The main objective of this paper is to extend certain fundamental inequalities from a single function to a family of orthonormal systems. In the first part of the paper, we consider a non-negative, self-adjoint operator L on \(L^2(X,\mu )\) , where \((X,\mu )\) is a measure space. Under the assumption that the kernel \(K_{it}(x,y)\) of the Schrödinger propagator \(e^{itL}\) satisfies a uniform \(L^\infty \) -decay estimate of the form \(\begin{aligned} \sup _{x,y\in X}|K_{it}(x,y)|\lesssim |t|^{-\frac{n}{2}},\,|t|<T_0, {\text { for some }}n\ge 1, \end{aligned}\) where \(T_0\in (0,+\infty ]\) , we establish Strichartz estimates for the Schrödinger propagator \(e^{itL}\) and using a duality principle argument by Frank-Sabin [11], we extend it for a system of infinitely many fermions on \(L^2(X)\) . We also obtain orthonormal Strichartz estimates for a class of dispersive semigroup \(U(t)=e^{it\phi (L)}\psi (\sqrt{L}),\) where \(\phi : \mathbb {R}^+\rightarrow \mathbb {R}\) is a smooth function and \(\psi \in C_c^\infty ([\frac{1}{2},2])\) . As an application of these orthonormal versions of Strichartz estimates, we prove the well-posedness for the Hartree equation in the Schatten spaces. In the next part of the paper, we obtain some new orthonormal Strichartz estimates, which extend prior work of Kenig-Ponce-Vega [22] for single functions. Using those orthonormal versions of Kenig-Ponce-Vega result, we prove the orthonormal restriction theorem for the Fourier transform on some particular noncompact hypersurface of the form \(S=\{(\xi , \phi (\xi )): \xi \in \mathbb {R}\}\) , where \(\phi \) satisfies certain growth condition.
