2014/03/04 by Ian D. Morris, Morris, Ian D.
Mathematics · #Advanced Operator Algebra Research #Dynamical Systems (math.DS) #FOS: Mathematics #Functional Analysis (math.FA) #Holomorphic and Operator Theory #Spectral Theory in Mathematical Physics #math.DS #math.FA
paper · pdf · doi:10.48550/arxiv.1403.0824
This paper has been withdrawn by the author due to a critical error: step 8 is written as if the image of the operator P(x) were \mathcal{U}(x), but it is actually \mathcal{W}(x). This error invalidates the proof of the main theorem and the entire article should be treated as incorrect
openalex publication_date 2014/03/04 · arxiv created 2015/12/21 · arxiv updated 2015/12/22 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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.