Generalization of some commutative perturbation results

We study the stability of certain spectra under some algebraic conditions weaker than commutativity and we generalize many known commutative perturbation results. In particular, in a complex unital Banach algebra $$\mathcal {A},$$ prove that if $$x \in \hbox {comm}_{w}(a)$$ is nilpotent, then $$\sigma (x)=\sigma (x+a).$$ Among other things, {comm}(xy)\cup {comm}(yx)$$ or $$y\in for all $$x,y \mathcal {A}$$ commutative. When {A}=L(X)$$ bounded linear operators acting on space X, _{p}(T){{\setminus }}\{0\}\subset \sigma _{p}(T+N){\setminus }\{0\},$$ where N nilpotent $$T\in {comm}(NT).$$