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Bulletin of the London Mathematical Society Advance Access originally published online on May 22, 2009
Bulletin of the London Mathematical Society 2009 41(4):663-668; doi:10.1112/blms/bdp040
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© 2009 London Mathematical Society

Non-linear factorization of linear operators

W. B. Johnson

Department of Mathematics
Texas A&M University
College Station, TX 77843
USA

B. Maurey

Laboratoire d’Analyse et de Mathématiques Appliquées
Université Paris-Est
UMR 8050
77454 Marne-la-Vallée cedex 2
France
maurey@univ-mlv.fr

G. Schechtman

Department of Mathematics
Weizmann Institute of Science
Rehovot
Israel
gideon@weizmann.ac.il

Received 9 January 2007. Revision received 30 May 2008.

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{infty}([0, 1], Z) to L1([0, 1], Z) (and thus, in particular, through each Lp(Z), for 1 ≤ p ≤ {infty}) with the same factorization constant. It follows that, for each 1 ≤ p ≤ {infty}, the class of Lp 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.

The first author is supported in part by NSF DMS-0200690 and DMS-0503688 and U.S.–Israel Binational Science Foundation. The third author is supported in part by Israel Science Foundation and U.S.–Israel Binational Science Foundation; participant, NSF Workshop in Analysis and Probability, Texas A&M University.

2000 Mathematics Subject Classification 46B20, 46E30.


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