6], [7], [8].

Section snippets

Nonlinear $p$ -nuclear operators

Suppose that $1 \leq p < \infty$ and that $u : X \to Y$ is a linear operator between Banach spaces. We say that $u$ is $p$ -summing if there is a constant $c \geq 0$ such that regardless of the natural number $n$ and regardless of the choice of $x_{1}, \dots, x_{n}$ in $X$ we have ${(\sum_{i = 1}^{n} {‖ u (x_{i}) ‖}^{p})}^{1 / p} \leq c \cdot sup {{(\sum_{i = 1}^{n} {| x^{*} (x_{i}) |}^{p})}^{1 / p} : x^{*} \in B_{X^{*}}},$ where $B_{X^{*}}$ is the unit ball of $X^{*}$ . The least constant $c$ for which this inequality always holds is denoted by $π_{p} (u)$ . We shall write $Π_{p} (X, Y)$ for the set of all $p$ -summing operators from $X$ to $Y$ .

We recall that a linear

Nonlinear $p$ -integral operators

We begin this section with the definitions of Lipschitz $p$ -summing operators and Lipschitz $p$ -integral operators [4] and some known results. Recall that a Lipschitz map $T$ from a metric space $(X, d_{X})$ into a metric space $(Y, d_{Y})$ is said to be Lipschitz $p$ -summing $(1 \leq p < \infty)$ if there is a constant $C > 0$ such that regardless of the natural number $n$ and regardless of the choices of ${x_{i}}_{i = 1}^{n}$ and ${y_{i}}_{i = 1}^{n}$ in $X$ , we have ${(\sum_{i = 1}^{n} d_{Y} {(T x_{i}, T y_{i})}^{p})}^{1 / p} \leq C sup_{f \in B_{X^{#}}} {(\sum_{i = 1}^{n} {| f (x_{i}) - f (y_{i}) |}^{p})}^{1 / p},$ the least constant $C$ for which the

Applications to Lipschitz mappings from a finite metric space into a Banach space

This section deals with the relationship between Lipschitz $p$ -integral and Lipschitz $p$ -nuclear operators from a finite metric space into a Banach space.

Theorem 4.1

Let $X$ be a finite metric space and $Y$ be a Banach space. Then, for any $1 \leq p < \infty$ and any mapping $T : X \to Y$ , we have

(a)
$ν_{p}^{L} (T) = ι_{p}^{L} (T)$ ;
(b)
$ν_{p}^{L} (T) = ι_{p}^{L} (T) \leq C \cdot {(log | X |)}^{2} \cdot Lip (T)$ , where $C$ is an absolute constant.

Proof

(a) Consider a typical $p$ -integral factorization $J_{Y} T : X \overset{B}{⟶} L_{\infty} (μ) \overset{i_{p}}{⟶} L_{p} (μ) \overset{A}{⟶} Y^{* *} .$ Fix $ε > 0$ . Then, there exists a finite dimensional subspace $E$ of $L_{\infty} (μ)$ , of dimension $N$

Acknowledgments

This work was done during the first author’s visit to Department of Mathematics, Texas A&M University. The authors are grateful to Professor W.B. Johnson, without whose ideas and encouragement, this work would not have taken place. The authors would also like to thank the referee for his valuable suggestions which made the current version of the paper more readable.

Dongyang Chen’s research was supported in part by the National Natural Science Foundation of China (Grant Nos. 10526034, and

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Lipschitz p-integral operators and Lipschitz p-nuclear operators

Abstract

Introduction

Section snippets

Nonlinear p-nuclear operators

Nonlinear p-integral operators

Applications to Lipschitz mappings from a finite metric space into a Banach space

Acknowledgments

Résumé de la théorie métrique des produits tensoriels topologiques

Bol. Soc. Mat. Sao Paulo

Absolut p-summierende Abbildungen in normierten Raumen

Studia Math.

Lipschitz p-summing operators

Proc. Amer. Math. Soc.

Remarks on Lipschitz p-summing operators

Proc. Amer. Math. Soc.

Absolutely summing operators

Lipschitz $p$ -integral operators and Lipschitz $p$ -nuclear operators

Nonlinear $p$ -nuclear operators

Nonlinear $p$ -integral operators

Absolut $p$ -summierende Abbildungen in normierten Raumen

Lipschitz $p$ -summing operators

Remarks on Lipschitz $p$ -summing operators