2 problems

Let $A$ and $C$ be operators with separated spectra satisfying … and let $c_{\mathrm{best}}>0$ denote the largest universal constant for which the smallness condition … guarantees…

Let $A$, $B$, $C$, and $D$ be the operator entries in the block-operator setting, with $D$ self-adjoint and with spectrum $\sigma_D=\sigma_D^{\rm ac}=\overline{\Delta}$, where…