Lipschitz Tensor Product

Document Type: Original Article

Authors

1 Departamento de Matematicas´ , Universidad de Almerıa, 04120 Almerıa, Spain.

2 Department of Mathematics, University of Oklahoma, Norman, Oklahoma, 7 3019, United States.

3 Departamento de Matematicas´ , Universidad de Cadiz´ , 11510 Puerto Real, Spain.

Abstract

Inspired by ideas of R. Schatten in his celebrated monograph [23] on a theory of cross-spaces, we introduce the notion of a Lipschitz tensor product $X\boxtimes E$ of a pointed metric space $X$ and a Banach space $E$ as a certain linear subspace of the algebraic dual of $\text{Lip}0(X,E^*)$. We prove that $\left\langle \text{Lip}0(X,E^*),X\boxtimes E\right\rangle$ forms a dual pair.We prove that $X\boxtimes E$ is linearly isomorphic to the linear space of all finite-rank continuous linear operators from $(X^\#,\tau_p)$ into $E$, where $X^\#$ denotes the space $\text{Lip}0(X,\mathbb{K})$ and $\tau_p$ is the topology of pointwise convergence of $X^\#$. The concept of Lipschitz tensor product of elements of $X^\#$ and $E^*$ yields the space $X^\#⧆ E^*$ as a certain linear subspace of the algebraic dual of $X\boxtimes E$. To ensure the good behavior of a norm on $X\boxtimes E$ with respect to the Lipschitz tensor product of Lipschitz functionals (mappings) and bounded linear functionals (operators), the concept of dualizable (respectively, uniform) Lipschitz cross-norm on $X\boxtimes E$ is defined. We show that the Lipschitz injective norm $\varepsilon$, the Lipschitz projective norm $\pi$ and the Lipschitz $p$-nuclear norm $d_p$ $(1\leq p\leq\infty)$ are uniform dualizable Lipschitz cross-norms on $X\boxtimes E$. In fact, $\varepsilon$ is the least dualizable Lipschitz cross-norm and $\pi$ is the greatest Lipschitz cross-norm on $X\boxtimes E$. Moreover, dualizable Lipschitz cross-norms $\alpha$ on $X\boxtimes E$ are characterized by satisfying the relation $\varepsilon\leq\alpha\leq\pi$.
In addition, the Lipschitz injective (projective) norm on $X\boxtimes E$ can be identified with the injective (respectively, projective) tensor norm on the Banach-space tensor product of the Lipschitz-free space over $X$ and $E$, but this identification does not hold for the Lipschitz $2$-nuclear norm and the corresponding Banach-space tensor norm. In terms of the space $X^\#⧆ E^*$, we describe the spaces of Lipschitz compact (finite-rank, approximable) operators from $X$ to $E^*$.

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