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Transactions of the American Mathematical Society

Published by the American Mathematical Society since 1900, Transactions of the American Mathematical Society is devoted to longer research articles in all areas of pure and applied mathematics.

ISSN 1088-6850 (online) ISSN 0002-9947 (print)

The 2020 MCQ for Transactions of the American Mathematical Society is 1.48 .

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Variational principles for spectral radius of weighted endomorphisms of $C(X,D)$
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by Bartosz Kosma Kwaśniewski and Andrei Lebedev PDF
Trans. Amer. Math. Soc. 373 (2020), 2659-2698 Request permission

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
  • ORCID: 0000-0002-5173-0519
  • 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
  • MR Author ID: 194196
  • Email: lebedev@bsu.by
  • 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
  • © Copyright 2019 American Mathematical Society
  • 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
  • MathSciNet review: 4069230