On phase-isometries between the positive cones of $${\varvec{c}}_{\textbf{0}}$$ c 0

Longfa Sun (), Yinghua Sun () and Duanxu Dai ()
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Longfa Sun: North China Electric Power University
Yinghua Sun: North China Electric Power University
Duanxu Dai: Jimei University

Indian Journal of Pure and Applied Mathematics, 2025, vol. 56, issue 1, 210-217

Abstract: Abstract Let $$c_0^+$$ c 0 + be the positive cone of $$c_0$$ c 0 , i.e., $$c_0^+=\{x=(x_n)_{n=1}^\infty \in c_0: x_n\ge 0,\; \textrm{for}\;\textrm{all}\;n\in {\mathbb {N}}\}$$ c 0 + = { x = ( x n ) n = 1 ∞ ∈ c 0 : x n ≥ 0 , for all n ∈ N } . A map $$f:c_0^+\rightarrow c_0^+$$ f : c 0 + → c 0 + is called a phase-isometry provided $$\begin{aligned} \{\Vert f(x)+f(y)\Vert ,\Vert f(x)-f(y)\Vert \}=\{\Vert x+y\Vert ,\Vert x-y\Vert \} \end{aligned}$$ { ‖ f ( x ) + f ( y ) ‖ , ‖ f ( x ) - f ( y ) ‖ } = { ‖ x + y ‖ , ‖ x - y ‖ } for all $$x,y\in c_0^+$$ x , y ∈ c 0 + . In this paper, we prove that every phase-isometry $$f:c_0^+\rightarrow c_0^+$$ f : c 0 + → c 0 + is actually an isometry. And there exists a bounded linear operator $$T:\overline{span} f(c_0^+)\rightarrow c_0$$ T : span ¯ f ( c 0 + ) → c 0 with $$\Vert T\Vert =1$$ ‖ T ‖ = 1 such that $$\begin{aligned} Tf=Id_{c_0^+}. \end{aligned}$$ T f = I d c 0 + . Furthermore, if f is almost surjective, then f is an additive isometry as the restriction of a surjective linear isometry from $$c_0$$ c 0 onto itself.

Keywords: Phase-isometries; Linear isometries; Banach space; Wigner’s theorem; Primary 46B04; 46B20 (search for similar items in EconPapers)
Date: 2025
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DOI: 10.1007/s13226-023-00472-5

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