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