The spectrum of an invertible element $x$ is $\sigma(x^{-1})=\{\lambda^{-1}: \lambda\in \sigma(x)\}$

Question

Suppose $x$ is invertible in the unital Banach algebra $A$.

How can I prove that $\sigma(x^{-1})=\{\lambda^{-1} : \lambda\in \sigma(x)\}$

Answer 1

If $x$ is invertible, $x \cdot v = \lambda v$ is equivalent to $v= x^{-1}x \cdot v= \lambda x^{-1} \cdot v$, that is $x^{-1} \cdot v=\lambda^{-1} v$.