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

Previous Article
Notes on monotone Lagrangian twist tori
ERA-MS Home
This Volume
Next Article
Longitudinal foliation rigidity and Lipschitz-continuous invariant forms for hyperbolic flows

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

Abstract
Related Papers

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.
[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.
[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.
[4]	E. Berkson and T. A. Gillespie, AC functions on the circle and spectral families, J. Operator Theory, 13 (1985), 33-47.
[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.
[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.
[7]	E. Berkson and T. A. Gillespie, Spectral decompositions and harmonic analysis on UMD spaces, Studia Math., 112 (1994), 13-49.
[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.
[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.
[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.
[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.
[12]	E. Berkson and T. A. Gillespie, Shifts as models for spectral decomposability on Hilbert space, J. Operator Theory, 50 (2003), 77-106.
[13]	E. Berkson and T. A. Gillespie, Operator means and spectral integration of Fourier multipliers, Houston J. Math., 30 (2004), 767-814.
[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.
[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.
[16]	D. Blagojevic, "Spectral Families and Geometry of Banach Spaces," PhD thesis, University of Edinburgh, 2007.
[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.
[18]	M. M. Day, Reflexive Banach spaces not isomorphic to uniformly convex spaces, Bull. Amer. Math. Soc., 47 (1941), 313-317.
[19]	P. Enflo, Banach spaces which can be given an equivalent uniformly convex norm, Israel J. Math., 13 (1972), 281-288.
[20]	G. H. Hardy, Weierstrass's non-differentiable function, Trans. Amer. Math. Soc., 17 (1916), 301-325. doi: 10.2307/1989005.
[21]	G. H. Hardy and J. E. Littlewood, Some properties of fractional integrals. I, Math. Zeitschrift, 27 (1928), 565-606. doi: 10.1007/BF01171116.
[22]	G. H. Hardy and J. E. Littlewood, A convergence criterion for Fourier series, Math. Zeitschrift, 28 (1928) 612-634. doi: 10.1007/BF01181186.
[23]	R. C. James, Super-reflexive spaces with bases, Pacific J. Math., 41 (1972), 409-419.
[24]	B. Maurey, Système de Haar, Séminaire Maurey-Schwartz, 1974-1975, (École Polytechnique, Paris, 1975), I.1-II.13.
[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.
[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.
[27]	J. L. Rubio de Francia, A Littlewood-Paley inequality for arbitrary intervals, Revista Mat. Iberoamericana, 1 (1985), 1-14.
[28]	P. G. Spain, On well-bounded operators of type $B$, Proc. Edinburgh Math. Soc. (2), 18 (1972), 35-48. doi: 10.1017/S0013091500026134.
[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.
[30]	L. C. Young, An inequality of the Hölder type, connected with Stieltjes integration, Acta Math., 67 (1936), 251-282. doi: 10.1007/BF02401743.
[31]	A. Zygmund, "Trigonometric Series," 2nd ed., vol. 1, Cambridge Univ. Press, London, 1959.

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.
[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.
[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.
[4]	E. Berkson and T. A. Gillespie, AC functions on the circle and spectral families, J. Operator Theory, 13 (1985), 33-47.
[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.
[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.
[7]	E. Berkson and T. A. Gillespie, Spectral decompositions and harmonic analysis on UMD spaces, Studia Math., 112 (1994), 13-49.
[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.
[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.
[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.
[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.
[12]	E. Berkson and T. A. Gillespie, Shifts as models for spectral decomposability on Hilbert space, J. Operator Theory, 50 (2003), 77-106.
[13]	E. Berkson and T. A. Gillespie, Operator means and spectral integration of Fourier multipliers, Houston J. Math., 30 (2004), 767-814.
[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.
[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.
[16]	D. Blagojevic, "Spectral Families and Geometry of Banach Spaces," PhD thesis, University of Edinburgh, 2007.
[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.
[18]	M. M. Day, Reflexive Banach spaces not isomorphic to uniformly convex spaces, Bull. Amer. Math. Soc., 47 (1941), 313-317.
[19]	P. Enflo, Banach spaces which can be given an equivalent uniformly convex norm, Israel J. Math., 13 (1972), 281-288.
[20]	G. H. Hardy, Weierstrass's non-differentiable function, Trans. Amer. Math. Soc., 17 (1916), 301-325. doi: 10.2307/1989005.
[21]	G. H. Hardy and J. E. Littlewood, Some properties of fractional integrals. I, Math. Zeitschrift, 27 (1928), 565-606. doi: 10.1007/BF01171116.
[22]	G. H. Hardy and J. E. Littlewood, A convergence criterion for Fourier series, Math. Zeitschrift, 28 (1928) 612-634. doi: 10.1007/BF01181186.
[23]	R. C. James, Super-reflexive spaces with bases, Pacific J. Math., 41 (1972), 409-419.
[24]	B. Maurey, Système de Haar, Séminaire Maurey-Schwartz, 1974-1975, (École Polytechnique, Paris, 1975), I.1-II.13.
[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.
[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.
[27]	J. L. Rubio de Francia, A Littlewood-Paley inequality for arbitrary intervals, Revista Mat. Iberoamericana, 1 (1985), 1-14.
[28]	P. G. Spain, On well-bounded operators of type $B$, Proc. Edinburgh Math. Soc. (2), 18 (1972), 35-48. doi: 10.1017/S0013091500026134.
[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.
[30]	L. C. Young, An inequality of the Hölder type, connected with Stieltjes integration, Acta Math., 67 (1936), 251-282. doi: 10.1007/BF02401743.
[31]	A. Zygmund, "Trigonometric Series," 2nd ed., vol. 1, Cambridge Univ. Press, London, 1959.

