Let $M\in M_n(\mathbb C)$ be a $n\times n$ matrix over the complex field. It can be written uniquely as $M=H+A$, where $H=H^*$ denotes its Hermitian part and $A=-A^*$ its anti-Hermitian part.

Its spectral norm is defined as \begin{equation} ||M||^2=\underset{x}{\text{max}}\,\frac{(Mx,Mx)}{(x,x)}\;, \end{equation} where the maximum is taken over all nonzero vectors $x\in \mathbb C^n$ and $(y,x)$ denotes the standard inner product in $\mathbb C^n$, i.e. $(y,x)\equiv \sum_{i=1}^n \bar y_i x_i$. The spectral norm is well-known to be equal to the square root of maximum eigenvalue of $M^* M$.

Consider matrices $M_U=U^*HU+A$, where $U$ is an $n\times n$ unitary matrices, i.e. they all have the same fixed anti-Hermitian part and the same fixed Hermitian part's spectrum.

What is the maximum possible value which $||M_U||$ can attain upon varying $U$?

By the triangle inequality, a trivial bound is $||H||+|||A||$, but this is not in general achievable. Instead, I am looking for an expression for $\text{max}_U\,||M_U||$, for general $H$ and $A$ (so as to avoid going through the maximization over all possible unitary matrices each time).