Suppose $A$ is an $n\times n$ matrix such that $A+A^H-\delta I_n$ is positive-semidefinite, for some $\delta>0$, then can we show a bound on the norm of $A^{-1}$ ? Can we show that this the norm of the inverse of $A$ is at most $2/\delta$ ? (The norm is the usual matrix 2-norm)
Let me briefly describe my approach. Please correct me if I am wrong.
By the positive semi-definiteness condition, we obtain that $v^H (A+A^H) v \geq \delta$, $\forall v$ of norm $1$. So the norm of $A+A^H$ is at least $\delta$, and using Cauchy-Schwarz inequality, we obtain a lower bound on the norm of $A$ as $\delta /2$. But this does not seem to help in upper bounding the norm of $A^{-1}$.