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Extension of multilinear forms and polynomials from subspaces of \({\mathcal L}_1\)-spaces. (English) Zbl 1139.46007

There is, in general, no Hahn-Banach theorem for multilinear mappings and homogeneous polynomials. Situations where every multilinear mapping or homogeneous polynomial on a Banach space extends to a superspace only occur in very exceptional circumstances. (A continuous mapping \(P\colon X\to {\mathbf K}\) is said to be an \(m\)-homogeneous polynomial if there is a continuous symmetric \(m\)-linear mapping \(T:X\times\dots \times X\to {\mathbf K}\) so that \(P(x)=T(x,\dots,x)\) for all \(x\in X\).) In this paper, the authors consider the case where \(Y\) is an \({\mathcal L}_1\) space and \(X\) is a subspace of \(Y\) with an unconditional basis. They show that there is a constant \(C>0\) such that, for each integer \(m\), each \(m\)-linear mapping \(T: X\times\dots \times X\to {\mathbf K}\) has an extension to an \(m\)-linear mapping \(\widetilde T: Y\times\ldots\times Y\to {\mathbf K}\) with \(\| \widetilde T\| \leq C^m \| T\| \) if and only if \(X\) is isomorphic to \(\ell_1\).
For homogeneous polynomials the result is not so clear-cut. The authors prove that if there is a constant \(C>0\) such that each \(m\)-homogeneous polynomial \(P\) on \(X\) has an extension to an \(m\)-homogeneous polynomial \(\widetilde P\) on \(Y\) with \(\| \widetilde P\| \leq C^m \| P\| \), then \(X\) is a subspace \(\ell_{1+\varepsilon}\) for every \(\varepsilon>0\). Such results are possible in the settings of \({\mathcal L}_1\) spaces as every subspace of an \({\mathcal L}_1\) space has the Gordon-Lewis property.
Given a finite dimensional Banach space \(X\), the Bohr radius of \(X\) is the supremum of all \(r\) such that, whenever \(| \sum_\alpha a_\alpha z^\alpha| \leq 1\) for \(z\) in the unit ball of \(X\), we have that \(\sum_\alpha | a_\alpha z^\alpha| \leq 1\) for \(z\) in the ball of radius \(r\). The authors assume that \(X\) is a Banach space with 1-unconditional basis \(\{e_k\}_k\) and let \(X_n\) denote the span of \(\{e_1,\dots, e_n\}\). They show that if the Bohr radii of the \(X_n\) do not converge to \(0\), then \(X\) is a subspace of \(\ell_{1+\varepsilon}\) for every \(\varepsilon>0\). To obtain the results in this paper, the authors use Gaussian and binomial random variables to find bounds for the unconditional basis constants of the bases of monomials in the spaces of full and symmetric tensor products.

MSC:

46B07 Local theory of Banach spaces
46G20 Infinite-dimensional holomorphy
46B09 Probabilistic methods in Banach space theory
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