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Banach spaces with the $2$-summing property

Alvaro Arias·Tadek Figiel·William B. Johnson·Gideon Schechtman
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Abstract

A Banach space $X$ has the $2$-summing property if the norm of every linear operator from $X$ to a Hilbert space is equal to the $2$-summing norm of the operator. Up to a point, the theory of spaces which have this property is independent of the scalar field: the property is self-dual and any space with the property is a finite dimensional space of maximal distance to the Hilbert space of the same dimension. In the case of real scalars only the real line and real $\ell_\infty^2$ have the $2$-summing property. In the complex case there are more examples; e.g., all subspaces of complex $\ell_\infty^3$ and their duals.

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Alvaro Arias, Tadek Figiel, William B. Johnson, Gideon Schechtman. 1994-03-02. Banach spaces with the $2$-summing property. https://arxiv.org/abs/math/9403206

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