A Sharp Unitarily Invariant Norm Bound for the Off-Diagonal Block Perturbation of a Hermitian Matrix

Let $$ A=\begin{bmatrix} H_1 & E^* \\ E & H_2 \end{bmatrix} \quad\text{and}\quad \widetilde A=\begin{bmatrix} H_1 & 0 \\ 0 & H_2 \end{bmatrix} $$ be two partitioned Hermitian matrices, where $\widetilde A$ is obtained from $A$ by simply dropping the off-diagonal blocks, and let $η$ be the gap betwee