American Institute of Mathematical Sciences

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  2010, 17: 90-103. doi: 10.3934/era.2010.17.90

Fourier analysis methods in operator ergodic theory on super-reflexive Banach spaces

Earl Berkson 1,

1. 

Department of Mathematics, University of Illinois, 1409 W. Green Street, Urbana, IL 61801, United States

Received  June 2010 Published  October 2010

On reflexive spaces trigonometrically well-bounded operators (abbreviated "twbo's'') have an operator-ergodic-theory characterization as the invertible operators $U$ whose rotates "transfer'' the discrete Hilbert averages $(C,1)$-boundedly. Twbo's permeate many settings of modern analysis, and this note treats advances in their spectral theory, Fourier analysis, and operator ergodic theory made possible by applying classical analysis techniques pioneered by Hardy-Littlewood and L.C. Young to the R.C. James inequalities for super-reflexive spaces. When the James inequalities are combined with spectral integration methods and Young-Stieltjes integration for the spaces $V_{p}(\mathbb{T}) $ of functions having bounded $p$-variation, it transpires that every twbo on a super-reflexive space $X$ has a norm-continuous $V_{p}(\mathbb{T}) $-functional calculus for a range of values of $p>1$, and we investigate the ways this outcome logically simplifies and simultaneously advances the structure theory of twbo's on $X$. In particular, on a super-reflexive space $X$ (but not on the general reflexive space) Tauberian-type theorems emerge which improve to their $(C,0) $ counterparts the $(C,1) $ averaging and convergence associated with twbo's.
Keywords: p-variation, Ergodic Hilbert transform, super-reflexive Banach space, trigonometrically well-bounded operator., spectral decomposition.
Mathematics Subject Classification: Primary: 26A45, 46B20, 47A35, 47B4.
Citation: Earl Berkson. Fourier analysis methods in operator ergodic theory on super-reflexive Banach spaces. Electronic Research Announcements, 2010, 17: 90-103. doi: 10.3934/era.2010.17.90
References:
[1]

D. J. Aldous, Unconditional bases and martingales in $Lp(F)$, Math. Proc. Cambridge Philos. Soc., 85 (1979), 117-123. doi: 10.1017/S0305004100055559.  Google Scholar

[2]

B. Beauzamy, "Introduction to Banach Spaces and Their Geometry," North-Holland Math. Studies 68 (Notas de Matemática 86), Elsevier Science, New York, New York, 1982.  Google Scholar

[3]

E. Berkson, J. Bourgain and T. A. Gillespie, On the almost everywhere convergence of ergodic averages for power-bounded operators on $L^p$-subspaces, Integral Equations and Operator Theory, 14 (1991), 678-715. doi: 10.1007/BF01200555.  Google Scholar

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E. Berkson and T. A. Gillespie, AC functions on the circle and spectral families, J. Operator Theory, 13 (1985), 33-47.  Google Scholar

[5]

E. Berkson and T. A. Gillespie, Fourier series criteria for operator decomposability, Integral Equations and Operator Theory, 9 (1986), 767-789. doi: 10.1007/BF01202516.  Google Scholar

[6]

E. Berkson and T. A. Gillespie, Stečkin's theorem, transference, and spectral decompositions, J. Functional Analysis, 70 (1987), 140-170. doi: 10.1016/0022-1236(87)90128-5.  Google Scholar

[7]

E. Berkson and T. A. Gillespie, Spectral decompositions and harmonic analysis on UMD spaces, Studia Math., 112 (1994), 13-49.  Google Scholar

[8]

E. Berkson and T. A. Gillespie, Mean-boundedness and Littlewood-Paley for separation-preserving operators, Trans. Amer. Math. Soc., 349 (1997), 1169-1189. doi: 10.1090/S0002-9947-97-01896-5.  Google Scholar

[9]

E. Berkson and T. A. Gillespie, The $q$-variation of functions and spectral integration of Fourier multipliers, Duke Math. J., 88 (1997), 103-132. doi: 10.1215/S0012-7094-97-08804-9.  Google Scholar

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E. Berkson and T. A. Gillespie, $Mean_{2}$-bounded operators on Hilbert space and weight sequences of positive operators, Positivity, 3 (1999), 101-133. doi: 10.1023/A:1009794510984.  Google Scholar

[11]

E. Berkson and T. A. Gillespie, Spectral decompositions, ergodic averages, and the Hilbert transform, Studia Math., 144 (2001), 39-61. doi: 10.4064/sm144-1-2.  Google Scholar

[12]

E. Berkson and T. A. Gillespie, Shifts as models for spectral decomposability on Hilbert space, J. Operator Theory, 50 (2003), 77-106.  Google Scholar

[13]

E. Berkson and T. A. Gillespie, Operator means and spectral integration of Fourier multipliers, Houston J. Math., 30 (2004), 767-814.  Google Scholar

[14]

E. Berkson and T. A. Gillespie, The $q$-variation of functions and spectral integration from dominated ergodic estimates, Journal of Fourier Analysis and Applications, 10 (2004), 149-177. doi: 10.1007/s00041-004-8009-z.  Google Scholar

