Abstract
Notions of weak and uniformly weak mixing (to zero) are defined for bounded sequences in arbitrary Banach spaces. Uniformly weak mixing for vector sequences is characterized by mean ergodic convergence properties. This characterization turns out to be useful in the study of multiple recurrence, where mixing properties of vector sequences, which are not orbits of linear operators, are investigated. For bounded sequences, which satisfy a certain domination condition, it is shown that weak mixing to zero is equivalent with uniformly weak mixing to zero.
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References
S. Banach and S. Saks, Sur la convergence forte dans les champs L p, Studia Math. 2 (1930), 51–57.
D. Berend and V. Bergelson, Mixing sequences in Hilbert spaces, Proc. Amer.Math. Soc. 98 (1986), 239–246.
J. Blum and D. Hanson, On the mean ergodic theorem for subsequences, Bull. Amer. Math. Soc. 66 (1960), 308–311.
J. Diestel, Sequences and Series in Banach Spaces, Graduate texts in mathematics 92, Springer-Verlag, 1984.
K. Floret, Weakly Compact Sets, Lecture Notes in Mathematics 801, Springer-Verlag, 1980.
H. Furstenberg, Ergodic behaviour of diagonal measures and a theorem of Szemerédi on arithmetic progressions, J. d’Analyse Math. 31 (1977), 204–256.
H. Furstenberg, Recurrence in Ergodic Theory and Combinatorial Number Theory, Princeton Univ. Press, New Jersey, 1981.
L.K. Jones, A mean ergodic theorem for weakly mixing operators, Adv. Math. 7 (1971), 211–216.
L.K. Jones, A generalization of the Mean Ergodic Theorem in Banach spaces, Z. Wahrscheinlichkeitstheorie verw. Geb. 27 (1973), 105–107.
L.K. Jones and M. Lin, Ergodic theorems of weak mixing type, Proc. Amer. Math. Soc. 57 (1976), 50–52.
S. Kakutani, Weak convergence in uniformly convex spaces, Tôhoku Math. J. 45 (1938), 188–193.
U. Krengel, Ergodic Theorems, Walter de Gruyter, Berlin-New York, 1985.
C. Niculescu, A. Ströh and L. Zsidó, Noncommutative extensions of classical and multiple recurrence theorems, J. Operator Theory 50 (2003), 3–52.
E. Szemerédi, On sets of integers containing no k elements in arithmetic progression, Acta Arith. 27 (1975), 199–245.
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Dedicated to the memory of our colleague Gert K. Pedersen
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© 2006 Birkhäuser Verlag Basel/Switzerland
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Zsidó, L. (2006). Weak Mixing Properties of Vector Sequences. In: Dritschel, M.A. (eds) The Extended Field of Operator Theory. Operator Theory: Advances and Applications, vol 171. Birkhäuser Basel. https://doi.org/10.1007/978-3-7643-7980-3_17
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DOI: https://doi.org/10.1007/978-3-7643-7980-3_17
Publisher Name: Birkhäuser Basel
Print ISBN: 978-3-7643-7979-7
Online ISBN: 978-3-7643-7980-3
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