Consider two skew-adjoint matrices $A$ and $A'$, i.e. $A^\*=-A$ and $A'^\*=-A'$. It is well-known that 


$e^{-tA}$ and $e^{-tA'}$ are unitary operators.


I would like to know:


Is it true that $\sup\_{t \in \mathbb{R}} \Vert e^{-tA}-e^{-tA'} \Vert = 2(1-\delta\_{A,A'})?$



Distance of parametrized skew hermitian exponentials
This is not true in general. E.g., let 
$$A:=\left(
\begin{array}{cc}
 -i & 0 \\
 0 & 0 \\
\end{array}
\right),\quad A':=\frac i2\,
\left(
\begin{array}{cc}
 -1 & 1 \\
 1 & -1 \\
\end{array}
\right).$$
Then, for real $t$,
$$e^{-tA}=\left(
\begin{array}{cc}
 e^{i t} & 0 \\
 0 & 1 \\
\end{array}
\right),\quad
e^{-tA'}=\frac12\,\left(
\begin{array}{cc}
 1+e^{i t} & 1-e^{i t} \\
 1-e^{i t} & 1+e^{i t} \\
\end{array}
\right),$$
and 
$$\|e^{-tA}-e^{-tA'}\|=\sqrt{1-\cos t}, $$
so that 
$$\sup\_{t\in\mathbb R}\|e^{-tA}-e^{-tA'}\|=\sqrt2\ne2(1-\delta\_{A,A'}).$$


