Property of spectrum

Question

Let $T$ be an operator on $X$ with non-empty resolvent set, and let $\lambda _0 \in \rho (T)$. Show that $\lambda \in \sigma (T)$ if and only if $(\lambda _0 − λ)^{−1} \in \sigma(R(\lambda _0; T))$

I tried to prove it using definiton of resolvent set but it didn't work.

Answer 1

Hint: $$(\lambda_0-\lambda)^{-1}-R(\lambda_0;T)=(\lambda_0-\lambda)^{-1}(\lambda-T)R(\lambda_0;T).$$