On the spectrum of infinite dimensional random products of compact operators

TL;DR

This paper investigates the spectral properties of infinite-dimensional random products of compact operators, showing that generically such cocycles exhibit either dominated or trivial spectra along typical orbits.

Contribution

It establishes a residual subset of cocycles in an infinite-dimensional setting where the Oseledets-Ruelle decomposition is either dominated or trivial for almost every point.

Findings

Residual subset of cocycles with specific spectral properties

Almost every orbit exhibits either dominated or trivial spectrum

Results extend spectral theory to infinite-dimensional compact operator cocycles

Abstract

We consider an infinite dimensional separable Hilbert space and its family of compact integrable cocycles over a dynamical system f. Assuming that f acts in a compact Hausdorff space X and preserves a Borel regular ergodic measure which is positive on non-empty open sets, we conclude that there is a residual subset of cocycles within which, for almost every x, either the Oseledets-Ruelle's decomposition along the orbit of x is dominated or has a trivial spectrum.