  The Strichartz inequality for the system of orthonormal functions for the
Hermite operator $H=-\Delta+|x|^2$ on $\mathbb{R}^n$ has been proved in
\cite{lee}, using the classical Strichartz estimates for the free Schr\"odinger
propagator for orthonormal systems \cite{frank, frank1} and the link between
the Schr\"odinger kernel and the Mehler kernel associated with the Hermite
semigroup \cite{SjT}.
  In this article, we give an alternative proof of the above result in
connection with the restriction theorem with respect to the Hermite transform
with an optimal behavior of the constant in the limit of a large number of
functions. As an application, we show the well-posedness results in Schatten
spaces for the nonlinear Hermite-Hartree equation.
