Generic properties of the Lyapunov spectrum of compact operator cocycles on Hilbert spaces

This paper is devoted to studying the stability of Lyapunov exponents and the simplicity of Lyapunov spectrum for compact operator cocycles on a separable infinite-dimensional Hilbert space from a generic point of view generated by the essential supremum norm. Firstly, we show the density of both the set of compact operator cocycles having a finite number of Lyapunov exponents and the set of compact operator cocycles having a countably infinite number of Lyapunov exponents. Meanwhile, the set of compact operator cocycles having no Lyapunov exponent is nowhere dense. Finally, for any k ∈ N , we show that the set of compact operator cocycles satisfying the condition that the Lyapunov spectra corresponding to their first k Lyapunov exponents are simple and continuous contains an open and dense set.