Non-linear factorization of linear operators
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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([0, 1], Z) to L1([0, 1], Z) (and thus, in particular, through each Lp(Z), for 1 p ) with the same factorization constant. It follows that, for each 1 p , the class of 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. 2009 London Mathematical Society.
