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Almost-Lp-projections and Lp isomorphisms

Published online by Cambridge University Press:  14 November 2011

Michael Cambern ,
Krzysztof Jarosz  and
Georg Wodinski
Michael Cambern
Affiliation:
Department of Mathematics, University of California, Santa Barbara, CA 93106, U.S.A.
Krzysztof Jarosz
Affiliation:
Department of Mathematics, Southern Illinois University at Edwardsville, Edwardsville, IL 62026, U.S.A.
Georg Wodinski
Affiliation:
Institut für Mathematik I, Freie Universität Berlin, Arnimalle 3, D-1000 Berlin 33, Germany
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Synopsis

Lp -summands and Lp -projections in Banach spaces have been studied by E. Behrends, who showed that for a fixed value of p, l ≦ p ≦ ∞, p ≠ 2, any two Lp -projections on a given Banach space E commute. Here we introduce the notion of almost-Lp -projections, and we establish a result which generalises Behrends' theorem, while also simplifying its proof. Almost-Lp-projections are then applied to the study of small-bound isomorphisms of Bochner LP -spaces. It is shown that if the Banach space E satisfies a geometric condition which, in the finite-dimensional case, reduces to the absence of non-trivial Lp-summands, then for separable measure spaces, the existence of a small-bound isomorphism between Lp (λ1, E) and LP(λ2, E) implies that these Bochner spaces are, in fact, isometric.

Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 113 , Issue 1-2 , 1989 , pp. 13 - 25
Copyright
Copyright © Royal Society of Edinburgh 1989

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