Remarks on the stability of Lie homomorphisms
type: Item
instance of: scholarly article
title: Remarks on the stability of Lie homomorphisms
zbMATH Open document ID: 1280.39016
author: Ajda Fošner
author: Janusz Brzdȩk
published in: Journal of Mathematical Analysis and Applications
publication date: 2013-03-11T00:00:00Z
review text: The authors consider the stability of Lie homomorphisms, in particular, the generalized Hyers-Ulam stability. They consider functions from a normed Lie algebra \(X\) into a Banach Lie algebra \(Y\), both over the real or complex field. A Lie homomorphism is an additive function \(h:X\rightarrow Y\) such that \(h([x,y])=[h(x),h(y)]\) holds for all \(x,y\in X\). In the first theorem, they provide conditions on \(\phi,\psi:X\times X\rightarrow[0,\infty]\) that ensure the existence of a unique Lie homomorphism \(H\) which is close to \(h\) when the latter satisfies for all \(x,y\in X\) \( \| h([x,y])-[h(x),h(y)]\| \leq\psi(x,y) \) and \( \| h(\lambda ax-by)-\lambda Ah(x)-Bh(y)\| \leq\phi(x,y) \), where \(\lambda\) ranges over a set that guarantees that every additive bounded function has to be continuous; \(a,b,A,B\) are some arbitrary constants in the field satisfying \(ab\neq 0\). Theorem 2 replaces the second condition with \( \| h(2x+y)+h(x+2y)-h(3x)-h(3y)\| \leq\phi(x,y) \) and the authors are thus able to relax the conditions on \(\phi,\psi\). The paper concludes with some results on superstability. Here the assumption is that \( \| h(x+y)-h(x)-h(y)\| \leq\left(\| x\| +\| y\| \right)^{r} \) and \( \| h([x,y])-[h(x),h(y)]\| \leq\psi(x,y) \) for all \(x,y\in X\) and some negative real number \(r\). The authors give a condition on \(\psi\) that guarantees that any \(h\) satisfying these two inequalities will indeed be a Lie homomorphism. Two further results on superstability are given.
reviewed by: Thomas Riedel
Mathematics Subject Classification ID: 39B82
Mathematics Subject Classification ID: 39B52
zbMATH DE Number: 6143237
zbMATH Keywords: Hyers-Ulam stability
zbMATH Keywords: Lie homomorphisms
zbMATH Keywords: Banach algebra
MaRDI profile type: Publication
full work available at URL: j.jmaa.2012.11.008
OpenAlex ID: W2092866341
DOI: 10.1016/J.JMAA.2012.11.008
description: scientific article; zbMATH DE number 6143237