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On Products of Conditional Expectation Operators

Published online by Cambridge University Press:  20 November 2018

Radu Zaharopol
Radu Zaharopol*
Affiliation:
School of Mathematics The Institute for Advanced Study Princeton, NJ 08540
*
Current address: SUNY at Binghampton Binghampton, NY., 13901
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Abstract

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Let (X, Σ, μ) be a probability space, let f1, f2, ..., Fk be k σ-subalgebras of Σ, and let p ∊ R be such that 1 < p < + ∞. Let Pi :LP(X, Σ, μ) → LP(X, Σ, μ) be the conditional expectation operator corresponding to fi for every i = 1,2,…, k, and set T = P1 . . . Pk. Our goal in the note is to give a new and simpler proof of the fact that for every f ∊ LP(X, Σ, μ), the sequence (Tnf)n∊N converges in the norm topology of LP(X, Σ, μ), and that its limit is the conditional expectation of f with respect to f1 ∩ f2 ∩ … ∩ Fk.

Keywords

47A3528D99
Type
Research Article
Information
Canadian Mathematical Bulletin , Volume 33 , Issue 3 , 01 September 1990 , pp. 257 - 260
Copyright
Copyright © Canadian Mathematical Society 1990

Footnotes

This work was done while the author was supported by a grant-in-aid from The Institute for Advanced Study. The grant-in-aid is part of a grant made to The Institute by the National Science Foundation. The author would like to express his deep gratitude to all the people who made his stay at The Institute possible.

I am indebted to Professor Harry Furstenberg for bringing to my attention the topic discussed in this note, and to Professor Mustafa A. Akcoglu, Professor Tsuyoshi Ando, Professor Gian-Carlo Rota and Professor Louis Sucheston for many instructive comments which were very useful to my work on the paper.

References

1. Akcoglu, M. A., and Sucheston, L., An alternating procedure for operators on Lp spaces, Proc. Amer. Math. Soc. 99 (1987), 555–558.Google Scholar
2. Akcoglu, M. A., and Sucheston, L., Pointwise convergence of alternating sequences, preprint.Google Scholar
3. Amemiya, I., and T. Ando, Convergence of random products of contractions in Hilbert space, Acta Sci. Math. (Szeged) 26 (1965), 239–244.Google Scholar
4. Burkholder, D. L., and Y. S. Chow, Iterates of conditional expectation operators, Proc. Amer. Math. Soc. 12 (1961), 490–495.Google Scholar
5. Halperin, I., The product of projection operators, Acta Sci. Math. (Szeged) 23 (1962), 96–99.Google Scholar
6. Hildebrandt, S., Über die alternierenden Verfahren von H. A. Schwarz und C. Neumann, J. Reine Angew. Math. 232 (1968), 136–155.Google Scholar
7. Hildebrandt, S., Unendliche Produkte von Kontraktionen, Indiana Univ. Math. J. 20 (1971), 909–911.Google Scholar
8. Hildebrandt, S., and B. Schmidt, Zur Konvergenz von Operatorprodukten im Hilbertraum, Math. Z. 105 (1968), 62–71.Google Scholar
9. Krengel, U., Ergodic Theorems, Walter de Gruyter, Berlin-New York, 1985.Google Scholar
10. Ornstein, D., On the pointwise behavior of iterates of a self-adjoint operator, J. Math. Mech. 18 (1968), 473–477.Google Scholar
11. Rota, G.-C., An “’ alter nier ende Verfahren” for general positive operators, Bull. Amer. Math. Soc. 68 (1962), 95–102.Google Scholar
12. Stein, E. M., On the maximal ergodic theorem, Proc. Nat. Acad. Sci. USA 47 (1961), 1894–1897.Google Scholar
13. Wittmann, R., Analogues of the “zero-two” law for positive linear contractions in LP and C(X), Israel J. Math. 59 (1987), 8–28.Google Scholar
14. Zbǎganu, G., Two inequalities concerning centered moments, Proc. Seventh Conf. Probab. Theory, Bra§ov, Romania, 1982, Editura Academiei R.S. Romania, Bucharest, 1984, pp. 515–518.Google Scholar