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

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.

### 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
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