Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)

 

 

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
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. Ana I. Alonso, Jialin Hong, and Rafael Obaya, Absolutely continuous dynamics and real coboundary cocycles in 𝐿^{𝑝}-spaces, 0<𝑝<∞, Studia Math. 138 (2000), no. 2, 121–134. MR 1749076
  • 2. I. Assani, A note on the equation 𝑌=(𝐼-𝑇)𝑋 in 𝐿¹, Proceedings of the Conference on Probability, Ergodic Theory, and Analysis (Evanston, IL, 1997), 1999, pp. 540–541. MR 1700608
  • 3. Michael Lin and Robert Sine, Ergodic theory and the functional equation (𝐼-𝑇)𝑥=𝑦, J. Operator Theory 10 (1983), no. 1, 153–166. MR 715565

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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: http://dx.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