The Segre cone of Banach spaces and multilinear operators

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 $S_{X_1,\dots ,X_n}^k$ of vectors of rank $\le k$. With this, we define the Segre cone of Banach spaces, ${\Sigma }_{X_1,\dots ,X_n},$ and state when $S_{X_1,\dots ,X_n}^k$ 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.

References in corpus (1)

• Noncommutative vector valued $L_p$-spaces and completely $p$-summing maps

Cited by in corpus (1)

• Multilinear Operators Factoring through Hilbert Spaces