Explicit bounds for separation between Oseledets subspaces

TL;DR

This paper establishes explicit bounds for the separation between Oseledets subspaces in Banach spaces, extending previous finite-dimensional and injective cases to more general non-injective operators without requiring a dynamical system or compactness.

Contribution

It provides explicit angle bounds between invariant subspaces for non-injective operators in Banach spaces, broadening the scope of previous results.

Findings

Explicit lower bounds on the angle between subspaces.

Extension of results to non-injective operators.

Quantitative estimates of subspace angles in Banach spaces.

Abstract

We consider a two-sided sequence of bounded operators in a Banach space which are not necessarily injective and satisfy two properties (SVG) and (FI). The singular value gap (SVG) property says that two successive singular values of the cocycle at some index $d$ admit a uniform exponential gap; the fast invertibility (FI) property says that the cocycle is uniformly invertible on the fastest $d$-dimensional direction. We prove the existence of a uniform equivariant splitting of the Banach space into a fast space of dimension $d$ and a slow space of co-dimension $d$. We compute an explicit constant lower bound on the angle between these two spaces using solely the constants defining the properties (SVG) and (FI). We extend the results obtained in the finite-dimensional case for bijective operators and the results obtained by Blumenthal and Morris in the infinite-dimensional case for injective norm-continuous cocycles, in the direction that the operators are not required to be globally injective, that no dynamical system is involved, and no compactness of the underlying system or smoothness of the cocycle is required. Moreover, we give quantitative estimates of the angle between the fast and slow spaces that are new even in the case of finite-dimensional bijective operators in Hilbert spaces.