On norm-preserving isomorphisms of $L^{p}(\mu,H)$

Year 2016, Volume: 45 Issue: 1, 33 - 41, 01.02.2016

B.a. Güntürk B. Cengiz M. Gürdal

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 $1\leq p<\infty$, $p\neq 2$) 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.

Keywords

Measure space, Bochner space, perfect measure, hyperstonean space, linear isometries

References

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