[1]	Doǧan Çömez. The modulated ergodic Hilbert transform. Discrete and Continuous Dynamical Systems - S, 2009, 2 (2) : 325-336. doi: 10.3934/dcdss.2009.2.325
[2]	Daniel Alpay, Mihai Putinar, Victor Vinnikov. A Hilbert space approach to bounded analytic extension in the ball. Communications on Pure and Applied Analysis, 2003, 2 (2) : 139-145. doi: 10.3934/cpaa.2003.2.139
[3]	Mostafa Mbekhta. Representation and approximation of the polar factor of an operator on a Hilbert space. Discrete and Continuous Dynamical Systems - S, 2021, 14 (8) : 3043-3054. doi: 10.3934/dcdss.2020463
[4]	Sébastien Gouëzel. An interval map with a spectral gap on Lipschitz functions, but not on bounded variation functions. Discrete and Continuous Dynamical Systems, 2009, 24 (4) : 1205-1208. doi: 10.3934/dcds.2009.24.1205
[5]	Rakesh Pilkar, Erik M. Bollt, Charles Robinson. Empirical mode decomposition/Hilbert transform analysis of postural responses to small amplitude anterior-posterior sinusoidal translations of varying frequencies. Mathematical Biosciences & Engineering, 2011, 8 (4) : 1085-1097. doi: 10.3934/mbe.2011.8.1085
[6]	Xue-Ping Luo, Yi-Bin Xiao, Wei Li. Strict feasibility of variational inclusion problems in reflexive Banach spaces. Journal of Industrial and Management Optimization, 2020, 16 (5) : 2495-2502. doi: 10.3934/jimo.2019065
[7]	Ren-You Zhong, Nan-Jing Huang. Strict feasibility for generalized mixed variational inequality in reflexive Banach spaces. Numerical Algebra, Control and Optimization, 2011, 1 (2) : 261-274. doi: 10.3934/naco.2011.1.261
[8]	Nigel Higson and Gennadi Kasparov. Operator K-theory for groups which act properly and isometrically on Hilbert space. Electronic Research Announcements, 1997, 3: 131-142.
[9]	Alexandre I. Danilenko, Mariusz Lemańczyk. Spectral multiplicities for ergodic flows. Discrete and Continuous Dynamical Systems, 2013, 33 (9) : 4271-4289. doi: 10.3934/dcds.2013.33.4271
[10]	Andi Kivinukk, Anna Saksa. On Rogosinski-type approximation processes in Banach space using the framework of the cosine operator function. Mathematical Foundations of Computing, 2022, 5 (3) : 197-218. doi: 10.3934/mfc.2021030
[11]	T. J. Sullivan. Well-posed Bayesian inverse problems and heavy-tailed stable quasi-Banach space priors. Inverse Problems and Imaging, 2017, 11 (5) : 857-874. doi: 10.3934/ipi.2017040
[12]	Cecilia González-Tokman, Anthony Quas. A concise proof of the multiplicative ergodic theorem on Banach spaces. Journal of Modern Dynamics, 2015, 9: 237-255. doi: 10.3934/jmd.2015.9.237
[13]	Xing Wang, Nan-Jing Huang. Stability analysis for set-valued vector mixed variational inequalities in real reflexive Banach spaces. Journal of Industrial and Management Optimization, 2013, 9 (1) : 57-74. doi: 10.3934/jimo.2013.9.57
[14]	Franco Obersnel, Pierpaolo Omari. Multiple bounded variation solutions of a capillarity problem. Conference Publications, 2011, 2011 (Special) : 1129-1137. doi: 10.3934/proc.2011.2011.1129
[15]	O. A. Veliev. Essential spectral singularities and the spectral expansion for the Hill operator. Communications on Pure and Applied Analysis, 2017, 16 (6) : 2227-2251. doi: 10.3934/cpaa.2017110
[16]	Xiongping Dai, Yu Huang, Mingqing Xiao. Realization of joint spectral radius via Ergodic theory. Electronic Research Announcements, 2011, 18: 22-30. doi: 10.3934/era.2011.18.22
[17]	Woochul Jung, Ngocthach Nguyen, Yinong Yang. Spectral decomposition for rescaling expansive flows with rescaled shadowing. Discrete and Continuous Dynamical Systems, 2020, 40 (4) : 2267-2283. doi: 10.3934/dcds.2020113
[18]	Keonhee Lee, Ngoc-Thach Nguyen, Yinong Yang. Topological stability and spectral decomposition for homeomorphisms on noncompact spaces. Discrete and Continuous Dynamical Systems, 2018, 38 (5) : 2487-2503. doi: 10.3934/dcds.2018103
[19]	Aihua Fan, Lingmin Liao, Jacques Peyrière. Generic points in systems of specification and Banach valued Birkhoff ergodic average. Discrete and Continuous Dynamical Systems, 2008, 21 (4) : 1103-1128. doi: 10.3934/dcds.2008.21.1103
[20]	Alex Blumenthal. A volume-based approach to the multiplicative ergodic theorem on Banach spaces. Discrete and Continuous Dynamical Systems, 2016, 36 (5) : 2377-2403. doi: 10.3934/dcds.2016.36.2377

2020 Impact Factor: 0.929

Tools

Metrics

Other articles
by authors

on AIMS
Earl Berkson
on Google Scholar
Earl Berkson

[Back to Top]