# When is the product of a hermitian unitary and another unitary hermitian?

I have a Hermitian unitary Ĥ and I want to know, if Û is some other unitary, when is Ĥ Û a Hermitian unitary? Specifically, what are the conditions on Û such that $(\hat{HU})^{\dagger}=\hat{HU}$?

I know that if Û is Hermitian then as long as they commute this is true. Also that if they commute, as long as Û is Hermitian this is also true.

So then does it have to be the case that they commute AND Û is Hermitian?

Thanks a lot for the help in advance!

$(HU)^* = U^* H^* = U^{-1} H$, so what you need is $HU = U^{-1} H$, i.e. $U H U = H$. $H$ and $U$ do not need to commute. For example, try the matrices
$$H = \pmatrix{1 & 0\cr 0 & -1\cr}, \ U = \pmatrix{\cos(t) & -\sin(t)\cr \sin(t) & \cos(t)\cr}$$ where $\sin(t) \ne 0$.