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

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 cannot be reduced to any of the multiplicative 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 its spectral radius is always a Lyapunov exponent in some direction $v\in F$ when $F$ is reflexive.