On a Conjecture Related to Eigenvalue Perturbations

In connection with eigenvalue perturbation bounds, Li [Linear Algebra Appl., 278 (1998), pp. 317-326] formulated three minimization problems involving Hadamard products. The third asks how to relate $$\underset{W\mathrm{n}\mathrm{o}\mathrm{n}\mathrm{s}\mathrm{i}\mathrm{n}\mathrm{g}\mathrm{u}\mathrm{l}\mathrm{a}\mathrm{r}}{\min }\mathrm{\parallel }W\circ G{\mathrm{\parallel }}_{\mathrm{u}\mathrm{i}}\mathrm{\parallel }{W}^{-1}\circ G^{\mathrm{T}}{\mathrm{\parallel }}_{\mathrm{u}\mathrm{i}}\text{quadand}\underset{P\mathrm{p}\mathrm{e}\mathrm{r}\mathrm{m}\mathrm{u}\mathrm{t}\mathrm{a}\mathrm{t}\mathrm{i}\mathrm{o}\mathrm{n}}{\min }\mathrm{\parallel }P\circ G{\mathrm{\parallel }}_{\mathrm{u}\mathrm{i}}^2,$$ where $\mathrm{\parallel }\cdot {\mathrm{\parallel }}_{\mathrm{u}\mathrm{i}}$ denotes a unitarily invariant norm and $\circ$ denotes the Hadamard product, i.e., entrywise multiplication. The first quantity never exceeds the second, and a reverse inequality with a constant independent of the matrix size was conjectured. Romeo and Tilli [Linear Algebra Appl., 326 (2001), pp. 161-172] showed that if such a bound holds for every $G$, then $\mathrm{\parallel }\cdot {\mathrm{\parallel }}_{\mathrm{u}\mathrm{i}}$ must be uniformly equivalent to the Frobenius norm $\mathrm{\parallel }\cdot {\mathrm{\parallel }}_{\mathrm{F}}$. We prove that the two quantities coincide when $\mathrm{\parallel }\cdot {\mathrm{\parallel }}_{\mathrm{u}\mathrm{i}}=\mathrm{\parallel }\cdot {\mathrm{\parallel }}_{\mathrm{F}}$: for every complex matrix $G$, $$\underset{W\mathrm{n}\mathrm{o}\mathrm{n}\mathrm{s}\mathrm{i}\mathrm{n}\mathrm{g}\mathrm{u}\mathrm{l}\mathrm{a}\mathrm{r}}{\min }\mathrm{\parallel }W\circ G{\mathrm{\parallel }}_{\mathrm{F}}\mathrm{\parallel }{W}^{-1}\circ G^{\mathrm{T}}{\mathrm{\parallel }}_{\mathrm{F}}=\underset{P\mathrm{p}\mathrm{e}\mathrm{r}\mathrm{m}\mathrm{u}\mathrm{t}\mathrm{a}\mathrm{t}\mathrm{i}\mathrm{o}\mathrm{n}}{\min }\mathrm{\parallel }P\circ G{\mathrm{\parallel }}_{\mathrm{F}}^2\mathrm{.}$$ The associated conjecture therefore holds with the optimal constant $c=1$.