Spectral norm of a Hadamard product Let $F$ be the $n\times n$ DFT matrix, i.e., $F_{i,j} = \exp\left(-\tfrac{2\pi(i-1)(j-1)}{n}\imath\right)$, for $1\le i,j\le n$ and $\imath =\sqrt{-1}$. Futhermore, let $\Vert \cdot\Vert$ and $\odot$ denote the spectral norm and the Hadamard product, respectively.
For any $X\in \mathbb{R}^{n\times n}$, is there an explicit formula for $$\Vert F\odot X\Vert, \tag{$\star$}$$ 
in terms of $\Vert X\Vert$ or other simple functions of $X$? If instead of the spectral norm we had the Frobenius norm then the answer simply would have been the Frobenius norm of $X$. For $(\star)$, howerver, I guess the answer is negative. If my guess is correct, is there a reasonably sharp upper bound for $(\star)$?
 A: I obtained the following upper bound. Let $D_z$ denote a diagonal matrix whose diagonal entries are given by vector $z$. Furthermore, write the SVD of $X$ as $X=UD_\sigma V^\mathrm{T}=\sum _{i=1}^r\sigma_i u_iv_i^\mathrm{T}$ where $r$ denotes the rank of $X$. Then we have
\begin{align*}
F\odot X &=\sum_{i=1}^r F\odot \left(\sigma_i u_i v_i^\mathrm{T}\right)\\
&= \left[\begin{array}{cccc}
D_{u_1}&D_{u_2}&\dotsm&D_{u_r}
\end{array}\right]
\left(D_\sigma \otimes F\right)\left[\begin{array}{c}
D_{v_1}\\
D_{v_2}\\
\vdots\\
D_{v_r}
\end{array}\right]\\
&=\left[\begin{array}{cccc}
\sigma_1D_{u_1}&\sigma_2D_{u_2}&\dotsm&\sigma_rD_{u_r}
\end{array}\right]
\left(I_{r\times r}\otimes F\right)\left[\begin{array}{c}
D_{v_1}\\
D_{v_2}\\
\vdots\\
D_{v_r}
\end{array}\right],
\end{align*}
where $\otimes$ denotes the Kronecker product. Now we can bound $\left\Vert F\odot X\right \Vert$ as
\begin{align*}
\left\Vert F\odot X\right\Vert & \leq \left\Vert\left[\begin{array}{cccc}
\sigma_1D_{u_1}&\sigma_2D_{u_2}&\dotsm&\sigma_rD_{u_r}
\end{array}\right]\right\Vert\left\Vert
\left(I_{r\times r}\otimes F\right)\left[\begin{array}{c}
D_{v_1}\\
D_{v_2}\\
\vdots\\
D_{v_r}
\end{array}\right]\right\Vert\\
&=\sqrt{n} \left\Vert UD_\sigma \right\Vert_{\infty,2}\left\Vert V\right\Vert_{\infty,2},
\end{align*}
where $\left\Vert\cdot\right\Vert_{\infty,2}$ is the maximum row-wise $\ell_2$-norm. There is also a simpler bound 
\begin{align*}
\left\Vert F\odot X\right\Vert&\leq \left\Vert F\odot X\right\Vert_F\\
&=\left\vert X\right\Vert_F\\
&=\left\Vert\sigma\right\Vert_2,
\end{align*}
which can be sharper.
