2018/12/11 by B. K. Kwaśniewski, Kwasniewski, B. K., А. В. Лебедев +1 · 1 citation
Mathematics · #37A99 (Primary) 37H15 #47A10 (Secondary) #47B48 #Dynamical Systems (math.DS) #FOS: Mathematics #Functional Analysis (math.FA) #Mathematical Analysis and Transform Methods #Mathematical Dynamics and Fractals #Operator Algebras (math.OA) #advanced mathematical theories
paper · pdf · doi:10.48550/arxiv.1812.04267
openalex publication_date 2018/12/11 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
We give formulas for the spectral radius of weighted endomorphisms aα: C(X,D)→ C(X,D), a∈ C(X,D), where X is a compact Hausdorff space and D is a unital Banach algebra. Under the assumption that α generates a partial dynamical system (X,φ), we establish two kinds of variational principles for r(aα): using linear extensions of (X,φ) and using Lyapunov exponents associated with ergodic measures for (X,φ). This requires considering (twisted) cocycles over (X,φ) 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 α: C(X,D)→ 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=B(F), for a Banach space F, and endomorphisms of B(F) induced by α are inner isometric. As a by product we obtain a dynamical variational principle for an arbitrary operator b∈ B(F) and that it's spectral radius is always a Lyapunov exponent in some direction v∈ F, when F is reflexive.