Weak almost periodicity of L 1 contractions and coboundaries of non-singular transformations

Isaac Kornfeld; Michael Lin

Studia Mathematica (2000)

  • Volume: 138, Issue: 3, page 225-240
  • ISSN: 0039-3223

Abstract

top
It is well known that a weakly almost periodic operator T in a Banach space is mean ergodic, and in the complex case, also λT is mean ergodic for every |λ|=1. We prove that a positive contraction on L 1 is weakly almost periodic if (and only if) it is mean ergodic. An example shows that without positivity the result is false. In order to construct a contraction T on a complex L 1 such that λT is mean ergodic whenever |λ|=1, but T is not weakly almost periodic, we prove the following: Let τ be an invertible weakly mixing non-singular transformation of a separable atomless probability space. Then there exists a complex function φ ∈ L ∞ with |φ(x)|=1 a.e. such that for every λ ∈ℂ with |λ|=1 the function ⨍ ≡ 0 is the only solution of the equation ⨍(τx)=λφ(x)⨍(x). Moreover, the set of such functions φ is residual in the set of all complex unimodular measurable functions (with the L 1 topology)

How to cite

top

Kornfeld, Isaac, and Lin, Michael. "Weak almost periodicity of $L_1$ contractions and coboundaries of non-singular transformations." Studia Mathematica 138.3 (2000): 225-240. <http://eudml.org/doc/216701>.

@article{Kornfeld2000,
abstract = {It is well known that a weakly almost periodic operator T in a Banach space is mean ergodic, and in the complex case, also λT is mean ergodic for every |λ|=1. We prove that a positive contraction on $L_1$ is weakly almost periodic if (and only if) it is mean ergodic. An example shows that without positivity the result is false. In order to construct a contraction T on a complex $L_1$ such that λT is mean ergodic whenever |λ|=1, but T is not weakly almost periodic, we prove the following: Let τ be an invertible weakly mixing non-singular transformation of a separable atomless probability space. Then there exists a complex function $φ ∈ L_∞$ with |φ(x)|=1 a.e. such that for every λ ∈ℂ with |λ|=1 the function ⨍ ≡ 0 is the only solution of the equation ⨍(τx)=λφ(x)⨍(x). Moreover, the set of such functions φ is residual in the set of all complex unimodular measurable functions (with the $L_1$ topology)},
author = {Kornfeld, Isaac, Lin, Michael},
journal = {Studia Mathematica},
keywords = {positive contraction; weakly almost periodic; mean ergodic; multiplicative coboundaries; weakly mixing transformation},
language = {eng},
number = {3},
pages = {225-240},
title = {Weak almost periodicity of $L_1$ contractions and coboundaries of non-singular transformations},
url = {http://eudml.org/doc/216701},
volume = {138},
year = {2000},
}

TY - JOUR
AU - Kornfeld, Isaac
AU - Lin, Michael
TI - Weak almost periodicity of $L_1$ contractions and coboundaries of non-singular transformations
JO - Studia Mathematica
PY - 2000
VL - 138
IS - 3
SP - 225
EP - 240
AB - It is well known that a weakly almost periodic operator T in a Banach space is mean ergodic, and in the complex case, also λT is mean ergodic for every |λ|=1. We prove that a positive contraction on $L_1$ is weakly almost periodic if (and only if) it is mean ergodic. An example shows that without positivity the result is false. In order to construct a contraction T on a complex $L_1$ such that λT is mean ergodic whenever |λ|=1, but T is not weakly almost periodic, we prove the following: Let τ be an invertible weakly mixing non-singular transformation of a separable atomless probability space. Then there exists a complex function $φ ∈ L_∞$ with |φ(x)|=1 a.e. such that for every λ ∈ℂ with |λ|=1 the function ⨍ ≡ 0 is the only solution of the equation ⨍(τx)=λφ(x)⨍(x). Moreover, the set of such functions φ is residual in the set of all complex unimodular measurable functions (with the $L_1$ topology)
LA - eng
KW - positive contraction; weakly almost periodic; mean ergodic; multiplicative coboundaries; weakly mixing transformation
UR - http://eudml.org/doc/216701
ER -

References

top
  1. [ÇL] D. Çömez and M. Lin, Mean ergodicity of L 1 contractions and pointwise ergodic theorems, in: Almost Everywhere Convergence II, A. Bellow and R. L. Jones (eds.), Academic Press, Boston, 1991, 113-126. Zbl0759.47004
  2. [ÇLO] D. Çömez, M. Lin, and J. Olsen, Weighted ergodic theorems for mean ergodic L 1 contractions, Trans. Amer. Math. Soc. 350 (1998), 101-117. Zbl0888.47006
  3. [DS] N. Dunford and J. Schwartz, Linear Operators, part I, Interscience, New York, 1958. 
  4. [E] R. Emilion, Mean bounded operators and mean ergodic theorems, J. Funct. Anal. 61 (1985), 1-14. Zbl0562.47007
  5. [HOkOs] T. Hamachi, Y. Oka, and M. Osikawa, A classification of ergodic non-singular transformation groups, Mem. Fac. Sci. Kyushu Univ. Ser. A 28 (1974), 113-133. Zbl0293.28011
  6. [He-1] H. Helson, Note on additive cocycles, J. London Math. Soc. 31 (1985), 473-477. 
  7. [He-2] H. Helson, The Spectral Theorem, Lecture Notes in Math. 1227, Springer, Berlin, 1986. Zbl0615.47021
  8. [Hi] E. Hille, Remarks on ergodic theorems, Trans. Amer. Math. Soc. 57 (1945), 246-269. Zbl0063.02017
  9. [IY] Y. Ito and M. Yoshida, Cocycles for non-singular transformations, Comment. Math. Univ. St. Paul. 44 (1995), 93-103. Zbl0842.28007
  10. [JP] R. Jones and W. Parry, Compact abelian group extensions of dynamical systems, Compositio Math. 25 (1972), 135-147. Zbl0243.54039
  11. [KP] S. Kakutani and W. Parry, Infinite measure preserving transformation with "mixing", Bull. Amer. Math. Soc. 69 (1963), 752-756. Zbl0126.31801
  12. [Ke] J. L. Kelly, General Topology, Van Nostrand, Princeton, 1955. 
  13. [Kr] U. Krengel, Ergodic Theorems, de Gruyter Stud. Math., de Gruyter, Berlin, 1985. 
  14. [MSc] C. Moore and K. Schmidt, Coboundaries and homomorphisms for non-singular actions and a problem of H. Helson, Proc. London Math. Soc. 40 (1980), 443-475. Zbl0428.28014
  15. [Sc] K. Schmidt, Cocycles on Ergodic Transformation Groups, Macmillan of India, 1977. 
  16. [W] B. Weiss, Orbit equivalence of non-singular actions, Enseign. Math. 29 (1981), 77-107. 
  17. [Y] K. Yosida, Functional Analysis, 3rd ed., Springer, Berlin, 1971. Zbl0217.16001

NotesEmbed ?

top

You must be logged in to post comments.

To embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.

Only the controls for the widget will be shown in your chosen language. Notes will be shown in their authored language.

Tells the widget how many notes to show per page. You can cycle through additional notes using the next and previous controls.

    
                

                
Note: Best practice suggests putting the JavaScript code just before the closing </body> tag.