# Variational principles for spectral radius of weighted endomorphisms of $C(X,D)$

## TL;DR

This paper develops variational formulas for the spectral radius of weighted endomorphisms on $C(X,D)$, extending to a broad class of operators and linking spectral radius to Lyapunov exponents in dynamical systems.

## Contribution

It introduces new variational principles for spectral radius involving cocycles over partial dynamical systems with values in Banach algebras, generalizing previous formulas.

## Key findings

- Formulas for spectral radius of weighted endomorphisms on $C(X,D)$.
- Variational principles involving Lyapunov exponents and linear extensions.
- Application to operators on Banach spaces, especially $B(F)$, with spectral radius as a Lyapunov exponent.

## Abstract

We give formulas for the spectral radius of weighted endomorphisms $a\alpha: C(X,D)\to C(X,D)$, $a\in C(X,D)$, where $X$ is a compact Hausdorff space and $D$ is a unital Banach algebra. Under the assumption that $\alpha$ generates a partial dynamical system $(X,\varphi)$, we establish two kinds of variational principles for $r(a\alpha)$: using linear extensions of $(X,\varphi)$ and using Lyapunov exponents associated with ergodic measures for $(X,\varphi)$. This requires considering (twisted) cocycles over $(X,\varphi)$ with values in an arbitrary Banach algebra $D$, and thus our analysis can not be reduced to any of mutliplicative ergodic theorems known so far. The established variational principles apply not only to weighted endomorphisms but also to a vast class of operators acting on Banach spaces that we call abstract weighted shifts associated with $\alpha: C(X,D)\to C(X,D)$. In particular, they are far reaching generalizations of formulas obtained by Kitover, Lebedev, Latushkin, Stepin and others. They are most efficient when $D=\mathcal{B}(F)$, for a Banach space $F$, and endomorphisms of $\mathcal{B}(F)$ induced by $\alpha$ are inner isometric. As a by product we obtain a dynamical variational principle for an arbitrary operator $b\in \mathcal{B}(F)$ and that it's spectral radius is always a Lyapunov exponent in some direction $v\in F$, when $F$ is reflexive.