Extension of phase-isometries between the unit spheres of complex L∞(Γ) spaces

Yarong Zhang, Lingen Zhu, Jianan Yang

Abstract


Let Γ be nonempty index set, and X, Y are complex L(Γ)-type spaces. f: SX, SY will denote their unit spheres. Give a surjective mapping f: SX → SY satisfying the functional equation

{||f(x) + f(y)||, ||f(x)− f(y)||} = {||x+y||, ||x−y||} (x, y ∈ SX )

We show that there exists a function ε: SX → {−1,1} such that ε f is an isometry. Moreover, this isometry is the restriction of a real linear isometry from X to Y.

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Published: 2021-03-24

How to Cite this Article:

Yarong Zhang, Lingen Zhu, Jianan Yang, Extension of phase-isometries between the unit spheres of complex L∞(Γ) spaces, Adv. Fixed Point Theory, 11 (2021), Article ID 7

Copyright © 2021 Yarong Zhang, Lingen Zhu, Jianan Yang. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Advances in Fixed Point Theory

ISSN: 1927-6303

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