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On non-linear ε\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\varepsilon$$\end{document}-isometries between the positive cones of certain continuous function spaces

On non-linear ε\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym}... Let X, Y be two w∗\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$w^*$$\end{document}-almost smooth Banach spaces, C(B(X∗),w∗)\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$C(B(X^*),w^*)$$\end{document} be the Banach space of all continuous real-valued functions on B(X∗)\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$B(X^*)$$\end{document} endowed with the supremum norm and C+(B(X∗),w∗)\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$C_+(B(X^*),w^*)$$\end{document} be the positive cone of C(B(X∗),w∗)\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$C(B(X^*),w^*)$$\end{document}. In this paper, we show that if F:C+(B(X∗),w∗)→C+(B(Y∗),w∗)\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$F: C_+(B(X^*),w^*)\rightarrow C_+(B(Y^*),w^*)$$\end{document} is a standard almost surjective ε\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$\varepsilon$$\end{document}-isometry, then there exists a homeomorphism τ:B(X∗)→B(Y∗)\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$\tau : B(X^*)\rightarrow B(Y^*)$$\end{document} in the w∗\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$w^*$$\end{document}-topology such that for any x∗∈B(X∗)\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$x^*\in B(X^*)$$\end{document}, we have |⟨δx∗,f⟩-⟨δτ(x∗),F(f)⟩|≤2ε,for allf∈C+(B(X∗),w∗).\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$\begin{aligned} |\langle \delta _{x^*}, f\rangle -\langle \delta _{\tau (x^*)}, F(f)\rangle |\le 2\varepsilon ,\quad \text{for all } f\in C_+(B(X^*),w^*). \end{aligned}$$\end{document}As its application, we show that if U:C(B(X∗),w∗)→C(B(Y∗),w∗)\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$U:C(B(X^*),w^*)\rightarrow C(B(Y^*),w^*)$$\end{document} is the canonical linear surjective isometry induced by the homeomorphism γ=τ-1:B(Y∗)→B(X∗)\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$\gamma =\tau ^{-1}:B(Y^*)\rightarrow B(X^*)$$\end{document} in the w∗\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$w^*$$\end{document}-topology, then ‖F(f)-U(f)‖≤2ε,for allf∈C+(B(X∗),w∗).\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$\begin{aligned} \Vert F(f)-U(f)\Vert \le 2\varepsilon , \quad \text{for all }f\in C_+(B(X^*),w^*). \end{aligned}$$\end{document} http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Annals of Functional Analysis Springer Journals

On non-linear ε\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\varepsilon$$\end{document}-isometries between the positive cones of certain continuous function spaces

Annals of Functional Analysis , Volume 12 (4) – Oct 1, 2021

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Publisher
Springer Journals
Copyright
Copyright © Tusi Mathematical Research Group (TMRG) 2021
ISSN
2639-7390
eISSN
2008-8752
DOI
10.1007/s43034-021-00141-w
Publisher site
See Article on Publisher Site

Abstract

Let X, Y be two w∗\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$w^*$$\end{document}-almost smooth Banach spaces, C(B(X∗),w∗)\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$C(B(X^*),w^*)$$\end{document} be the Banach space of all continuous real-valued functions on B(X∗)\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$B(X^*)$$\end{document} endowed with the supremum norm and C+(B(X∗),w∗)\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$C_+(B(X^*),w^*)$$\end{document} be the positive cone of C(B(X∗),w∗)\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$C(B(X^*),w^*)$$\end{document}. In this paper, we show that if F:C+(B(X∗),w∗)→C+(B(Y∗),w∗)\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$F: C_+(B(X^*),w^*)\rightarrow C_+(B(Y^*),w^*)$$\end{document} is a standard almost surjective ε\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$\varepsilon$$\end{document}-isometry, then there exists a homeomorphism τ:B(X∗)→B(Y∗)\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$\tau : B(X^*)\rightarrow B(Y^*)$$\end{document} in the w∗\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$w^*$$\end{document}-topology such that for any x∗∈B(X∗)\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$x^*\in B(X^*)$$\end{document}, we have |⟨δx∗,f⟩-⟨δτ(x∗),F(f)⟩|≤2ε,for allf∈C+(B(X∗),w∗).\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$\begin{aligned} |\langle \delta _{x^*}, f\rangle -\langle \delta _{\tau (x^*)}, F(f)\rangle |\le 2\varepsilon ,\quad \text{for all } f\in C_+(B(X^*),w^*). \end{aligned}$$\end{document}As its application, we show that if U:C(B(X∗),w∗)→C(B(Y∗),w∗)\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$U:C(B(X^*),w^*)\rightarrow C(B(Y^*),w^*)$$\end{document} is the canonical linear surjective isometry induced by the homeomorphism γ=τ-1:B(Y∗)→B(X∗)\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$\gamma =\tau ^{-1}:B(Y^*)\rightarrow B(X^*)$$\end{document} in the w∗\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$w^*$$\end{document}-topology, then ‖F(f)-U(f)‖≤2ε,for allf∈C+(B(X∗),w∗).\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$\begin{aligned} \Vert F(f)-U(f)\Vert \le 2\varepsilon , \quad \text{for all }f\in C_+(B(X^*),w^*). \end{aligned}$$\end{document}

Journal

Annals of Functional AnalysisSpringer Journals

Published: Oct 1, 2021

Keywords: ε\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\varepsilon$$\end{document}-isometry; Hyers–Ulam stability; Banach–Stone theorem; Continuous function space; 46B04; 46B20; 46E15

References