The isometries of LP(Ω, X)

AbstractLet (Ω, ∑, μ) be a finite measure space and X a separable Banach space. We characterize the linear isometries of Lp(Ω, X) onto itself for 1 ⩽ p < ∞, p ≠ 2 under the condition that X is not the lp-direct sum of two nonzero spaces (for the same p). It is shown that T is such an isometry if and only if (Tf)(·) = S(·)h(·)(Φ(f))(·), where Φ is a set isomorphism of ∑ onto itself, S is a strongly measurable operator-valued map such that S(t) is a.e. an isometry of X onto itself, and h is a scalar function which is related to Φ. It is further shown that for a big class of measure spaces (perhaps all nontrivial ones) the condition on X is also a necessary condition for the above conclusion to hold. In the case when X is a Hilbert space the injective isometries of Lp(Ω, X) are also characterized. They have the same form as above, except that Φ and S(t) are not necessarily onto