Approximate local isometries on spaces of absolutely continuous functions
type: Item
instance of: scholarly article
title: Approximate local isometries on spaces of absolutely continuous functions
zbMATH Open document ID: 1472.46008
author: Maliheh Hosseini
author: A. Jiménez-Vargas
published in: Results in Mathematics
publication date: 2021-07-07T00:00:00Z
review text: Let \(X,Y\) be compact sets of real numbers. Let \(AC(X)$ $(AC(Y))\) denote space of complex absolutely continous functions, equipped with the norm \(\|f\|_{\Sigma}= \|f\|_{\infty}+V(f)\), for \(f \in AC(X)\), where \(V(f)\) denotes the variation of~\(f\). A description of the surjective isometries between such spaces has long been known (see [\textit{V. D. Pathak}, Can. J. Math. 34, 298--306 (1982; Zbl 0464.46029)]). The authors first note that the group of isometries is not topologically reflexive and go on to provide a description of the objects in the topological closure of the isometry group (Theorem~2).
reviewed by: T.S.S.R.K. Rao
Mathematics Subject Classification ID: 46B04
Mathematics Subject Classification ID: 46J15
Mathematics Subject Classification ID: 47B48
Mathematics Subject Classification ID: 46E15
Mathematics Subject Classification ID: 47B38
zbMATH DE Number: 7369084
zbMATH Keywords: algebraic reflexivity
zbMATH Keywords: topological reflexivity
zbMATH Keywords: local isometry
zbMATH Keywords: 2-local isometry
zbMATH Keywords: absolutely continuous function
MaRDI profile type: Publication
full work available at URL: s00025-021-01384-8
OpenAlex ID: W3138164025
cites work: Functions of bounded variation on compact subsets of the plane
cites work: Q2723195
cites work: Algebraic reflexivity of C(X,E) and Cambern's theorem
cites work: Q3653818
cites work: Algebraic and topological reflexivity of spaces of Lipschitz functions
cites work: Local isometries on spaces of continuous functions
cites work: Reflexivity of the isometry group of some classical spaces.
cites work: Isometries of certain Banach algebras
cites work: Isomorphisms of AC(σ) spaces
cites work: Q4537892
cites work: 2-local isometries on function spaces
cites work: Isometries on spaces of absolutely continuous vector-valued functions
cites work: Algebraic reflexivity of sets of bounded linear operators on absolutely continuous function spaces
cites work: 2-local isometries between spaces of functions of bounded variation
cites work: Isometries Between Function Spaces
cites work: Weak-2-local isometries on uniform algebras and Lipschitz algebras
cites work: Some characterizations of the automorphisms of \(B(H)\) and \(C(X)\)
cites work: Reflexivity of the group of surjective isometries on some Banach spaces
cites work: Linear Isometries of Spaces of Absolutely Continuous Functions
cites work: Linear isometries of some function spaces
cites work: Absolute continuity and existence of solutions to dynamic inclusions in time scales
cites work: Q3901719
DOI: 10.1007/S00025-021-01384-8
description: scientific article; zbMATH DE number 7369084