Remarks on the stability of Lie homomorphisms. (English) Zbl 1280.39016

Brzdȩk, Janusz; Fošner, Ajda

J. Math. Anal. Appl. 400, No. 2, 585-596 (2013).

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.

Reviewer: Thomas Riedel (Louisville)

MSC:

39B82	Stability, separation, extension, and related topics for functional equations
39B52	Functional equations for functions with more general domains and/or ranges

Keywords:

Hyers-Ulam stability; Lie homomorphisms; Banach algebra