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arXiv · 1804.10641

The Segre cone of Banach spaces and multilinear operators

Maite Fernández-Unzueta
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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 $S^k_{X_1,\ldots, X_n}$ of vectors of rank $\leq k$. With this, we define the Segre cone of Banach spaces, $\Sigma_{X_1,\ldots, X_n},$ and state when $S^k_{X_1,\ldots, X_n}$ is closed. We introduce an auxiliary mapping (a $\Sigma$-operator) that allows us to study multilinear mappings with a geometrical point of view. We use the isometric correspondence between multilinear mappings and Lipschitz $\Sigma$-operators, to have a strategy to generalize ideal properties from the linear to the multilinear settitng.

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Maite Fernández-Unzueta. 2018-04-27. The Segre cone of Banach spaces and multilinear operators. https://arxiv.org/abs/1804.10641

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