Vol. 55, No. 1, 1974

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The isometries of Lp (X, K)

Michael James Cambern

Vol. 55 (1974), No. 1, 9–17

Let (X,Σ) 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 < ,p2. 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.

Mathematical Subject Classification 2000
Primary: 46E40
Received: 3 July 1973
Published: 1 November 1974
Michael James Cambern