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A remark on real coboundary cocycles in $L^{\infty}$-space


Author: Ryotaro Sato
Journal: Proc. Amer. Math. Soc. 131 (2003), 231-233
MSC (2000): Primary 37A20, 28D05, 47A35
DOI: https://doi.org/10.1090/S0002-9939-02-06756-4
Published electronically: June 27, 2002
MathSciNet review: 1929042
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Abstract | References | Similar Articles | Additional Information

Abstract: Let $T$ be an ergodic automorphism of a probability measure space $(\Omega, {\mathcal A},m)$ and let $f$ be a real-valued measurable function on $\Omega$. We deduce a necessary and sufficient condition for the existence of $L^{\infty}$-solutions of the cohomology equation $f=h\circ T-h$, by using the recent result of Alonso, Hong and Obaya.


References [Enhancements On Off] (What's this?)

  • 1. A. I. Alonso, J. Hong and R. Obaya, Absolutely continuous dynamics and real coboundary cocycles in $L^{p}$-spaces, $0<p<\infty$, Studia Math. 138 (2000), 121-134. MR 2001i:37010
  • 2. I. Assani, A note on the equation $Y=(I-T)X$ in $L^{1}$, Illinois J. Math. 43 (1999), 540-541. MR 2000j:47016
  • 3. M. Lin and R. Sine, Ergodic theory and the functional equation $(I-T)x=y$, J. Operator Theory 10 (1983), 153-166. MR 84m:47015

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Additional Information

Ryotaro Sato
Affiliation: Department of Mathematics, Okayama University, Okayama, 700-8530 Japan
Email: satoryot@math.okayama-u.ac.jp

DOI: https://doi.org/10.1090/S0002-9939-02-06756-4
Keywords: Ergodic automorphism, additive real cocycle, cohomology equation, coboundary
Received by editor(s): August 31, 2001
Published electronically: June 27, 2002
Communicated by: Michael Handel
Article copyright: © Copyright 2002 American Mathematical Society

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