The Segre cone of Banach spaces and multilinear operators

2018/04/27 by Fernández-Unzueta, Maite
#FOS: Mathematics #Functional Analysis (math.FA)

paper · doi:10.48550/arxiv.1804.10641

Abstract

We prove that any pair of reasonable cross norms defined on the tensor product of n Banach spaces induce (2k)^n-1-Lipschitz equivalent metrics (and thus, a unique topology) on the set S^k_{X1,…, Xn} of vectors of rank ≤ k. With this, we define the Segre cone of Banach spaces, Σ_{X1,…, Xn}, and state when S^k_{X1,…, Xn} is closed. We introduce an auxiliary mapping (a Σ-operator) that allows us to study multilinear mappings with a geometrical point of view. We use the isometric correspondence between multilinear mappings and Lipschitz Σ-operators, to have a strategy to generalize ideal properties from the linear to the multilinear settitng.

Related

• Lipschitz tensor product
• Polynomials as Lipschitz maps on the Veronese cone
• Isotone Cones in Banach Spaces and Applications to Best Approximations of Operators without Continuity Conditions
• Equivalent norms with the property (β) of Rolewicz
• Continuous operators from spaces of Lipschitz functions