Relation between the distance from a point to the resolvent set of a self-adjoint operator and the norm of a related operator
Solution 1:
For any continuous function $F$ on the spectrum of $A$, $\|F(A)\|=\sup_{\lambda\in\sigma(A)}|F(\lambda)|$. Let $F(\lambda)=\frac{1}{\lambda-\lambda_0}$. Then $F(A)=(A-\lambda_0 I)^{-1}$ and $$ \|(A-\lambda_0 I)^{-1}\| = \sup_{\lambda\in\sigma(A)}\left|\frac{1}{\lambda-\lambda_0}\right|=\frac{1}{\inf_{\lambda\in\sigma(A)}|\lambda-\lambda_0|}=\frac{1}{\mbox{dist}(\lambda_0,\sigma(A))}. $$