A well known result by Varah states that if $A$ is a strictly diagonally dominant matrix of dimension $n$, then $\|A^{-1}\|_{\infty} \le \max_i\frac{1}{|a_{kk}|-\sum_{j \neq k}|a_{kj}|}$, where the infinity norm here is the induced $\ell_{\infty}$, i.e. $\max_{k}\sum_{j}|a_{kj}|$.

Let us assume that $A$ is real and Hermitian, with eigenvalues in $[-a,a]$, all distinct. I am trying to bound the infinity norm of the following resolvent:

$$ B = \left( \delta i I - A \right)^{-1} $$

where $\delta$ is small, so $\delta i I -A$ is not necessarily diagonally dominant. Naturally, $\delta i I -A$ is always invertible. I denote $i$ as the imaginary unit and $I$ the identity matrix.

A trivial upper bound would be $\frac{\sqrt{n}}{\delta}$. Do you see any circumstances for which a better bound may be attained?

Thank you.

**Edit**: A bound of $\epsilon$ to the above question is equivalent to asking whether for all matrices $E$ with $\|E\|_{\infty} \le \epsilon$, $i\delta$ is not an eigenvalue of $A+E$.

Generally, $\epsilon$ will be a function of both $\delta$, $\|E\|$ and the condition number of the eigenvalue matrix of $A$. However, subject to the above restrictions, do you see a better bound?