Let $X$ be a Banach space and let $T$ be a bounded linear operator on $X$. How to prove that: If $\lambda \pi(I)-\pi(T)$ is invertible in the Calkin algebra $C(X)$, then $\pi(T)$ is quasinilpotent in $C(X)$. Where, $\pi$ is the (canonical) quotient map of the set of bounded linear operators on $X$ onto $C(X)$.
Definition : an operator $T \in \mathcal{L}(X)$ is said to be a quasinilpotent if the spectral radius $r(T)=\lim_{n \rightarrow \infty}\|T^{n}|| ^{\frac{1}{n}}=0$.