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Lipschitz tensor product (1408.1874v2)

Published 8 Aug 2014 in math.FA

Abstract: Inspired by ideas of R. Schatten in his celebrated monograph on a theory of cross-spaces, we introduce the notion of a Lipschitz tensor product X\boxtimes E of a pointed metric space and a Banach space E as a certain linear subspace of the algebraic dual of Lipo(X,E*). We prove that <Lipo(X,E^*),X\boxtimes E> 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#,T) into E, where X# denotes the space Lipo(X,K) and T is the topology of pointwise convergence of X#. The concept of Lipschitz tensor product of elements of X# and E* yields the space X#\boxast 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 epsilon, the Lipschitz projective norm pi and the Lipschitz p-nuclear norm d_p (1<=p<=infty) are uniform dualizable Lipschitz cross-norms on X\boxtimes E. In fact, epsilon 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 epsilon<=alpha<=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 between the Lipschitz-free space over X and E. In terms of the space X#\boxast E*, we describe the spaces of Lipschitz compact (finite-rank, approximable) operators from X to E$.

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