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- W2904376834 abstract "We give formulas for the spectral radius of weighted endomorphisms $aalpha: C(X,D)to C(X,D)$, $ain C(X,D)$, 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 $r(aalpha)$: 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 $bin mathcal{B}(F)$ and that it's spectral radius is always a Lyapunov exponent in some direction $vin F$, when $F$ is reflexive." @default.
- W2904376834 created "2018-12-22" @default.
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- W2904376834 date "2018-12-11" @default.
- W2904376834 modified "2023-10-14" @default.
- W2904376834 title "Variational principles for spectral radius of weighted endomorphisms of $C(X,D)$" @default.
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- W2904376834 doi "https://doi.org/10.48550/arxiv.1812.04267" @default.
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