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Non-linear factorization of linear operators

Abstract : We show, in particular, that a linear operator between finite-dimensional normed spaces, which factors through a third Banach space Z via Lipschitz maps, factors linearly through the identity from L(infinity)([0, 1], Z) to L(1)([0, 1], Z) (and thus, in particular, through each L(p)(Z), for 1 < p < infinity) with the same factorization constant. It follows that, for each 1 < p < infinity, the class of L(p) spaces is closed under uniform (and even coarse) equivalences. The case p = 1 is new and solves a problem raised by Heinrich and Mankiewicz in 1982. The proof is based on a simple local-global linearization idea.
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Submitted on : Tuesday, May 1, 2012 - 7:43:55 PM
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W. B. Johnson, Bernard Maurey, G. Schechtman. Non-linear factorization of linear operators. Bulletin of the London Mathematical Society, London Mathematical Society, 2009, 41 (?), pp.663--668. ⟨10.1112/blms/bdp040⟩. ⟨hal-00693044⟩



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