Spectrum of the inverse operator?

Question

How can you prove that the spectrum of the inverse of an operator $A^{-1}$ is given by all $\frac{1}{\lambda}$ for all $\lambda \in \sigma(A)\backslash \{0\}$?

Answer 1

We suppose that $A$ is invertible . By $\rho(T)$ we denote the resolvent set of an operator $T$. Let $ \lambda \ne 0$.

Then:

$$ (*) \quad A^{-1}- \lambda I=A^{-1}(I-\lambda A)=\lambda A^{-1}(\frac{1}{\lambda}I-A).$$

From $(*)$ we see:

$\lambda \in \rho(A^{-1})$ iff $\frac{1}{\lambda} \in \rho(A)$