Dominated splittings for semi-invertible operator cocycles on Hilbert space

TL;DR

This paper extends the singular value criterion for dominated splittings from invertible to semi-invertible operator cocycles on Hilbert spaces, unifying and generalizing previous results in the field.

Contribution

It introduces a modified singular value criterion that characterizes dominated splittings for semi-invertible cocycles on Hilbert spaces, broadening the scope of prior theorems.

Findings

Modified criterion equivalent to dominated splitting

Extends to semi-invertible cocycles of bounded operators

Generalizes previous results by Yoccoz, Bochi, Gourmelon

Abstract

A theorem of J. Bochi and N. Gourmelon states that an invertible linear cocycle admits a dominated splitting if and only if the singular values of its iterates become separated at a uniform exponential rate. It is not difficult to show that for cocycles of non-invertible linear maps over an invertible dynamical system -- which we refer to as semi-invertible cocycles -- this criterion fails to imply the existence of a dominated splitting. In this article we show that a simple modification of Bochi and Gourmelon's singular value criterion is equivalent to the existence of a dominated splitting in both the invertible and the semi-invertible cases. This result extends to the more general context of semi-invertible cocycles of bounded linear operators acting on a Hilbert space, and generalises previous results due to J.-C. Yoccoz, J. Bochi and N. Gourmelon, and the present author.