Suppose $H$ be a Hilbert space and $U,V\in B(H)$ bounded linear operators. If operator $V$ is closed range and injective, under which conditions operator $UV$ is closed range too?
The range of $V$ is a Banach space since it is a closed subspace of $H$. So you only need to answer, when is the range of a bounded operator between two Banach spaces closed.
That was already answered here When is the image of a linear operator closed?