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Transactions of the American Mathematical Society

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


Authors: Bartosz Kosma Kwaśniewski and Andrei Lebedev
Journal: Trans. Amer. Math. Soc. 373 (2020), 2659-2698
MSC (2010): Primary 47B48, 37A99; Secondary 37H15, 47A10
DOI: https://doi.org/10.1090/tran/7993
Published electronically: January 7, 2020
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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.


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Additional Information

Bartosz Kosma Kwaśniewski
Affiliation: Faculty of Mathematics, University of Białystok, K. Ciołkowskiego 1M, 15-245 Białystok, Poland
Email: bartoszk@math.uwb.edu.pl

Andrei Lebedev
Affiliation: Belorussian State University, Nesavisimosti av., 4, Minsk, Belarus; and Faculty of Mathematics, University of Białystok, K. Ciołkowskiego 1M, 15-245 Białystok, Poland
Email: lebedev@bsu.by

DOI: https://doi.org/10.1090/tran/7993
Keywords: Variational principle, spectral radius, endomorphism, Lyapunov exponent, cocycle
Received by editor(s): March 18, 2019
Received by editor(s) in revised form: August 21, 2019
Published electronically: January 7, 2020
Additional Notes: The research leading to these results has received funding from the European Union’s Seventh Framework Programme (FP7/2007-2013) under grant agreement number 621724, as well as Polish National Science Centre grant number DEC-2011/01/D/ST1/04112
Article copyright: © Copyright 2019 American Mathematical Society