Vol. 55, No. 1, 1974

Download this article
Download this article. For screen
For printing
Recent Issues
Vol. 328: 1  2
Vol. 327: 1  2
Vol. 326: 1  2
Vol. 325: 1  2
Vol. 324: 1  2
Vol. 323: 1  2
Vol. 322: 1  2
Vol. 321: 1  2
Online Archive
Volume:
Issue:
     
The Journal
About the journal
Ethics and policies
Peer-review process
 
Submission guidelines
Submission form
Editorial board
Officers
 
Subscriptions
 
ISSN: 1945-5844 (e-only)
ISSN: 0030-8730 (print)
 
Special Issues
Author index
To appear
 
Other MSP journals
The isometries of Lp (X, K)

Michael James Cambern

Vol. 55 (1974), No. 1, 9–17
Abstract

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
Milestones
Received: 3 July 1973
Published: 1 November 1974
Authors
Michael James Cambern