Given a matrix $A\in M_n(\mathbb{C})$, I will denote by $||A||_\infty$ the operator norm of $A$, as seen acting on the Hilbert space $\mathbb{C}^n$. This makes $M_n(\mathbb{C})$ into a Banach space (actually a Banach algebra, actually a $C^*$ algebra).

We will define a new norm on $M_n(\mathbb{C})$ in the following way: A matrix $T$ acts on $M_n(\mathbb{C})$ by pointwise multiplication (i.e. Schur product), giving rise to an operator $M_T:M_n(\mathbb{C})\longrightarrow M_n(\mathbb{C})$. For example for

$$T=\left[\begin{array}{cc} 1 & 2 \\ 3 & 4 \end{array}\right]$$ we have

$$M_T\left(\left[\begin{array}{cc} a & b \\ c & d \end{array}\right]\right)= \left[\begin{array}{cc} a & 2b \\ 3c & 4d \end{array}\right]$$

We will denote by $||T||_{Schur}$ the norm of the operator $M_T$, as an operator of the Banach space $M_n(\mathbb{C})$. How hard is it to compute/estimate this norm? Are there polynomial algorithms (in $n$ and the approximation error) that can do it? Is it actually NP?