Orthonormal Strichartz inequalities and their applications on abstract measure spaces

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 Schr\"{o}dinger propagator $e^{itL}$ satisfies a uniform $L^\infty$-decay estimate of the form \begin{equation*} \sup_{x,y\in X}|K_{it}(x,y)|\lesssim |t|^{-\frac{n}{2}},\,|t|