On Norm-Preserving Isomorphisms of L-P (Mu, H)

Abstract

Given an arbitrary positive measure space (X, A, mu) and a Hilbert space H. In this article we give a new proof for the characterization theorem of the surjective linear isometries of the space L-p (mu, H) (for 1 <= p < infinity p not equal 2) which is essentially different from the existing one, and depends on the p-projections of L-p (mu, H). We generalize the known characterization of the p-projections of L-p (mu, H) for sigma-finite measure to the arbitrary case. They are proved to be the multiplication operations by the characteristic functions of the locally measurable sets, or that of the clopen (closed-open) subsets of the hyperstonean space the measure mu determines.