Spectral Theory of an operator

Question

If we define the spectrum of a bounded linear operator $T$ by $$\sigma(T)=\{\lambda\in \mathbb C:\ T-\lambda I \ \text{ has no inverse} \}.$$ What about $\sigma(T^{-1})$?

Answer 1

If you are talking about $T^{-1}$, it means you are assuming that $T$ is invertible. So $0\not\in\sigma(T)\cup\sigma(T^{-1})$

You have $$ T^{-1}-\lambda I=-\lambda T^{-1}(T-\lambda^{-1} I). $$ So $\lambda\in\sigma(T^{-1})$ if and only if $1/\lambda\in\sigma(T)$. That is $$ \sigma(T^{-1})=\{1/\lambda:\ \lambda\in\sigma(T)\}. $$