The generalized Berger-Wang formula and the spectral radius of linear cocycles

TL;DR

This paper extends the Berger-Wang formula to Banach space operator sets, linking joint spectral radii and spectral properties of linear cocycles, and answers an open question on spectral radius limits.

Contribution

It generalizes the Berger-Wang formula to a broader setting and provides new insights into the spectral behavior of linear cocycles.

Findings

Derived new formulas relating joint spectral radii in Banach spaces

Reproved a known formula using ergodic theory methods

Answered an open question on spectral radius limits of matrix cocycles

Abstract

Using multiplicative ergodic theory we prove two formulae describing the relationships between different joint spectral radii for sets of bounded linear operators acting on a Banach space. In particular we recover a formula previously proved by V. S. Shulman and Yu. V. Turovski\u{\i} using operator-theoretic ideas. As a byproduct of our method we answer a question of J. E. Cohen on the limiting behaviour of the spectral radius of a measurable matrix cocycle.