[15]

E. Berkson, T. A. Gillespie and P. S. Muhly, Abstract spectral decompositions guaranteed by the Hilbert transform, Proc. London Math. Soc. (3), 53 (1986), 489-517. doi: 10.1112/plms/s3-53.3.489.  Google Scholar

[16]

D. Blagojevic, "Spectral Families and Geometry of Banach Spaces," PhD thesis, University of Edinburgh, 2007. Google Scholar

[17]

J. Bourgain, Some remarks on Banach spaces in which martingale difference sequences are unconditional, Arkiv för Mat., 21 (1983), 163-168. doi: 10.1007/BF02384306.  Google Scholar

[18]

M. M. Day, Reflexive Banach spaces not isomorphic to uniformly convex spaces, Bull. Amer. Math. Soc., 47 (1941), 313-317.  Google Scholar

[19]

P. Enflo, Banach spaces which can be given an equivalent uniformly convex norm, Israel J. Math., 13 (1972), 281-288.  Google Scholar

[20]

G. H. Hardy, Weierstrass's non-differentiable function, Trans. Amer. Math. Soc., 17 (1916), 301-325. doi: 10.2307/1989005.  Google Scholar

[21]

G. H. Hardy and J. E. Littlewood, Some properties of fractional integrals. I, Math. Zeitschrift, 27 (1928), 565-606. doi: 10.1007/BF01171116.  Google Scholar

[22]

G. H. Hardy and J. E. Littlewood, A convergence criterion for Fourier series, Math. Zeitschrift, 28 (1928) 612-634. doi: 10.1007/BF01181186.  Google Scholar

[23]

R. C. James, Super-reflexive spaces with bases, Pacific J. Math., 41 (1972), 409-419.  Google Scholar

[24]

B. Maurey, Système de Haar, Séminaire Maurey-Schwartz, 1974-1975, (École Polytechnique, Paris, 1975), I.1-II.13.  Google Scholar

[25]

G. Pisier, Un exemple concernant la super-ré flexivité, Séminaire Maurey-Schwartz 1974-1975: Espaces $L^p$ applications radonifiantes et géométrie des espaces de Banach, Annexe No. 2, 12 pp, Centre Math. école Polytech., Paris, 1975.  Google Scholar

[26]

J. Porter, Helly's selection principle for functions of bounded $p$-variation, Rocky Mountain J. Math., 35 (2005), 675-679. doi: 10.1216/rmjm/1181069753.  Google Scholar

[27]

J. L. Rubio de Francia, A Littlewood-Paley inequality for arbitrary intervals, Revista Mat. Iberoamericana, 1 (1985), 1-14.  Google Scholar

[28]

P. G. Spain, On well-bounded operators of type $B$, Proc. Edinburgh Math. Soc. (2), 18 (1972), 35-48. doi: 10.1017/S0013091500026134.  Google Scholar

[29]

S. Treil and A. Volberg, Wavelets and the angle between past and future, Journal of Functional Analysis, 143 (1997), 269-308. doi: 10.1006/jfan.1996.2986.  Google Scholar

[30]

L. C. Young, An inequality of the Hölder type, connected with Stieltjes integration, Acta Math., 67 (1936), 251-282. doi: 10.1007/BF02401743.  Google Scholar

[31]

A. Zygmund, "Trigonometric Series," 2nd ed., vol. 1, Cambridge Univ. Press, London, 1959. Google Scholar

show all references

References:
[1]

D. J. Aldous, Unconditional bases and martingales in $Lp(F)$, Math. Proc. Cambridge Philos. Soc., 85 (1979), 117-123. doi: 10.1017/S0305004100055559.  Google Scholar

[2]

B. Beauzamy, "Introduction to Banach Spaces and Their Geometry," North-Holland Math. Studies 68 (Notas de Matemática 86), Elsevier Science, New York, New York, 1982.  Google Scholar

[3]

E. Berkson, J. Bourgain and T. A. Gillespie, On the almost everywhere convergence of ergodic averages for power-bounded operators on $L^p$-subspaces, Integral Equations and Operator Theory, 14 (1991), 678-715. doi: 10.1007/BF01200555.  Google Scholar

[4]

E. Berkson and T. A. Gillespie, AC functions on the circle and spectral families, J. Operator Theory, 13 (1985), 33-47.  Google Scholar

[5]

E. Berkson and T. A. Gillespie, Fourier series criteria for operator decomposability, Integral Equations and Operator Theory, 9 (1986), 767-789. doi: 10.1007/BF01202516.  Google Scholar

[6]

E. Berkson and T. A. Gillespie, Stečkin's theorem, transference, and spectral decompositions, J. Functional Analysis, 70 (1987), 140-170. doi: 10.1016/0022-1236(87)90128-5.  Google Scholar

[7]

E. Berkson and T. A. Gillespie, Spectral decompositions and harmonic analysis on UMD spaces, Studia Math., 112 (1994), 13-49.  Google Scholar

