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

Question

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!

Answer 1

$(HU)^* = U^* H^* = U^{-1} H$, 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 $\sin(t) \ne 0$.