Consider two skew-adjoint matrices / operators (possibly unbounded) $A$ 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 $\sup_{t \in \mathbb{R}} \Vert e^{-tA}-e^{-tA'} \Vert = 2(1-\delta_{A,A'})?$

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 $\sup_{t \in \mathbb{R}} \Vert e^{-tA}-e^{-tA'} \Vert = 2(1-\delta_{A,A'})?$