Let (X,Σ,μ) be a finite
measure space, and denote by Lp(X,K) the Banach space of measurable functions F
defined on X and taking values in a separable Hilbert space K, such that
∥F(x)∥p is integrable. In this article a characterization is given of the linear
isometries of Lp(X,K) onto itself, for 1 ≦ p < ∞,p≠2. It is shown that T
is such an isometry iff T is of the form (T(F))(x) = U(x)h(x)(Φ(F))(x),
where Φ is a set isomorphism of Σ onto itself, U is a weakly measurable
operator-valued function such that U(x) is a.e. an isometry of K onto itself, and h is
a scalar function which is related to Φ via a formula involving Radon-Nikodym
derivatives.
