Proposition 29.4.9.label Let $A$ be a unital Banach algebra and $x, y \in A$, then:
- (1)
$\sigma(xy) \cup \bracs{0}= \sigma(yx) \cup \bracs{0}$.
- (2)
$[xy]_{sp}= [yx]_{sp}$.
Proof. (1): Let $\lambda \in \complex \setminus \bracs{0}$, then by Proposition 29.2.4, $\lambda - xy \in G(A)$ if and only if $\lambda - yx \in G(A)$.$\square$
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