[8]

E. Berkson and T. A. Gillespie, Mean-boundedness and Littlewood-Paley for separation-preserving operators, Trans. Amer. Math. Soc., 349 (1997), 1169-1189. doi: 10.1090/S0002-9947-97-01896-5.  Google Scholar

[9]

E. Berkson and T. A. Gillespie, The $q$-variation of functions and spectral integration of Fourier multipliers, Duke Math. J., 88 (1997), 103-132. doi: 10.1215/S0012-7094-97-08804-9.  Google Scholar

[10]

E. Berkson and T. A. Gillespie, $Mean_{2}$-bounded operators on Hilbert space and weight sequences of positive operators, Positivity, 3 (1999), 101-133. doi: 10.1023/A:1009794510984.  Google Scholar

[11]

E. Berkson and T. A. Gillespie, Spectral decompositions, ergodic averages, and the Hilbert transform, Studia Math., 144 (2001), 39-61. doi: 10.4064/sm144-1-2.  Google Scholar

[12]

E. Berkson and T. A. Gillespie, Shifts as models for spectral decomposability on Hilbert space, J. Operator Theory, 50 (2003), 77-106.  Google Scholar

[13]

E. Berkson and T. A. Gillespie, Operator means and spectral integration of Fourier multipliers, Houston J. Math., 30 (2004), 767-814.  Google Scholar

[14]

E. Berkson and T. A. Gillespie, The $q$-variation of functions and spectral integration from dominated ergodic estimates, Journal of Fourier Analysis and Applications, 10 (2004), 149-177. doi: 10.1007/s00041-004-8009-z.  Google Scholar

[15]

E. Berkson, T. A. Gillespie and P. S. Muhly, Abstract spectral decompositions guaranteed by the Hilbert transform, Proc. London Math. Soc. (3), 53 (1986), 489-517. doi: 10.1112/plms/s3-53.3.489.  Google Scholar

[16]

D. Blagojevic, "Spectral Families and Geometry of Banach Spaces," PhD thesis, University of Edinburgh, 2007. Google Scholar

[17]

J. Bourgain, Some remarks on Banach spaces in which martingale difference sequences are unconditional, Arkiv för Mat., 21 (1983), 163-168. doi: 10.1007/BF02384306.  Google Scholar

[18]

M. M. Day, Reflexive Banach spaces not isomorphic to uniformly convex spaces, Bull. Amer. Math. Soc., 47 (1941), 313-317.  Google Scholar

[19]

P. Enflo, Banach spaces which can be given an equivalent uniformly convex norm, Israel J. Math., 13 (1972), 281-288.  Google Scholar

[20]

G. H. Hardy, Weierstrass's non-differentiable function, Trans. Amer. Math. Soc., 17 (1916), 301-325. doi: 10.2307/1989005.  Google Scholar

[21]

G. H. Hardy and J. E. Littlewood, Some properties of fractional integrals. I, Math. Zeitschrift, 27 (1928), 565-606. doi: 10.1007/BF01171116.  Google Scholar

[22]

G. H. Hardy and J. E. Littlewood, A convergence criterion for Fourier series, Math. Zeitschrift, 28 (1928) 612-634. doi: 10.1007/BF01181186.  Google Scholar

[23]

R. C. James, Super-reflexive spaces with bases, Pacific J. Math., 41 (1972), 409-419.  Google Scholar

[24]

B. Maurey, Système de Haar, Séminaire Maurey-Schwartz, 1974-1975, (École Polytechnique, Paris, 1975), I.1-II.13.  Google Scholar

[25]

G. Pisier, Un exemple concernant la super-ré flexivité, Séminaire Maurey-Schwartz 1974-1975: Espaces $L^p$ applications radonifiantes et géométrie des espaces de Banach, Annexe No. 2, 12 pp, Centre Math. école Polytech., Paris, 1975.  Google Scholar

[26]

J. Porter, Helly's selection principle for functions of bounded $p$-variation, Rocky Mountain J. Math., 35 (2005), 675-679. doi: 10.1216/rmjm/1181069753.  Google Scholar

[27]

J. L. Rubio de Francia, A Littlewood-Paley inequality for arbitrary intervals, Revista Mat. Iberoamericana, 1 (1985), 1-14.  Google Scholar

[28]

P. G. Spain, On well-bounded operators of type $B$, Proc. Edinburgh Math. Soc. (2), 18 (1972), 35-48. doi: 10.1017/S0013091500026134.  Google Scholar

[29]

S. Treil and A. Volberg, Wavelets and the angle between past and future, Journal of Functional Analysis, 143 (1997), 269-308. doi: 10.1006/jfan.1996.2986.  Google Scholar

[30]

L. C. Young, An inequality of the Hölder type, connected with Stieltjes integration, Acta Math., 67 (1936), 251-282. doi: 10.1007/BF02401743.  Google Scholar

[31]

A. Zygmund, "Trigonometric Series," 2nd ed., vol. 1, Cambridge Univ. Press, London, 1959. Google Scholar

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