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Almost-\(L^ p\)-projections and \(L^ p\) isomorphisms. (English) Zbl 0688.46019

Summary: \(L^ p\)-summands and \(L^ p\)-projections in Banach spaces have been studied by E. Behrends, who showed that for a fixed value p, \(1\leq p\leq \infty\), \(p\neq 2\), any two \(L^ p\)-projections on a given Banach space E commute. Here we introduce the notion of almost-\(L^ p\)-projections, and we establish a result which generalizes Behrends’ theorem, while also simplifying its proof. Almost-\(L^ p\)-projections are then applied to the study of small-bound isomorphisms of Bochner \(L^ p\)-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 \(L^ p\)-summands, then for separable measure spaces, the existence of a small-bound isomorphism between \(L^ p(\mu_ 1,E)\) and \(L^ p(\mu_ 2,E)\) implies that these Bochner spaces are, in fact, isometric.

MSC:

46E40 Spaces of vector- and operator-valued functions
47B38 Linear operators on function spaces (general)
46E20 Hilbert spaces of continuous, differentiable or analytic functions
46E30 Spaces of measurable functions (\(L^p\)-spaces, Orlicz spaces, Köthe function spaces, Lorentz spaces, rearrangement invariant spaces, ideal spaces, etc.)
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References:

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