Norm of the Resolvent when an equality holds?

Question

Let X is Banach space and $T:X \rightarrow X$ is linear operator,

then $||R_\lambda(T)||≥\frac{1}{d(\lambda,σ(T))}\;\;\;$ where $\;\;\;R_\lambda(T) = (T-\lambda I)^{-1}$.

I would like to know when the equality holds.

In other words: what assumptions on operator $\;T\;$ I need for

$||R_\lambda(T)||=\frac{1}{d(\lambda,σ(T))}$.