Self-adjoint operators on real Banach spaces
type: Item
instance of: scholarly article
title: Self-adjoint operators on real Banach spaces
zbMATH Open document ID: 1262.39038
author: Paweł Wójcik
published in: Nonlinear Analysis. Theory, Methods \& Applications. Series A: Theory and Methods
publication date: 2013-03-04T00:00:00Z
review text: Suppose that \((X,\|\cdot\|)\) is a real normed space.The normed derivatives \({\rho}_+^{'},{\rho}_-^{'} :X\times X \to \mathbb{R}\) are defined by \[ {\rho}_{\pm}^{'}(x,y):=\lim_{t\to 0^{\pm}} \frac{\|x+ty\|^2-\|x\|^2}{2t}=\|x\|.\lim_{t\to 0^{\pm}} \frac{\|x+ty\|-\|x\|}{t}. \] In this paper the author shows that if \(X\) is a Banach space and \(f:X\to X\) is a function such that \({\rho}_{+}^{'}(f(x),y)={\rho}_{+}^{'}(x,f(y)),\) then \(f\) is linear and continuous. See page 177 of [\textit{C. Alsina, J. Sikorska} and \textit{M. S. Tomás}, Norm derivatives and characterizations of inner product spaces. Hackensack, NJ: World Scientific. (2010; Zbl 1196.46001)]. He also solves the question,when \(X=C(M)\) and derives a new characterizations of inner product spaces.
reviewed by: Maryam Amyari
Mathematics Subject Classification ID: 39B52
Mathematics Subject Classification ID: 46C50
Mathematics Subject Classification ID: 47A05
Mathematics Subject Classification ID: 46C15
zbMATH DE Number: 6140763
zbMATH Keywords: norm derivatives
zbMATH Keywords: self-adjoint operator
zbMATH Keywords: functional equation
zbMATH Keywords: characterization of inner product spaces
MaRDI profile type: Publication
full work available at URL: j.na.2012.12.013
OpenAlex ID: W2086244247
DOI: 10.1016/J.NA.2012.12.013
description: scientific article; zbMATH DE number 6140763