Operators between subspaces and quotients of L1

Gilles Godefroy (1) , Nigel Kalton (2) , Daniel Li (3)

1 IMJ - Institut de Mathématiques de Jussieu
2 Department of Mathematics [New York]
3 LML - Laboratoire de Mathématiques de Lens

We provide an unified approach of results of L. Dor on the complementation of the range, and of D. Alspach on the nearness from isometries, of small into isomorphisms of L1. We introduce the notion of small subspace of L1 and show lifting theorems for operators between quotients of L1 by small subspaces. We construct a subspace of L1 which shows that extension of isometries from subspaces of L1 to the whole space are no longer true for isomorphisms, and that nearly isometric isomorphisms from subspaces of L1 into L1 need not be near from any isometry.

Keywords: integral representation of operators on $L^1$; small into-isomorphisms of $L^1$; complemented subspace of $L^1$; isometries of $L^1$; small subspaces of $L^1$; nicely placed subspaces of $L^1$; semi-Riesz sets; strong Enflo operators; Daugavet operators; lifting of operators; $p$-stable random variables

Domains: Functional Analysis [math.FA]

Gilles Godefroy, Nigel Kalton, Daniel Li. Operators between subspaces and quotients of L1. Indiana University Mathematics Journal, 2000, 49 (1), pp.245-286. ⟨hal-00441551⟩