Weakly wandering vectors and weakly independent partitions

Krengel, Ulrich

doi:10.1090/S0002-9947-1972-0290168-X

Weakly wandering vectors and weakly independent partitions
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Trans. Amer. Math. Soc. 164 (1972), 199-226 Request permission

Abstract:

We first characterize continuous spectrum and purely discrete spectrum of an isometry U of a Hilbert space geometrically by the existence of a spanning system, resp. by the absence, of vectors with infinitely many orthogonal images under powers of U. We then characterize weak mixing and discrete spectrum of an invertible measure preserving transformation of a probability space in terms of the null sets of the space. Finally for two-fold weakly mixing transformations the result on isometries is strengthened by proving the density of the set of partitions with infinitely many mutually independent images in the set of all finite partitions.

References

Yael Naim Dowker, On measurable transformations in finite measure spaces, Ann. of Math. (2) 62 (1955), 504–516. MR 72929, DOI 10.2307/1970076
Arshag B. Hajian and Shizuo Kakutani, Weakly wandering sets and invariant measures, Trans. Amer. Math. Soc. 110 (1964), 136–151. MR 154961, DOI 10.1090/S0002-9947-1964-0154961-1
Paul R. Halmos, Shifts on Hilbert spaces, J. Reine Angew. Math. 208 (1961), 102–112. MR 152896, DOI 10.1515/crll.1961.208.102
A. Hanen and J. Neveu, Atomes conditionnels d’un espace de probabilité, Acta Math. Acad. Sci. Hungar. 17 (1966), 443–449 (French). MR 205285, DOI 10.1007/BF01894889

Über Potenzreihen mit positivem, reellem Teil im Einheitskreis

63

Edwin Hewitt and Karl Stromberg, Real and abstract analysis. A modern treatment of the theory of functions of a real variable, Springer-Verlag, New York, 1965. MR 0188387

Ergodentheorie

10

Konrad Jacobs, Neuere Methoden und Ergebnisse der Ergodentheorie, Ergebnisse der Mathematik und ihrer Grenzgebiete, (N.F.), Heft 29, Springer-Verlag, Berlin-Göttingen-Heidelberg, 1960 (German). MR 0123676, DOI 10.1007/978-3-642-94777-3
S. Kakutani and W. Parry, Infinite measure preserving transformations with “mixing”, Bull. Amer. Math. Soc. 69 (1963), 752–756. MR 153815, DOI 10.1090/S0002-9904-1963-11022-8
U. Krengel and L. Sucheston, On mixing in infinite measure spaces, Z. Wahrscheinlichkeitstheorie und Verw. Gebiete 13 (1969), 150–164. MR 254215, DOI 10.1007/BF00537021
K. Krickeberg, Recent results on mixing in topological measure spaces, Probability and Information Theory (Proc. Internat. Sympos., McMaster Univ., Hamilton, Ont., 1968) Springer, Berlin, 1969, pp. 178–185. MR 0269809
A. G. Kušnirenko, Metric invariants of entropy type, Uspehi Mat. Nauk 22 (1967), no. 5 (137), 57–65 (Russian). MR 0217257
V. P. Leonov, The use of the characteristic functional and semi-invariants in the ergodic theory of stationary processes, Soviet Math. Dokl. 1 (1960), 878–881. MR 0125638

Théorie ergodique

Atomes conditionnels d’espaces de probabilité et théorie de l’information

36

A. Rényi, On mixing sequences of sets, Acta Math. Acad. Sci. Hungar. 9 (1958), 215–228. MR 98161, DOI 10.1007/BF02023873
V. A. Rohlin, On the fundamental ideas of measure theory, Mat. Sbornik N.S. 25(67) (1949), 107–150 (Russian). MR 0030584
L. Sucheston, On mixing and the zero-one law, J. Math. Anal. Appl. 6 (1963), 447–456. MR 149524, DOI 10.1016/0022-247X(63)90024-6
Louis Sucheston, On existence of finite invariant measures, Math. Z. 86 (1964), 327–336. MR 190293, DOI 10.1007/BF01110407
László Zsidó, Conditional expectation and the theorem of Sinaĭ, Stud. Cerc. Mat. 20 (1968), 1045–1111 (Romanian, with English summary). MR 274723
James W. England and N. F. G. Martin, On weak mixing metric automorphisms, Bull. Amer. Math. Soc. 74 (1968), 505–507. MR 222251, DOI 10.1090/S0002-9904-1968-11982-2

A short proof of Sucheston’s characterization of mixing