On the entangled ergodic theorem
Annali della Scuola Normale Superiore di Pisa - Classe di Scienze, Série 5, Tome 12 (2013) no. 1, pp. 141-156.

We study the convergence of the so-called entangled ergodic averages

1 N k n 1 ,...,n k =1 N T m n α(m) A m-1 T m-1 n α(m-1) A m-2 ...A 1 T 1 n α(1) ,

where km and α:1,...,m1,...,k is a surjective map. We show that, on general Banach spaces and without any restriction on the partition α, the above averages converge strongly as N under some quite weak compactness assumptions on the operators T j and A j . A formula for the limit based on the spectral analysis of the operators T j and the continuous version of the result are presented as well.

Publié le :
Classification : 47A35, 37A30
Eisner, Tanja 1 ; Kunszenti-Kovács, Dávid 2

1 KdV Institute for Mathematics University of Amsterdam P.O. Box 94248 1090 GE, Amsterdam, The Netherlands
2 Institute of Mathematics University of Tübingen Auf der Morgenstelle 10 72076 Tübingen, Germany
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Eisner, Tanja; Kunszenti-Kovács, Dávid. On the entangled ergodic theorem. Annali della Scuola Normale Superiore di Pisa - Classe di Scienze, Série 5, Tome 12 (2013) no. 1, pp. 141-156. http://archive.numdam.org/item/ASNSP_2013_5_12_1_141_0/

[1] L. Accardi, Yu. Hashimoto and N. Obata, Notions of independence related to the free group, Infin. Dimens. Anal. Quantum Probab. Relat. Top. 1 (1998), 201–220. | MR | Zbl

[2] T. Austin, T. Eisner and T. Tao, Nonconventional ergodic averages and multiple recurrence for von Neumann dynamical systems, Pacific J. Math. 250 (2011), 1–60. | MR | Zbl

[3] D. Berend, M. Lin, J. Rosenblatt and A. Tempelman, Modulated and subsequential ergodic theorems in Hilbert and Banach spaces, Ergodic Theory Dynam. Systems 22 (2002), 1653–1665. | MR | Zbl

[4] C. Beyers, R. Duvenhage and A. Ströh, The Szemerédi property in ergodic W * -dynamical systems, J. Operator Theory 64 (2010), 35–67. | MR | Zbl

[5] N. Dunford and J. T. Schwartz, “Linear Operators” I., Interscience Publishers, Inc., New York; Interscience Publishers, Ltd., London, 1958. | MR | Zbl

[6] R. Duvenhage, Bergelson’s theorem for weakly mixing C * -dynamical systems, Studia Math. 192 (2009), 235–257. | EuDML | MR | Zbl

[7] T. Eisner, “Stability of Operators and Operator Semigroups”, Operator Theory: Advances and Applications, 209, Birkhäuser Verlag, Basel, 2010. | MR | Zbl

[8] K.-J. Engel and R. Nagel, “One-parameter Semigroups for Linear Evolution Equations”, Graduate Texts in Mathematics, Vol. 194, Springer-Verlag, New York, 2000. | MR | Zbl

[9] F. Fidaleo, On the entangled ergodic theorem, Infin. Dimens. Anal. Quantum Probab. Relat. Top. 10 (2007), 67–77. | MR | Zbl

[10] F. Fidaleo, An ergodic theorem for quantum diagonal measures, Infin. Dimens. Anal. Quantum Probab. Relat. Top. 12 (2009), 307–320. | MR | Zbl

[11] F. Fidaleo, The entangled ergodic theorem in the almost periodic case, Linear Algebra Appl. 432 (2010), 526–535. | MR | Zbl

[12] R. V. Kadison and J. R. Ringrose, “Fundamentals of the Theory of Operator Algebras”, Vol. I. Academic Press, 1983. | MR | Zbl

[13] B. Kra, Ergodic methods in additive combinatorics, In: “Additive combinatorics”, CRM Proc. Lecture Notes, Vol. 43, Amer. Math. Soc., Providence, RI, 2007, 103–143. | MR | Zbl

[14] U. Krengel, “Ergodic Theorems”, de Gruyter Studies in Mathematics, de Gruyter, Berlin, 1985. | MR | Zbl

[15] V. Liebscher, Note on entangled ergodic theorems, Infin. Dimens. Anal. Quantum Probab. Relat. Top. 2 (1999), 301–304. | MR | Zbl

[16] C. P. Niculescu, A. Ströh and L. Zsidó, Noncommutative extensions of classical and multiple recurrence theorems, J. Operator Theory 50 (2003), 3–52. | MR | Zbl

[17] K. Petersen, “Ergodic Theory”, Cambridge Studies in Advanced Mathematics, Cambridge University Press, 1983. | MR | Zbl

[18] H. H. Schaefer, “Banach Lattices and Positive Operators”, Springer-Verlag, 1974. | MR | Zbl

[19] M. Takesaki, “Theory of Operator Algebras I”, Springer-Verlag, 1979. | MR | Zbl