On the spectral radius of compact operator cocycles

TL;DR

This paper extends spectral radius concepts to Banach space cocycles, establishing a Berger-Wang formula for compact operator cocycles and demonstrating the continuity of spectral quantities.

Contribution

It introduces a generalized spectral radius framework for Banach space cocycles and proves a Berger-Wang type formula specific to compact operator cocycles.

Findings

Spectral radii depend continuously on the cocycle

A Berger-Wang formula is established for compact operator cocycles

Extension of spectral radius notions to Banach space cocycles

Abstract

We extend the notions of joint and generalized spectral radii to cocycles acting on Banach spaces and obtain a version of Berger-Wang's formula when restricted to the space of cocycles taking values in the space of compact operators. Moreover, we observe that the previous quantities depends continuously on the underlying cocycle.