Dominated splittings and periodic data for quasi-compact operator cocycles

For infinite-dimensional quasi-compact cocycles over a map satisfying a certain closing condition, we show that periodic orbits carry enough information to guarantee the existence of a dominated splitting. More precisely, we establish that if the moduli of the $(k+1)$-largest eigenvalues of the cocycle are $e^{{\lambda }_{1}n}\ge e^{{\lambda }_{2}n}\ge \dots \ge e^{{\lambda }_{k}n}\ge e^{{\lambda }_{k+1}n}$ at every periodic point of period $n$, and ${\lambda }_k>{\lambda }_{k+1}$, then the cocycle admits a dominated splitting of index $k$. As a consequence, if ${\lambda }_k>0>{\lambda }_{k+1}$ then the cocycle is uniformly hyperbolic. Furthermore, we are able to obtain these same conclusions even when the eigenvalues are only close to constant, not strictly constant.