Non-linear modular Birkhoff-James orthogonality preservers between spaces of continuous functions

Ryotaro Tanaka

doi:10.1016/j.jmaa.2020.124744

Non-linear modular Birkhoff-James orthogonality preservers between spaces of continuous functions

Ryotaro Tanaka

Liberal Arts

Research output: Contribution to journal › Article › peer-review

Abstract

A new notion of non-linear isomorphisms between Banach modules is introduced based on modular Birkhoff-James orthogonality. It is shown that a (possibly non-linear) bijective modular Birkhoff-James orthogonality preserver in both direction between spaces of continuous functions (vanishing at infinity) induces a homeomorphism between underlying locally compact Hausdorff spaces. Moreover, characterizations of linear and additive modular Birkhoff-James orthogonality preservers between spaces of continuous functions are also given in terms of the induced homeomorphisms and continuous functions having constant absolute values.

Original language	English
Article number	124744
Journal	Journal of Mathematical Analysis and Applications
Volume	495
Issue number	2
DOIs	https://doi.org/10.1016/j.jmaa.2020.124744
Publication status	Accepted/In press - 2020

Keywords

(Modular) Birkhoff-James orthogonality
Continuous function
Locally compact Hausdorff space
Non-linear isomorphism

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title = "Non-linear modular Birkhoff-James orthogonality preservers between spaces of continuous functions",

abstract = "A new notion of non-linear isomorphisms between Banach modules is introduced based on modular Birkhoff-James orthogonality. It is shown that a (possibly non-linear) bijective modular Birkhoff-James orthogonality preserver in both direction between spaces of continuous functions (vanishing at infinity) induces a homeomorphism between underlying locally compact Hausdorff spaces. Moreover, characterizations of linear and additive modular Birkhoff-James orthogonality preservers between spaces of continuous functions are also given in terms of the induced homeomorphisms and continuous functions having constant absolute values.",

keywords = "(Modular) Birkhoff-James orthogonality, Continuous function, Locally compact Hausdorff space, Non-linear isomorphism",

author = "Ryotaro Tanaka",

note = "Funding Information: The author was supported in part by Grants-in-Aid for Scientific Research Grant Number 19K14561 , Japan Society for the Promotion of Science . The author would like to thank the referees for their careful reading and valuable comments.",

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doi = "10.1016/j.jmaa.2020.124744",

language = "English",

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journal = "Journal of Mathematical Analysis and Applications",

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AU - Tanaka, Ryotaro

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AB - A new notion of non-linear isomorphisms between Banach modules is introduced based on modular Birkhoff-James orthogonality. It is shown that a (possibly non-linear) bijective modular Birkhoff-James orthogonality preserver in both direction between spaces of continuous functions (vanishing at infinity) induces a homeomorphism between underlying locally compact Hausdorff spaces. Moreover, characterizations of linear and additive modular Birkhoff-James orthogonality preservers between spaces of continuous functions are also given in terms of the induced homeomorphisms and continuous functions having constant absolute values.

KW - (Modular) Birkhoff-James orthogonality

KW - Continuous function

KW - Locally compact Hausdorff space

KW - Non-linear isomorphism

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